MATLAB ® The Language of Technical Computing Mathematics Version 7 How to Contact The MathWorks: www.mathworks.com comp.soft-sys.matlab Web Newsgroup [email protected] Technical support Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 Mail [email protected] [email protected] [email protected] [email protected] [email protected] For contact information about worldwide offices, see the MathWorks Web site. MATLAB Mathematics © COPYRIGHT 1984 — 2005 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. 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Revision History: June 2004 First printing October 2004 March 2005 Online only Online only New for MATLAB 7.0 (Release 14) Formerly part of Using MATLAB Revised for Version 7.0.1 (Release 14SP1) Revised for Version 7.0.4 (Release 14SP2) Contents Matrices and Linear Algebra 1 Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Matrices in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Creating Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Adding and Subtracting Matrices . . . . . . . . . . . . . . . . . . . . . . . 1-6 Vector Products and Transpose . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Multiplying Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8 The Identity Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10 The Kronecker Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . 1-11 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12 Solving Linear Systems of Equations . . . . . . . . . . . . . . . . . . Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Square Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Underdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13 1-13 1-15 1-15 1-18 1-20 Inverses and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 Pseudoinverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24 Cholesky, LU, and QR Factorizations . . . . . . . . . . . . . . . . . . Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-28 1-28 1-30 1-31 Matrix Powers and Exponentials . . . . . . . . . . . . . . . . . . . . . . 1-35 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 1-43 i Polynomials and Interpolation 2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . Representing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution and Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Fraction Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 2-2 2-3 2-3 2-4 2-4 2-5 2-5 2-6 2-7 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9 Interpolation Function Summary . . . . . . . . . . . . . . . . . . . . . . . . 2-9 One-Dimensional Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 2-10 Two-Dimensional Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Comparing Interpolation Methods . . . . . . . . . . . . . . . . . . . . . . 2-13 Interpolation and Multidimensional Arrays . . . . . . . . . . . . . . 2-15 Triangulation and Interpolation of Scattered Data . . . . . . . . . 2-18 Tessellation and Interpolation of Scattered Data in Higher Dimensions 2-26 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37 Data Analysis and Statistics 3 Column-Oriented Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Basic Data Analysis Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 Covariance and Correlation Coefficients . . . . . . . . . . . . . . . . . 3-10 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11 Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13 Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13 ii Contents Removing Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15 Regression and Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear-in-the-Parameters Regression . . . . . . . . . . . . . . . . . . . . Multiple Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16 3-17 3-18 3-20 Case Study: Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomial Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyzing Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Basic Fitting Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21 3-21 3-23 3-25 3-27 3-28 Difference Equations and Filtering . . . . . . . . . . . . . . . . . . . . 3-39 Fourier Analysis and the Fast Fourier Transform (FFT) . Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnitude and Phase of Transformed Data . . . . . . . . . . . . . . FFT Length Versus Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-42 3-42 3-43 3-47 3-49 Function Functions 4 Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Representing Functions in MATLAB . . . . . . . . . . . . . . . . . . . . 4-3 Plotting Mathematical Functions . . . . . . . . . . . . . . . . . . . . . . . 4-5 Minimizing Functions and Finding Zeros . . . . . . . . . . . . . . . 4-8 Minimizing Functions of One Variable . . . . . . . . . . . . . . . . . . . . 4-8 Minimizing Functions of Several Variables . . . . . . . . . . . . . . . . 4-9 Fitting a Curve to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Setting Minimization Options . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13 Output Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14 iii Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21 Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25 Numerical Integration (Quadrature) . . . . . . . . . . . . . . . . . . . 4-27 Example: Computing the Length of a Curve . . . . . . . . . . . . . . 4-27 Example: Double Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28 Parameterizing Functions Called by Function Functions 4-30 Providing Parameter Values Using Nested Functions . . . . . . 4-30 Providing Parameter Values to Anonymous Functions . . . . . . 4-31 Differential Equations 5 Initial Value Problems for ODEs and DAEs . . . . . . . . . . . . . . 5-2 ODE Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Introduction to Initial Value ODE Problems . . . . . . . . . . . . . . . 5-4 Solvers for Explicit and Linearly Implicit ODEs . . . . . . . . . . . . 5-5 Examples: Solving Explicit ODE Problems . . . . . . . . . . . . . . . . 5-9 Solver for Fully Implicit ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Example: Solving a Fully Implicit ODE Problem . . . . . . . . . . 5-16 Changing ODE Integration Properties . . . . . . . . . . . . . . . . . . . 5-17 Examples: Applying the ODE Initial Value Problem Solvers . 5-18 Questions and Answers, and Troubleshooting . . . . . . . . . . . . . 5-39 Initial Value Problems for DDEs . . . . . . . . . . . . . . . . . . . . . . . DDE Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Initial Value DDE Problems . . . . . . . . . . . . . . DDE Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving DDE Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Changing DDE Integration Properties . . . . . . . . . . . . . . . . . . . 5-45 5-45 5-46 5-47 5-49 5-53 5-56 Boundary Value Problems for ODEs . . . . . . . . . . . . . . . . . . . 5-57 BVP Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-58 Introduction to Boundary Value ODE Problems . . . . . . . . . . . 5-59 iv Contents Boundary Value Problem Solver . . . . . . . . . . . . . . . . . . . . . . . . Solving BVP Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Continuation to Make a Good Initial Guess . . . . . . . . . Solving Singular BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Multi-Point BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Changing BVP Integration Properties . . . . . . . . . . . . . . . . . . . 5-60 5-63 5-68 5-75 5-79 5-79 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . PDE Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to PDE Problems . . . . . . . . . . . . . . . . . . . . . . . . . MATLAB Partial Differential Equation Solver . . . . . . . . . . . . Solving PDE Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluating the Solution at Specific Points . . . . . . . . . . . . . . . . Changing PDE Integration Properties . . . . . . . . . . . . . . . . . . . Example: Electrodynamics Problem . . . . . . . . . . . . . . . . . . . . . 5-81 5-81 5-82 5-83 5-86 5-91 5-92 5-92 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-98 Sparse Matrices 6 Function Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 Sparse Matrix Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 General Storage Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6 Creating Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7 Importing Sparse Matrices from Outside MATLAB . . . . . . . . 6-11 Viewing Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Information About Nonzero Elements . . . . . . . . . . . . . . . . . . . Viewing Sparse Matrices Graphically . . . . . . . . . . . . . . . . . . . The find Function and Sparse Matrices . . . . . . . . . . . . . . . . . . 6-12 6-12 6-14 6-15 Adjacency Matrices and Graphs . . . . . . . . . . . . . . . . . . . . . . . 6-16 Introduction to Adjacency Matrices . . . . . . . . . . . . . . . . . . . . . 6-16 Graphing Using Adjacency Matrices . . . . . . . . . . . . . . . . . . . . 6-17 v The Bucky Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17 An Airflow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-22 Sparse Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . Standard Mathematical Operations . . . . . . . . . . . . . . . . . . . . . Permutation and Reordering . . . . . . . . . . . . . . . . . . . . . . . . . . . Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simultaneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . Eigenvalues and Singular Values . . . . . . . . . . . . . . . . . . . . . . . 6-24 6-24 6-24 6-25 6-29 6-35 6-38 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-40 Nondouble Data Types 7 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 Integer Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4 Integer Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4 Largest and Smallest Values for Integer Data Types . . . . . . . . 7-5 Integer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6 Example — Digitized Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8 Warnings for Integer Data Types . . . . . . . . . . . . . . . . . . . . . . . 7-15 Single-Precision Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . Data Type single . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Precision Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Function eps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example — Writing M-Files for Different Data Types . . . . . . Largest and Smallest Numbers of Type double and single . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Contents 7-17 7-17 7-18 7-19 7-21 7-23 7-25 Index vii viii Contents 1 Matrices and Linear Algebra Function Summary (p. 1-2) Summarizes the MATLAB® linear algebra functions Matrices in MATLAB (p. 1-4) Explains the use of matrices and basic matrix operations in MATLAB Solving Linear Systems of Equations (p. 1-13) Discusses the solution of simultaneous linear equations in MATLAB, including square systems, overdetermined systems, and underdetermined systems Inverses and Determinants (p. 1-23) Explains the use in MATLAB of inverses, determinants, and pseudoinverses in the solution of systems of linear equations Cholesky, LU, and QR Factorizations (p. 1-28) Discusses the solution in MATLAB of systems of linear equations that involve triangular matrices, using Cholesky factorization, Gaussian elimination, and orthogonalization Matrix Powers and Exponentials (p. 1-35) Explains the use of MATLAB notation to obtain various matrix powers and exponentials Eigenvalues (p. 1-39) Explains eigenvalues and describes eigenvalue decomposition in MATLAB Singular Value Decomposition (p. 1-43) Describes singular value decomposition of a rectangular matrix in MATLAB 1 Matrices and Linear Algebra Function Summary The linear algebra functions are located in the MATLAB matfun directory. Function Summary Category Function Description Matrix analysis norm Matrix or vector norm. normest Estimate the matrix 2-norm. rank Matrix rank. det Determinant. trace Sum of diagonal elements. null Null space. orth Orthogonalization. rref Reduced row echelon form. subspace Angle between two subspaces. \ and / Linear equation solution. inv Matrix inverse. cond Condition number for inversion. condest 1-norm condition number estimate. chol Cholesky factorization. cholinc Incomplete Cholesky factorization. linsolve Solve a system of linear equations. lu LU factorization. luinc Incomplete LU factorization. qr Orthogonal-triangular decomposition. Linear equations 1-2 Function Summary Function Summary (Continued) Category Eigenvalues and singular values Matrix functions Function Description lsqnonneg Nonnegative least-squares. pinv Pseudoinverse. lscov Least squares with known covariance. eig Eigenvalues and eigenvectors. svd Singular value decomposition. eigs A few eigenvalues. svds A few singular values. poly Characteristic polynomial. polyeig Polynomial eigenvalue problem. condeig Condition number for eigenvalues. hess Hessenberg form. qz QZ factorization. schur Schur decomposition. expm Matrix exponential. logm Matrix logarithm. sqrtm Matrix square root. funm Evaluate general matrix function. 1-3 1 Matrices and Linear Algebra Matrices in MATLAB A matrix is a two-dimensional array of real or complex numbers. Linear algebra defines many matrix operations that are directly supported by MATLAB. Linear algebra includes matrix arithmetic, linear equations, eigenvalues, singular values, and matrix factorizations. For more information about creating and working with matrices, see Data Structures in the MATLAB Programming documentation. This section describes the following topics: • “Creating Matrices” on page 1-4 • “Adding and Subtracting Matrices” on page 1-6 • “Vector Products and Transpose” on page 1-7 • “Vector Products and Transpose” on page 1-7 • “Multiplying Matrices” on page 1-8 • “The Identity Matrix” on page 1-10 • “The Kronecker Tensor Product” on page 1-11 • “Vector and Matrix Norms” on page 1-12 Creating Matrices Informally, the terms matrix and array are often used interchangeably. More precisely, a matrix is a two-dimensional rectangular array of real or complex numbers that represents a linear transformation. The linear algebraic operations defined on matrices have found applications in a wide variety of technical fields. (The optional Symbolic Math Toolbox extends the capabilities of MATLAB to operations on various types of nonnumeric matrices.) MATLAB has dozens of functions that create different kinds of matrices. Two of them can be used to create a pair of 3-by-3 example matrices for use throughout this chapter. The first example is symmetric: A = pascal(3) A = 1 1 1 1-4 1 2 3 1 3 6 Matrices in MATLAB The second example is not symmetric: B = magic(3) B = 8 3 4 1 5 9 6 7 2 Another example is a 3-by-2 rectangular matrix of random integers: C = fix(10*rand(3,2)) C = 9 2 6 4 8 7 A column vector is an m-by-1 matrix, a row vector is a 1-by-n matrix and a scalar is a 1-by-1 matrix. The statements u = [3; 1; 4] v = [2 0 -1] s = 7 produce a column vector, a row vector, and a scalar: u = 3 1 4 v = 2 0 -1 s = 7 1-5 1 Matrices and Linear Algebra Adding and Subtracting Matrices Addition and subtraction of matrices is defined just as it is for arrays, element-by-element. Adding A to B and then subtracting A from the result recovers B: A = pascal(3); B = magic(3); X = A + B X = 9 4 5 2 7 12 7 10 8 1 5 9 6 7 2 Y = X - A Y = 8 3 4 Addition and subtraction require both matrices to have the same dimension, or one of them be a scalar. If the dimensions are incompatible, an error results: C = fix(10*rand(3,2)) X = A + C Error using ==> + Matrix dimensions must agree. w = v + s w = 9 1-6 7 6 Matrices in MATLAB Vector Products and Transpose A row vector and a column vector of the same length can be multiplied in either order. The result is either a scalar, the inner product, or a matrix, the outer product: u = [3; 1; 4]; v = [2 0 -1]; x = v*u x = 2 X = u*v X = 6 2 8 0 0 0 -3 -1 -4 For real matrices, the transpose operation interchanges a ij and a ji . MATLAB uses the apostrophe (or single quote) to denote transpose. The example matrix A is symmetric, so A' is equal to A. But B is not symmetric: B = magic(3); X = B' X = 8 1 6 3 5 7 4 9 2 Transposition turns a row vector into a column vector: x = v' x = 2 0 -1 1-7 1 Matrices and Linear Algebra If x and y are both real column vectors, the product x*y is not defined, but the two products x'*y and y'*x are the same scalar. This quantity is used so frequently, it has three different names: inner product, scalar product, or dot product. For a complex vector or matrix, z, the quantity z' denotes the complex conjugate transpose, where the sign of the complex part of each element is reversed. The unconjugated complex transpose, where the complex part of each element retains its sign, is denoted by z.'. So if z = [1+2i 3+4i] then z' is 1-2i 3-4i while z.' is 1+2i 3+4i For complex vectors, the two scalar products x'*y and y'*x are complex conjugates of each other and the scalar product x'*x of a complex vector with itself is real. Multiplying Matrices Multiplication of matrices is defined in a way that reflects composition of the underlying linear transformations and allows compact representation of systems of simultaneous linear equations. The matrix product C = AB is defined when the column dimension of A is equal to the row dimension of B, or when one of them is a scalar. If A is m-by-p and B is p-by-n, their product C is m-by-n. The product can actually be defined using MATLAB for loops, colon notation, and vector dot products: 1-8 Matrices in MATLAB A = B = m = for pascal(3); magic(3); 3; n = 3; i = 1:m for j = 1:n C(i,j) = A(i,:)*B(:,j); end end MATLAB uses a single asterisk to denote matrix multiplication. The next two examples illustrate the fact that matrix multiplication is not commutative; AB is usually not equal to BA: X = A*B X = 15 26 41 15 38 70 15 26 39 28 34 28 47 60 43 Y = B*A Y = 15 15 15 A matrix can be multiplied on the right by a column vector and on the left by a row vector: u = [3; 1; 4]; x = A*u x = 8 17 30 v = [2 0 -1]; 1-9 1 Matrices and Linear Algebra y = v*B y = 12 -7 10 Rectangular matrix multiplications must satisfy the dimension compatibility conditions: C = fix(10*rand(3,2)); X = A*C X = 17 31 51 19 41 70 Y = C*A Error using ==> * Inner matrix dimensions must agree. Anything can be multiplied by a scalar: s = 7; w = s*v w = 14 0 -7 The Identity Matrix Generally accepted mathematical notation uses the capital letter I to denote identity matrices, matrices of various sizes with ones on the main diagonal and zeros elsewhere. These matrices have the property that AI = A and IA = A whenever the dimensions are compatible. The original version of MATLAB could not use I for this purpose because it did not distinguish between upper and lowercase letters and i already served double duty as a subscript and as the complex unit. So an English language pun was introduced. The function eye(m,n) 1-10 Matrices in MATLAB returns an m-by-n rectangular identity matrix and eye(n) returns an n-by-n square identity matrix. The Kronecker Tensor Product The Kronecker product, kron(X,Y), of two matrices is the larger matrix formed from all possible products of the elements of X with those of Y. If X is m-by-n and Y is p-by-q, then kron(X,Y) is mp-by-nq. The elements are arranged in the following order: [X(1,1)*Y X(1,2)*Y X(m,1)*Y X(m,2)*Y . . . . . . . . . X(1,n)*Y X(m,n)*Y] The Kronecker product is often used with matrices of zeros and ones to build up repeated copies of small matrices. For example, if X is the 2-by-2 matrix X = 1 3 2 4 and I = eye(2,2) is the 2-by-2 identity matrix, then the two matrices kron(X,I) and kron(I,X) are 1 0 3 0 0 1 0 3 2 0 4 0 0 2 0 4 1 3 0 0 2 4 0 0 0 0 1 3 0 0 2 4 and 1-11 1 Matrices and Linear Algebra Vector and Matrix Norms The p-norm of a vector x x p = ⎛ ⎝ Σ p xi ⎞ ⎠ 1⁄p is computed by norm(x,p). This is defined by any value of p > 1, but the most common values of p are 1, 2, and ∞ . The default value is p = 2, which corresponds to Euclidean length: v = [2 0 -1]; [norm(v,1) norm(v) norm(v,inf)] ans = 3.0000 2.2361 2.0000 The p-norm of a matrix A, Ax --------------pA p = max x x p can be computed for p = 1, 2, and ∞ by norm(A,p). Again, the default value is p = 2. C = fix(10*rand(3,2)); [norm(C,1) norm(C) norm(C,inf)] ans = 19.0000 1-12 14.8015 13.0000 Solving Linear Systems of Equations Solving Linear Systems of Equations This section describes • Computational considerations • The general solution to a system It also discusses particular solutions to • Square systems • Overdetermined systems • Underdetermined systems Computational Considerations One of the most important problems in technical computing is the solution of simultaneous linear equations. In matrix notation, this problem can be stated as follows. Given two matrices A and B, does there exist a unique matrix X so that AX = B or XA = B? It is instructive to consider a 1-by-1 example. Does the equation 7x = 21 have a unique solution ? The answer, of course, is yes. The equation has the unique solution x = 3. The solution is easily obtained by division: x = 21 ⁄ 7 = 3 The solution is not ordinarily obtained by computing the inverse of 7, that is 7-1 = 0.142857…, and then multiplying 7-1 by 21. This would be more work and, if 7-1 is represented to a finite number of digits, less accurate. Similar considerations apply to sets of linear equations with more than one unknown; MATLAB solves such equations without computing the inverse of the matrix. Although it is not standard mathematical notation, MATLAB uses the division terminology familiar in the scalar case to describe the solution of a general system of simultaneous equations. The two division symbols, slash, /, and 1-13 1 Matrices and Linear Algebra backslash, \, are used for the two situations where the unknown matrix appears on the left or right of the coefficient matrix: X = A\B Denotes the solution to the matrix equation AX = B. X = B/A Denotes the solution to the matrix equation XA = B. You can think of “dividing” both sides of the equation AX = B or XA = B by A. The coefficient matrix A is always in the “denominator.” The dimension compatibility conditions for X = A\B require the two matrices A and B to have the same number of rows. The solution X then has the same number of columns as B and its row dimension is equal to the column dimension of A. For X = B/A, the roles of rows and columns are interchanged. In practice, linear equations of the form AX = B occur more frequently than those of the form XA = B. Consequently, backslash is used far more frequently than slash. The remainder of this section concentrates on the backslash operator; the corresponding properties of the slash operator can be inferred from the identity (B/A)' = (A'\B') The coefficient matrix A need not be square. If A is m-by-n, there are three cases: m=n Square system. Seek an exact solution. m>n Overdetermined system. Find a least squares solution. m<n Underdetermined system. Find a basic solution with at most m nonzero components. The backslash operator employs different algorithms to handle different kinds of coefficient matrices. The various cases, which are diagnosed automatically by examining the coefficient matrix, include • Permutations of triangular matrices • Symmetric, positive definite matrices • Square, nonsingular matrices • Rectangular, overdetermined systems • Rectangular, underdetermined systems 1-14 Solving Linear Systems of Equations General Solution The general solution to a system of linear equations AX = b describes all possible solutions. You can find the general solution by 1 Solving the corresponding homogeneous system AX = 0. Do this using the null command, by typing null(A). This returns a basis for the solution space to AX = 0. Any solution is a linear combination of basis vectors. 2 Finding a particular solution to the non-homogeneous system AX = b. You can then write any solution to AX = b as the sum of the particular solution to AX = b, from step 2, plus a linear combination of the basis vectors from step 1. The rest of this section describes how to use MATLAB to find a particular solution to AX = b, as in step 2. Square Systems The most common situation involves a square coefficient matrix A and a single right-hand side column vector b. Nonsingular Coefficient Matrix If the matrix A is nonsingular, the solution, x = A\b, is then the same size as b. For example, A = pascal(3); u = [3; 1; 4]; x = A\u x = 10 -12 5 It can be confirmed that A*x is exactly equal to u. 1-15 1 Matrices and Linear Algebra If A and B are square and the same size, then X = A\B is also that size: B = magic(3); X = A\B X = 19 -17 6 -3 4 0 -1 13 -6 It can be confirmed that A*X is exactly equal to B. Both of these examples have exact, integer solutions. This is because the coefficient matrix was chosen to be pascal(3), which has a determinant equal to one. A later section considers the effects of roundoff error inherent in more realistic computations. Singular Coefficient Matrix A square matrix A is singular if it does not have linearly independent columns. If A is singular, the solution to AX = B either does not exist, or is not unique. The backslash operator, A\B, issues a warning if A is nearly singular and raises an error condition if it detects exact singularity. If A is singular and AX = b has a solution, you can find a particular solution that is not unique, by typing P = pinv(A)*b P is a pseudoinverse of A. If AX = b does not have an exact solution, pinv(A) returns a least-squares solution. For example, A = [ 1 -1 1 3 4 10 7 4 18 ] is singular, as you can verify by typing det(A) ans = 0 1-16 Solving Linear Systems of Equations Note For information about using pinv to solve systems with rectangular coefficient matrices, see “Pseudoinverses” on page 1-24. Exact Solutions. For b =[5;2;12], the equation AX = b has an exact solution, given by pinv(A)*b ans = 0.3850 -0.1103 0.7066 You can verify that pinv(A)*b is an exact solution by typing A*pinv(A)*b ans = 5.0000 2.0000 12.0000 Least Squares Solutions. On the other hand, if b = [3;6;0], then AX = b does not have an exact solution. In this case, pinv(A)*b returns a least squares solution. If you type A*pinv(A)*b ans = -1.0000 4.0000 2.0000 you do not get back the original vector b. 1-17 1 Matrices and Linear Algebra You can determine whether AX = b has an exact solution by finding the row reduced echelon form of the augmented matrix [A b]. To do so for this example, enter rref([A b]) ans = 1.0000 0 0 0 1.0000 0 2.2857 1.5714 0 0 0 1.0000 Since the bottom row contains all zeros except for the last entry, the equation does not have a solution. In this case, pinv(A) returns a least-squares solution. Overdetermined Systems Overdetermined systems of simultaneous linear equations are often encountered in various kinds of curve fitting to experimental data. Here is a hypothetical example. A quantity y is measured at several different values of time, t, to produce the following observations: t y 0.0 0.82 0.3 0.72 0.8 0.63 1.1 0.60 1.6 0.55 2.3 0.50 Enter the data into MATLAB with the statements t = [0 .3 .8 1.1 1.6 2.3]'; y = [.82 .72 .63 .60 .55 .50]'; Try modeling the data with a decaying exponential function: y ( t ) ≈ c1 + c2 e 1-18 –t Solving Linear Systems of Equations The preceding equation says that the vector y should be approximated by a linear combination of two other vectors, one the constant vector containing all ones and the other the vector with components e-t. The unknown coefficients, c1 and c2, can be computed by doing a least squares fit, which minimizes the sum of the squares of the deviations of the data from the model. There are six equations in two unknowns, represented by the 6-by-2 matrix: E = [ones(size(t)) exp(-t)] E = 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.7408 0.4493 0.3329 0.2019 0.1003 Use the backslash operator to get the least squares solution: c = E\y c = 0.4760 0.3413 In other words, the least squares fit to the data is y ( t ) ≈ 0.4760 + 0.3413 e –t The following statements evaluate the model at regularly spaced increments in t, and then plot the result, together with the original data: T = (0:0.1:2.5)'; Y = [ones(size(T)) exp(-T)]*c; plot(T,Y,'-',t,y,'o') You can see that E*c is not exactly equal to y, but that the difference might well be less than measurement errors in the original data. 1-19 1 Matrices and Linear Algebra 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.5 1 1.5 2 2.5 A rectangular matrix A is rank deficient if it does not have linearly independent columns. If A is rank deficient, the least squares solution to AX = B is not unique. The backslash operator, A\B, issues a warning if A is rank deficient and produces a least squares solution that has at most rank(A) nonzeros. Underdetermined Systems Underdetermined linear systems involve more unknowns than equations. When they are accompanied by additional constraints, they are the purview of linear programming. By itself, the backslash operator deals only with the unconstrained system. The solution is never unique. MATLAB finds a basic solution, which has at most m nonzero components, but even this may not be unique. The particular solution actually computed is determined by the QR factorization with column pivoting (see a later section on the QR factorization). 1-20 Solving Linear Systems of Equations Here is a small, random example: R = fix(10*rand(2,4)) R = 6 3 8 5 7 4 3 1 b = fix(10*rand(2,1)) b = 1 2 The linear system Rx = b involves two equations in four unknowns. Since the coefficient matrix contains small integers, it is appropriate to use the format command to display the solution in rational format. The particular solution is obtained with format rat p = R\b p = 0 5/7 0 -11/7 One of the nonzero components is p(2) because R(:,2) is the column of R with largest norm. The other nonzero component is p(4) because R(:,4) dominates after R(:,2) is eliminated. The complete solution to the underdetermined system can be characterized by adding an arbitrary vector from the null space, which can be found using the null function with an option requesting a “rational” basis: Z = null(R,'r') Z = -1/2 -1/2 1 0 -7/6 1/2 0 1 1-21 1 Matrices and Linear Algebra It can be confirmed that R*Z is zero and that any vector x where x = p + Z*q for an arbitrary vector q satisfies R*x = b. 1-22 Inverses and Determinants Inverses and Determinants This section provides • An overview of the use of inverses and determinants for solving square nonsingular systems of linear equations • A discussion of the Moore-Penrose pseudoinverse for solving rectangular systems of linear equations Overview If A is square and nonsingular, the equations AX = I and XA = I have the same solution, X. This solution is called the inverse of A, is denoted by A-1, and is computed by the function inv. The determinant of a matrix is useful in theoretical considerations and some types of symbolic computation, but its scaling and roundoff error properties make it far less satisfactory for numeric computation. Nevertheless, the function det computes the determinant of a square matrix: A = pascal(3) A = 1 1 1 1 2 3 1 3 6 -3 5 -2 1 -2 1 d = det(A) X = inv(A) d = 1 X = 3 -3 1 1-23 1 Matrices and Linear Algebra Again, because A is symmetric, has integer elements, and has determinant equal to one, so does its inverse. On the other hand, B = magic(3) B = 8 3 4 1 5 9 6 7 2 d = det(B) X = inv(B) d = -360 X = 0.1472 -0.0611 -0.0194 -0.1444 0.0222 0.1889 0.0639 0.1056 -0.1028 Closer examination of the elements of X, or use of format rat, would reveal that they are integers divided by 360. If A is square and nonsingular, then without roundoff error, X = inv(A)*B would theoretically be the same as X = A\B and Y = B*inv(A) would theoretically be the same as Y = B/A. But the computations involving the backslash and slash operators are preferable because they require less computer time, less memory, and have better error detection properties. Pseudoinverses Rectangular matrices do not have inverses or determinants. At least one of the equations AX = I and XA = I does not have a solution. A partial replacement for 1-24 Inverses and Determinants the inverse is provided by the Moore-Penrose pseudoinverse, which is computed by the pinv function: C = fix(10*rand(3,2)); X = pinv(C) X = 0.0401 0.0110 -0.1492 0.1657 0.1050 -0.0055 The matrix Q = X*C Q = 1.0000 0.0000 0.0000 1.0000 is the 2-by-2 identity, but the matrix P = C*X P = 0.2044 0.0663 0.3978 0.0663 0.9945 -0.0331 0.3978 -0.0331 0.8011 is not the 3-by-3 identity. However, P acts like an identity on a portion of the space in the sense that P is symmetric, P*C is equal to C and X*P is equal to X. Solving a Rank-Deficient System If A is m-by-n with m > n and full rank n, then each of the three statements x = A\b x = pinv(A)*b x = inv(A'*A)*A'*b theoretically computes the same least squares solution x, although the backslash operator does it faster. However, if A does not have full rank, the solution to the least squares problem is not unique. There are many vectors x that minimize norm(A*x -b) 1-25 1 Matrices and Linear Algebra The solution computed by x = A\b is a basic solution; it has at most r nonzero components, where r is the rank of A. The solution computed by x = pinv(A)*b is the minimal norm solution because it minimizes norm(x). An attempt to compute a solution with x = inv(A'*A)*A'*b fails because A'*A is singular. Here is an example that illustrates the various solutions: A = [ 1 2 3 4 5 6 7 8 9 10 11 12 ] does not have full rank. Its second column is the average of the first and third columns. If b = A(:,2) is the second column, then an obvious solution to A*x = b is x = [0 1 0]'. But none of the approaches computes that x. The backslash operator gives x = A\b Warning: Rank deficient, rank = 2. x = 0.5000 0 0.5000 This solution has two nonzero components. The pseudoinverse approach gives y = pinv(A)*b y = 0.3333 0.3333 0.3333 1-26 Inverses and Determinants There is no warning about rank deficiency. But norm(y) = 0.5774 is less than norm(x) = 0.7071. Finally z = inv(A'*A)*A'*b fails completely: Warning: Matrix is singular to working precision. z = Inf Inf Inf 1-27 1 Matrices and Linear Algebra Cholesky, LU, and QR Factorizations The MATLAB linear equation capabilities are based on three basic matrix factorizations: • Cholesky factorization for symmetric, positive definite matrices • LU factorization (Gaussian elimination) for general square matrices • QR (orthogonal) for rectangular matrices These three factorizations are available through the chol, lu, and qr functions. All three of these factorizations make use of triangular matrices where all the elements either above or below the diagonal are zero. Systems of linear equations involving triangular matrices are easily and quickly solved using either forward or back substitution. Cholesky Factorization The Cholesky factorization expresses a symmetric matrix as the product of a triangular matrix and its transpose A = R′R where R is an upper triangular matrix. Not all symmetric matrices can be factored in this way; the matrices that have such a factorization are said to be positive definite. This implies that all the diagonal elements of A are positive and that the offdiagonal elements are “not too big.” The Pascal matrices provide an interesting example. Throughout this chapter, the example matrix A has been the 3-by-3 Pascal matrix. Temporarily switch to the 6-by-6: A = pascal(6) A = 1 1 1 1 1 1 1-28 1 2 3 4 5 6 1 3 6 10 15 21 1 4 10 20 35 56 1 5 15 35 70 126 1 6 21 56 126 252 Cholesky, LU, and QR Factorizations The elements of A are binomial coefficients. Each element is the sum of its north and west neighbors. The Cholesky factorization is R = chol(A) R = 1 0 0 0 0 0 1 1 0 0 0 0 1 2 1 0 0 0 1 3 3 1 0 0 1 4 6 4 1 0 1 5 10 10 5 1 The elements are again binomial coefficients. The fact that R'*R is equal to A demonstrates an identity involving sums of products of binomial coefficients. Note The Cholesky factorization also applies to complex matrices. Any complex matrix which has a Cholesky factorization satisfies A' = A and is said to be Hermitian positive definite. The Cholesky factorization allows the linear system Ax = b to be replaced by R′Rx = b Because the backslash operator recognizes triangular systems, this can be solved in MATLAB quickly with x = R\(R'\b) If A is n-by-n, the computational complexity of chol(A) is O(n3), but the complexity of the subsequent backslash solutions is only O(n2). 1-29 1 Matrices and Linear Algebra LU Factorization LU factorization, or Gaussian elimination, expresses any square matrix A as the product of a permutation of a lower triangular matrix and an upper triangular matrix A = LU where L is a permutation of a lower triangular matrix with ones on its diagonal and U is an upper triangular matrix. The permutations are necessary for both theoretical and computational reasons. The matrix 0 1 1 0 cannot be expressed as the product of triangular matrices without interchanging its two rows. Although the matrix ε 1 1 0 can be expressed as the product of triangular matrices, when ε is small the elements in the factors are large and magnify errors, so even though the permutations are not strictly necessary, they are desirable. Partial pivoting ensures that the elements of L are bounded by one in magnitude and that the elements of U are not much larger than those of A. For example [L,U] = lu(B) L = 1.0000 0.3750 0.5000 0 0.5441 1.0000 0 1.0000 0 8.0000 0 0 1.0000 8.5000 0 6.0000 -1.0000 5.2941 U = 1-30 Cholesky, LU, and QR Factorizations The LU factorization of A allows the linear system A*x = b to be solved quickly with x = U\(L\b) Determinants and inverses are computed from the LU factorization using det(A) = det(L)*det(U) and inv(A) = inv(U)*inv(L) You can also compute the determinants using det(A) = prod(diag(U)), though the signs of the determinants may be reversed. QR Factorization An orthogonal matrix, or a matrix with orthonormal columns, is a real matrix whose columns all have unit length and are perpendicular to each other. If Q is orthogonal, then Q′Q = 1 The simplest orthogonal matrices are two-dimensional coordinate rotations: cos ( θ ) – sin ( θ ) sin ( θ ) cos ( θ ) For complex matrices, the corresponding term is unitary. Orthogonal and unitary matrices are desirable for numerical computation because they preserve length, preserve angles, and do not magnify errors. The orthogonal, or QR, factorization expresses any rectangular matrix as the product of an orthogonal or unitary matrix and an upper triangular matrix. A column permutation may also be involved: A = QR or A P= QR 1-31 1 Matrices and Linear Algebra where Q is orthogonal or unitary, R is upper triangular, and P is a permutation. There are four variants of the QR factorization– full or economy size, and with or without column permutation. Overdetermined linear systems involve a rectangular matrix with more rows than columns, that is m-by-n with m > n. The full size QR factorization produces a square, m-by-m orthogonal Q and a rectangular m-by-n upper triangular R: [Q,R] = qr(C) Q = -0.8182 -0.1818 -0.5455 0.3999 -0.8616 -0.3126 R = -11.0000 0 0 -8.5455 -7.4817 0 -0.4131 -0.4739 0.7777 In many cases, the last m - n columns of Q are not needed because they are multiplied by the zeros in the bottom portion of R. So the economy size QR factorization produces a rectangular, m-by-n Q with orthonormal columns and a square n-by-n upper triangular R. For the 3-by-2 example, this is not much of a saving, but for larger, highly rectangular matrices, the savings in both time and memory can be quite important: [Q,R] = qr(C,0) 1-32 Q = -0.8182 -0.1818 -0.5455 0.3999 -0.8616 -0.3126 R = -11.0000 0 -8.5455 -7.4817 Cholesky, LU, and QR Factorizations In contrast to the LU factorization, the QR factorization does not require any pivoting or permutations. But an optional column permutation, triggered by the presence of a third output argument, is useful for detecting singularity or rank deficiency. At each step of the factorization, the column of the remaining unfactored matrix with largest norm is used as the basis for that step. This ensures that the diagonal elements of R occur in decreasing order and that any linear dependence among the columns is almost certainly be revealed by examining these elements. For the small example given here, the second column of C has a larger norm than the first, so the two columns are exchanged: [Q,R,P] = qr(C) Q = -0.3522 -0.7044 -0.6163 0.8398 -0.5285 0.1241 R = -11.3578 0 0 -8.2762 7.2460 0 -0.4131 -0.4739 0.7777 P = 0 1 1 0 When the economy size and column permutations are combined, the third output argument is a permutation vector, rather than a permutation matrix: [Q,R,p] = qr(C,0) Q = -0.3522 -0.7044 -0.6163 0.8398 -0.5285 0.1241 R = -11.3578 0 -8.2762 7.2460 1-33 1 Matrices and Linear Algebra p = 2 1 The QR factorization transforms an overdetermined linear system into an equivalent triangular system. The expression norm(A*x - b) is equal to norm(Q*R*x - b) Multiplication by orthogonal matrices preserves the Euclidean norm, so this expression is also equal to norm(R*x - y) where y = Q'*b. Since the last m-n rows of R are zero, this expression breaks into two pieces: norm(R(1:n,1:n)*x - y(1:n)) and norm(y(n+1:m)) When A has full rank, it is possible to solve for x so that the first of these expressions is zero. Then the second expression gives the norm of the residual. When A does not have full rank, the triangular structure of R makes it possible to find a basic solution to the least squares problem. 1-34 Matrix Powers and Exponentials Matrix Powers and Exponentials This section tells you how to obtain the following matrix powers and exponentials in MATLAB: • Positive integer • Inverse and fractional • Element-by-element • Exponentials Positive Integer Powers If A is a square matrix and p is a positive integer, then A^p effectively multiplies A by itself p-1 times. For example, A = [1 1 1;1 2 3;1 3 6] A = 1 1 1 1 2 3 1 3 6 6 14 25 10 25 46 X = A^2 X = 3 6 10 Inverse and Fractional Powers If A is square and nonsingular, then A^(-p) effectively multiplies inv(A) by itself p-1 times: Y = A^(-3) 1-35 1 Matrices and Linear Algebra Y = 145.0000 -207.0000 81.0000 -207.0000 298.0000 -117.0000 81.0000 -117.0000 46.0000 Fractional powers, like A^(2/3), are also permitted; the results depend upon the distribution of the eigenvalues of the matrix. Element-by-Element Powers The .^ operator produces element-by-element powers. For example, X = A.^2 A = 1 1 1 1 4 9 1 9 36 Exponentials The function sqrtm(A) computes A^(1/2) by a more accurate algorithm. The m in sqrtm distinguishes this function from sqrt(A) which, like A.^(1/2), does its job element-by-element. A system of linear, constant coefficient, ordinary differential equations can be written dx ⁄ dt = Ax where x = x(t) is a vector of functions of t and A is a matrix independent of t. The solution can be expressed in terms of the matrix exponential, x(t) = e tA x(0) The function expm(A) 1-36 Matrix Powers and Exponentials computes the matrix exponential. An example is provided by the 3-by-3 coefficient matrix A = 0 6 -5 -6 2 20 -1 -16 -10 and the initial condition, x(0) x0 = 1 1 1 The matrix exponential is used to compute the solution, x(t), to the differential equation at 101 points on the interval 0 ≤ t ≤ 1 with X = []; for t = 0:.01:1 X = [X expm(t*A)*x0]; end A three-dimensional phase plane plot obtained with plot3(X(1,:),X(2,:),X(3,:),'-o') shows the solution spiraling in towards the origin. This behavior is related to the eigenvalues of the coefficient matrix, which are discussed in the next section. 1-37 1 Matrices and Linear Algebra 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 1.5 1 1 0.8 0.5 0.6 0.4 0 0.2 −0.5 1-38 0 Eigenvalues Eigenvalues An eigenvalue and eigenvector of a square matrix A are a scalar λ and a nonzero vector v that satisfy Av = λ v This section explains: • Eigenvalue decomposition • Problems associated with defective (not diagonalizable) matrices • The use of Schur decomposition to avoid problems associated with eigenvalue decomposition Eigenvalue Decomposition With the eigenvalues on the diagonal of a diagonal matrix Λ and the corresponding eigenvectors forming the columns of a matrix V, you have AV = VΛ If V is nonsingular, this becomes the eigenvalue decomposition A = VΛV –1 A good example is provided by the coefficient matrix of the ordinary differential equation in the previous section: A = 0 6 -5 -6 2 20 -1 -16 -10 The statement lambda = eig(A) produces a column vector containing the eigenvalues. For this matrix, the eigenvalues are complex: lambda = -3.0710 -2.4645+17.6008i -2.4645-17.6008i 1-39 1 Matrices and Linear Algebra λt The real part of each of the eigenvalues is negative, so e approaches zero as t increases. The nonzero imaginary part of two of the eigenvalues, ± ω , contributes the oscillatory component, sin ( ωt ) , to the solution of the differential equation. With two output arguments, eig computes the eigenvectors and stores the eigenvalues in a diagonal matrix: [V,D] = eig(A) V = -0.8326 -0.3553 -0.4248 0.2003 - 0.1394i -0.2110 - 0.6447i -0.6930 D = -3.0710 0 0 0.2003 + 0.1394i -0.2110 + 0.6447i -0.6930 0 -2.4645+17.6008i 0 0 0 -2.4645-17.6008i The first eigenvector is real and the other two vectors are complex conjugates of each other. All three vectors are normalized to have Euclidean length, norm(v,2), equal to one. The matrix V*D*inv(V), which can be written more succinctly as V*D/V, is within roundoff error of A. And, inv(V)*A*V, or V\A*V, is within roundoff error of D. Defective Matrices Some matrices do not have an eigenvector decomposition. These matrices are defective, or not diagonalizable. For example, A = [ 6 -9 4 12 -20 9 For this matrix [V,D] = eig(A) 1-40 19 -33 15 ] Eigenvalues produces V = -0.4741 0.8127 -0.3386 -0.4082 0.8165 -0.4082 -0.4082 0.8165 -0.4082 0 1.0000 0 0 0 1.0000 D = -1.0000 0 0 There is a double eigenvalue at λ = 1 . The second and third columns of V are the same. For this matrix, a full set of linearly independent eigenvectors does not exist. The optional Symbolic Math Toolbox extends the capabilities of MATLAB by connecting to Maple, a powerful computer algebra system. One of the functions provided by the toolbox computes the Jordan Canonical Form. This is appropriate for matrices like the example given here, which is 3-by-3 and has exactly known, integer elements: [X,J] = jordan(A) X = -1.7500 3.0000 -1.2500 1.5000 -3.0000 1.5000 2.7500 -3.0000 1.2500 J = -1 0 0 0 1 0 0 1 1 The Jordan Canonical Form is an important theoretical concept, but it is not a reliable computational tool for larger matrices, or for matrices whose elements are subject to roundoff errors and other uncertainties. 1-41 1 Matrices and Linear Algebra Schur Decomposition in MATLAB Matrix Computations The MATLAB advanced matrix computations do not require eigenvalue decompositions. They are based, instead, on the Schur decomposition A = U S UT where U is an orthogonal matrix and S is a block upper triangular matrix with 1-by-1 and 2-by-2 blocks on the diagonal. The eigenvalues are revealed by the diagonal elements and blocks of S, while the columns of U provide a basis with much better numerical properties than a set of eigenvectors. The Schur decomposition of this defective example is [U,S] = schur(A) U = -0.4741 0.8127 -0.3386 0.6648 0.0782 -0.7430 0.5774 0.5774 0.5774 S = -1.0000 0 0 20.7846 1.0000 0 -44.6948 -0.6096 1.0000 The double eigenvalue is contained in the lower 2-by-2 block of S. Note If A is complex, schur returns the complex Schur form, which is upper triangular with the eigenvalues of A on the diagonal. 1-42 Singular Value Decomposition Singular Value Decomposition A singular value and corresponding singular vectors of a rectangular matrix A are a scalar σ and a pair of vectors u and v that satisfy Av = σu T A u = σv With the singular values on the diagonal of a diagonal matrix Σ and the corresponding singular vectors forming the columns of two orthogonal matrices U and V, you have AV = UΣ T A U = VΣ Since U and V are orthogonal, this becomes the singular value decomposition A = UΣV T The full singular value decomposition of an m-by-n matrix involves an m-by-m U, an m-by-n Σ , and an n-by-n V. In other words, U and V are both square and Σ is the same size as A. If A has many more rows than columns, the resulting U can be quite large, but most of its columns are multiplied by zeros in Σ . In this situation, the economy sized decomposition saves both time and storage by producing an m-by-n U, an n-by-n Σ and the same V. The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself, as it does for an ordinary differential equation. On the other hand, the singular value decomposition is the appropriate tool for analyzing a mapping from one vector space into another vector space, possibly with a different dimension. Most systems of simultaneous linear equations fall into this second category. If A is square, symmetric, and positive definite, then its eigenvalue and singular value decompositions are the same. But, as A departs from symmetry and positive definiteness, the difference between the two decompositions increases. In particular, the singular value decomposition of a real matrix is always real, but the eigenvalue decomposition of a real, nonsymmetric matrix might be complex. 1-43 1 Matrices and Linear Algebra For the example matrix A = 9 6 2 4 8 7 the full singular value decomposition is [U,S,V] = svd(A) U = -0.6105 -0.6646 -0.4308 S = 14.9359 0 0 0.7174 -0.2336 -0.6563 0.3355 -0.7098 0.6194 0 5.1883 0 V = -0.6925 -0.7214 0.7214 -0.6925 You can verify that U*S*V' is equal to A to within roundoff error. For this small problem, the economy size decomposition is only slightly smaller: [U,S,V] = svd(A,0) U = -0.6105 -0.6646 -0.4308 S = 14.9359 0 0.7174 -0.2336 -0.6563 0 5.1883 V = -0.6925 -0.7214 0.7214 -0.6925 Again, U*S*V' is equal to A to within roundoff error. 1-44 2 Polynomials and Interpolation Polynomials (p. 2-2) Functions for standard polynomial operations. Additional topics include curve fitting and partial fraction expansion. Interpolation (p. 2-9) Two- and multi-dimensional interpolation techniques, taking into account speed, memory, and smoothness considerations. Selected Bibliography (p. 2-37) Published materials that support concepts implemented in “Polynomials and Interpolation” 2 Polynomials and Interpolation Polynomials This section provides • A summary of the MATLAB polynomial functions • Instructions for representing polynomials in MATLAB It also describes the MATLAB polynomial functions that • Calculate the roots of a polynomial • Calculate the coefficients of the characteristic polynomial of a matrix • Evaluate a polynomial at a specified value • Convolve (multiply) and deconvolve (divide) polynomials • Compute the derivative of a polynomial • Fit a polynomial to a set of data • Convert between partial fraction expansion and polynomial coefficients Polynomial Function Summary MATLAB provides functions for standard polynomial operations, such as polynomial roots, evaluation, and differentiation. In addition, there are functions for more advanced applications, such as curve fitting and partial fraction expansion. The polynomial functions reside in the MATLAB polyfun directory. Polynomial Function Summary 2-2 Function Description conv Multiply polynomials. deconv Divide polynomials. poly Polynomial with specified roots. polyder Polynomial derivative. polyfit Polynomial curve fitting. polyval Polynomial evaluation. Polynomials Polynomial Function Summary (Continued) Function Description polyvalm Matrix polynomial evaluation. residue Partial-fraction expansion (residues). roots Find polynomial roots. The Symbolic Math Toolbox contains additional specialized support for polynomial operations. Representing Polynomials MATLAB represents polynomials as row vectors containing coefficients ordered by descending powers. For example, consider the equation 3 p ( x ) = x – 2x – 5 This is the celebrated example Wallis used when he first represented Newton’s method to the French Academy. To enter this polynomial into MATLAB, use p = [1 0 -2 -5]; Polynomial Roots The roots function calculates the roots of a polynomial: r = roots(p) r = 2.0946 -1.0473 + -1.0473 - 1.1359i 1.1359i By convention, MATLAB stores roots in column vectors. The function poly returns to the polynomial coefficients: p2 = poly(r) p2 = 1 8.8818e-16 -2 -5 2-3 2 Polynomials and Interpolation poly and roots are inverse functions, up to ordering, scaling, and roundoff error. Characteristic Polynomials The poly function also computes the coefficients of the characteristic polynomial of a matrix: A = [1.2 3 -0.9; 5 1.75 6; 9 0 1]; poly(A) ans = 1.0000 -3.9500 -1.8500 -163.2750 The roots of this polynomial, computed with roots, are the characteristic roots, or eigenvalues, of the matrix A. (Use eig to compute the eigenvalues of a matrix directly.) Polynomial Evaluation The polyval function evaluates a polynomial at a specified value. To evaluate p at s = 5, use polyval(p,5) ans = 110 It is also possible to evaluate a polynomial in a matrix sense. In this case 3 3 p ( s ) = x – 2x – 5 becomes p ( X ) = X – 2X – 5I , where X is a square matrix and I is the identity matrix. For example, create a square matrix X and evaluate the polynomial p at X: X = [2 4 5; -1 0 3; 7 1 5]; Y = polyvalm(p,X) Y = 377 111 490 2-4 179 81 253 439 136 639 Polynomials Convolution and Deconvolution Polynomial multiplication and division correspond to the operations convolution and deconvolution. The functions conv and deconv implement these operations. 2 2 Consider the polynomials a ( s ) = s + 2s + 3 and b ( s ) = 4s + 5s + 6 . To compute their product, a = [1 2 3]; b = [4 5 6]; c = conv(a,b) c = 4 13 28 27 18 Use deconvolution to divide a ( s ) back out of the product: [q,r] = deconv(c,a) q = 4 5 6 0 0 0 r = 0 0 Polynomial Derivatives The polyder function computes the derivative of any polynomial. To obtain the derivative of the polynomial p = [1 0 -2 -5], q = polyder(p) q = 3 0 -2 polyder also computes the derivative of the product or quotient of two polynomials. For example, create two polynomials a and b: a = [1 3 5]; b = [2 4 6]; 2-5 2 Polynomials and Interpolation Calculate the derivative of the product a*b by calling polyder with a single output argument: c = polyder(a,b) c = 8 30 56 38 Calculate the derivative of the quotient a/b by calling polyder with two output arguments: [q,d] = polyder(a,b) q = -2 -8 -2 4 16 40 d = 48 36 q/d is the result of the operation. Polynomial Curve Fitting polyfit finds the coefficients of a polynomial that fits a set of data in a least-squares sense: p = polyfit(x,y,n) x and y are vectors containing the x and y data to be fitted, and n is the order of the polynomial to return. For example, consider the x-y test data x = [1 2 3 4 5]; y = [5.5 43.1 128 290.7 498.4]; A third order polynomial that approximately fits the data is p = polyfit(x,y,3) p = -0.1917 2-6 31.5821 -60.3262 35.3400 Polynomials Compute the values of the polyfit estimate over a finer range, and plot the estimate over the real data values for comparison: x2 = 1:.1:5; y2 = polyval(p,x2); plot(x,y,'o',x2,y2) grid on 500 450 400 350 300 250 200 150 100 50 0 1 1.5 2 2.5 3 3.5 4 4.5 5 To use these functions in an application example, see the “Data Analysis and Statistics” chapter. Partial Fraction Expansion residue finds the partial fraction expansion of the ratio of two polynomials. This is particularly useful for applications that represent systems in transfer function form. For polynomials b and a, if there are no multiple roots, rn b ( s )- = r 1 + r 2 + ------------------------ --------------- … + -------------- + ks a(s) s – p 1 s – p2 s – pn 2-7 2 Polynomials and Interpolation where r is a column vector of residues, p is a column vector of pole locations, and k is a row vector of direct terms. Consider the transfer function – 4s + 8 ---------------------------2 s + 6s + 8 b = [-4 8]; a = [1 6 8]; [r,p,k] = residue(b,a) r = -12 8 p = -4 -2 k = [] Given three input arguments (r, p, and k), residue converts back to polynomial form: [b2,a2] = residue(r,p,k) b2 = -4 8 1 6 a2 = 2-8 8 Interpolation Interpolation Interpolation is a process for estimating values that lie between known data points. It has important applications in areas such as signal and image processing. This section • Provides a summary of the MATLAB interpolation functions • Discusses one-dimensional interpolation • Discusses two-dimensional interpolation • Uses an example to compare nearest neighbor, bilinear, and bicubic interpolation methods • Discusses interpolation of multidimensional data • Discusses triangulation and interpolation of scattered data Interpolation Function Summary MATLAB provides a number of interpolation techniques that let you balance the smoothness of the data fit with speed of execution and memory usage. The interpolation functions reside in the MATLAB polyfun directory. Interpolation Function Summary Function Description griddata Data gridding and surface fitting. griddata3 Data gridding and hypersurface fitting for three-dimensional data. griddatan Data gridding and hypersurface fitting (dimension >= 3). interp1 One-dimensional interpolation (table lookup). interp2 Two-dimensional interpolation (table lookup). interp3 Three-dimensional interpolation (table lookup). interpft One-dimensional interpolation using FFT method. 2-9 2 Polynomials and Interpolation Interpolation Function Summary (Continued) Function Description interpn N-dimensional interpolation (table lookup). mkpp Make a piecewise polynomial pchip Piecewise Cubic Hermite Interpolating Polynomial (PCHIP). ppval Piecewise polynomial evaluation spline Cubic spline data interpolation unmkpp Piecewise polynomial details One-Dimensional Interpolation There are two kinds of one-dimensional interpolation in MATLAB: • Polynomial interpolation • FFT-based interpolation Polynomial Interpolation The function interp1 performs one-dimensional interpolation, an important operation for data analysis and curve fitting. This function uses polynomial techniques, fitting the supplied data with polynomial functions between data points and evaluating the appropriate function at the desired interpolation points. Its most general form is yi = interp1(x,y,xi,method) y is a vector containing the values of a function, and x is a vector of the same length containing the points for which the values in y are given. xi is a vector containing the points at which to interpolate. method is an optional string specifying an interpolation method: • Nearest neighbor interpolation (method = 'nearest'). This method sets the value of an interpolated point to the value of the nearest existing data point. • Linear interpolation (method = 'linear'). This method fits a different linear function between each pair of existing data points, and returns the value of 2-10 Interpolation the relevant function at the points specified by xi. This is the default method for the interp1 function. • Cubic spline interpolation (method = 'spline'). This method fits a different cubic function between each pair of existing data points, and uses the spline function to perform cubic spline interpolation at the data points. • Cubic interpolation (method = 'pchip' or 'cubic'). These methods are identical. They use the pchip function to perform piecewise cubic Hermite interpolation within the vectors x and y. These methods preserve monotonicity and the shape of the data. If any element of xi is outside the interval spanned by x, the specified interpolation method is used for extrapolation. Alternatively, yi = interp1(x,Y,xi,method,extrapval) replaces extrapolated values with extrapval. NaN is often used for extrapval. All methods work with nonuniformly spaced data. Speed, Memory, and Smoothness Considerations When choosing an interpolation method, keep in mind that some require more memory or longer computation time than others. However, you may need to trade off these resources to achieve the desired smoothness in the result: • Nearest neighbor interpolation is the fastest method. However, it provides the worst results in terms of smoothness. • Linear interpolation uses more memory than the nearest neighbor method, and requires slightly more execution time. Unlike nearest neighbor interpolation its results are continuous, but the slope changes at the vertex points. • Cubic spline interpolation has the longest relative execution time, although it requires less memory than cubic interpolation. It produces the smoothest results of all the interpolation methods. You may obtain unexpected results, however, if your input data is non-uniform and some points are much closer together than others. • Cubic interpolation requires more memory and execution time than either the nearest neighbor or linear methods. However, both the interpolated data and its derivative are continuous. The relative performance of each method holds true even for interpolation of two-dimensional or multidimensional data. For a graphical comparison of 2-11 2 Polynomials and Interpolation interpolation methods, see the section “Comparing Interpolation Methods” on page 2-13. FFT-Based Interpolation The function interpft performs one-dimensional interpolation using an FFT-based method. This method calculates the Fourier transform of a vector that contains the values of a periodic function. It then calculates the inverse Fourier transform using more points. Its form is y = interpft(x,n) x is a vector containing the values of a periodic function, sampled at equally spaced points. n is the number of equally spaced points to return. Two-Dimensional Interpolation The function interp2 performs two-dimensional interpolation, an important operation for image processing and data visualization. Its most general form is ZI = interp2(X,Y,Z,XI,YI,method) Z is a rectangular array containing the values of a two-dimensional function, and X and Y are arrays of the same size containing the points for which the values in Z are given. XI and YI are matrices containing the points at which to interpolate the data. method is an optional string specifying an interpolation method. There are three different interpolation methods for two-dimensional data: • Nearest neighbor interpolation (method = 'nearest'). This method fits a piecewise constant surface through the data values. The value of an interpolated point is the value of the nearest point. • Bilinear interpolation (method = 'linear'). This method fits a bilinear surface through existing data points. The value of an interpolated point is a combination of the values of the four closest points. This method is piecewise bilinear, and is faster and less memory-intensive than bicubic interpolation. • Bicubic interpolation (method = 'cubic'). This method fits a bicubic surface through existing data points. The value of an interpolated point is a combination of the values of the sixteen closest points. This method is piecewise bicubic, and produces a much smoother surface than bilinear interpolation. This can be a key advantage for applications like image 2-12 Interpolation processing. Use bicubic interpolation when the interpolated data and its derivative must be continuous. All of these methods require that X and Y be monotonic, that is, either always increasing or always decreasing from point to point. You should prepare these matrices using the meshgrid function, or else be sure that the “pattern” of the points emulates the output of meshgrid. In addition, each method automatically maps the input to an equally spaced domain before interpolating. If X and Y are already equally spaced, you can speed execution time by prepending an asterisk to the method string, for example, '*cubic'. Comparing Interpolation Methods This example compares two-dimensional interpolation methods on a 7-by-7 matrix of data: 1 Generate the peaks function at low resolution: [x,y] = meshgrid(-3:1:3); z = peaks(x,y); surf(x,y,z) 6 4 2 0 −2 −4 −6 3 2 3 1 2 0 1 0 −1 −1 −2 −2 −3 −3 2-13 2 Polynomials and Interpolation 2 Generate a finer mesh for interpolation: [xi,yi] = meshgrid(-3:0.25:3); 3 Interpolate using nearest neighbor interpolation: zi1 = interp2(x,y,z,xi,yi,'nearest'); 4 Interpolate using bilinear interpolation: zi2 = interp2(x,y,z,xi,yi,'bilinear'); 5 Interpolate using bicubic interpolation: zi3 = interp2(x,y,z,xi,yi,'bicubic'); 6 Compare the surface plots for the different interpolation methods. 6 6 6 4 4 4 2 2 2 0 0 0 −2 −2 −2 −4 −4 −4 −6 3 −6 3 2 3 1 2 0 1 0 −1 −1 −2 −2 −3 −3 surf(xi,yi,zi1) % nearest 2-14 −6 3 2 3 1 2 0 1 0 −1 −1 −2 −2 −3 −3 surf(xi,yi,zi2) % bilinear 2 3 1 2 0 1 0 −1 −1 −2 −2 −3 −3 surf(xi,yi,zi3) % bicubic Interpolation 7 Compare the contour plots for the different interpolation methods. 3 3 3 2 2 2 1 1 1 0 0 0 −1 −1 −1 −2 −2 −2 −3 −3 −3 −3 −2 −1 0 1 2 3 contour(xi,yi,zi1) % nearest −2 −1 0 1 2 3 −3 −3 contour(xi,yi,zi2) % bilinear −2 −1 0 1 2 3 contour(xi,yi,zi3) % bicubic Notice that the bicubic method, in particular, produces smoother contours. This is not always the primary concern, however. For some applications, such as medical image processing, a method like nearest neighbor may be preferred because it doesn’t generate any “new” data values. Interpolation and Multidimensional Arrays Several interpolation functions operate specifically on multidimensional data. Interpolation Functions for Multidimensional Data Function Description interp3 Three-dimensional data interpolation. interpn Multidimensional data interpolation. ndgrid Multidimensional data gridding (elmat directory). This section discusses • Interpolation of three-dimensional data • Interpolation of higher dimensional data • Multidimensional data gridding 2-15 2 Polynomials and Interpolation Interpolation of Three-Dimensional Data The function interp3 performs three-dimensional interpolation, finding interpolated values between points of a three-dimensional set of samples V. You must specify a set of known data points: • X, Y, and Z matrices specify the points for which values of V are given. • A matrix V contains values corresponding to the points in X, Y, and Z. The most general form for interp3 is VI = interp3(X,Y,Z,V,XI,YI,ZI,method) XI, YI, and ZI are the points at which interp3 interpolates values of V. For out-of-range values, interp3 returns NaN. There are three different interpolation methods for three-dimensional data: • Nearest neighbor interpolation (method = 'nearest'). This method chooses the value of the nearest point. • Trilinear interpolation (method = 'linear'). This method uses piecewise linear interpolation based on the values of the nearest eight points. • Tricubic interpolation (method = 'cubic'). This method uses piecewise cubic interpolation based on the values of the nearest sixty-four points. All of these methods require that X, Y, and Z be monotonic, that is, either always increasing or always decreasing in a particular direction. In addition, you should prepare these matrices using the meshgrid function, or else be sure that the “pattern” of the points emulates the output of meshgrid. Each method automatically maps the input to an equally spaced domain before interpolating. If x is already equally spaced, you can speed execution time by prepending an asterisk to the method string, for example, '*cubic'. Interpolation of Higher Dimensional Data The function interpn performs multidimensional interpolation, finding interpolated values between points of a multidimensional set of samples V. The most general form for interpn is VI = interpn(X1,X2,X3...,V,Y1,Y2,Y3,...,method) 1, 2, 3, ... are matrices that specify the points for which values of V are given. V is a matrix that contains the values corresponding to these points. 1, 2, 3, ... 2-16 Interpolation are the points for which interpn returns interpolated values of V. For out-of-range values, interpn returns NaN. Y1, Y2, Y3, ... must be either arrays of the same size, or vectors. If they are vectors of different sizes, interpn passes them to ndgrid and then uses the resulting arrays. There are three different interpolation methods for multidimensional data: • Nearest neighbor interpolation (method = 'nearest'). This method chooses the value of the nearest point. • Linear interpolation (method = 'linear'). This method uses piecewise linear interpolation based on the values of the nearest two points in each dimension. • Cubic interpolation (method = 'cubic'). This method uses piecewise cubic interpolation based on the values of the nearest four points in each dimension. All of these methods require that X1, X2,X3 be monotonic. In addition, you should prepare these matrices using the ndgrid function, or else be sure that the “pattern” of the points emulates the output of ndgrid. Each method automatically maps the input to an equally spaced domain before interpolating. If X is already equally spaced, you can speed execution time by prepending an asterisk to the method string; for example, '*cubic'. Multidimensional Data Gridding The ndgrid function generates arrays of data for multidimensional function evaluation and interpolation. ndgrid transforms the domain specified by a series of input vectors into a series of output arrays. The ith dimension of these output arrays are copies of the elements of input vector xi. The syntax for ndgrid is [X1,X2,X3,...] = ndgrid(x1,x2,x3,...) For example, assume that you want to evaluate a function of three variables over a given range. Consider the function ( –x1 –x2 –x3 ) 2 z = x2 e 2 2 2-17 2 Polynomials and Interpolation for – 2π ≤ x 1 ≤ 0 , 2π ≤ x 2 ≤ 4π , and 0 ≤ x 3 ≤ 2π . To evaluate and plot this function, x1 = -2:0.2:2; x2 = -2:0.25:2; x3 = -2:0.16:2; [X1,X2,X3] = ndgrid(x1,x2,x3); z = X2.*exp(-X1.^2 -X2.^2 -X3.^2); slice(X2,X1,X3,z,[-1.2 0.8 2],2,[-2 0.2]) 2 1 0 −1 −2 2 2 1 1 0 0 −1 −1 −2 −2 Triangulation and Interpolation of Scattered Data MATLAB provides routines that aid in the analysis of closest-point problems and geometric analysis. Functions for Analysis of Closest-Point Problems and Geometric Analysis 2-18 Function Description convhull Convex hull. delaunay Delaunay triangulation. Interpolation Functions for Analysis of Closest-Point Problems and Geometric Analysis Function Description delaunay3 3-D Delaunay tessellation. dsearch Nearest point search of Delaunay triangulation. inpolygon True for points inside polygonal region. polyarea Area of polygon. rectint Area of intersection for two or more rectangles. tsearch Closest triangle search. voronoi Voronoi diagram. This section applies the following techniques to the seamount data set supplied with MATLAB: • Convex hulls • Delaunay triangulation • Voronoi diagrams See also “Tessellation and Interpolation of Scattered Data in Higher Dimensions” on page 2-26. Note Examples in this section use the MATLAB seamount data set. Seamounts are underwater mountains. They are valuable sources of information about marine geology. The seamount data set represents the surface, in 1984, of the seamount designated LR148.8W located at 48.2°S, 148.8°W on the Louisville Ridge in the South Pacific. For more information about the data and its use, see Parker [2]. The seamount data set provides longitude (x), latitude (y) and depth-in-feet (z) data for 294 points on the seamount LR148.8W. 2-19 2 Polynomials and Interpolation Convex Hulls The convhull function returns the indices of the points in a data set that comprise the convex hull for the set. Use the plot function to plot the output of convhull. This example loads the seamount data and plots the longitudinal (x) and latitudinal (y) data as a scatter plot. It then generates the convex hull and uses plot to plot the convex hull: load seamount plot(x,y,'.','markersize',10) k = convhull(x,y); hold on, plot(x(k),y(k),'-r'), hold off grid on −47.95 −48 −48.05 −48.1 −48.15 −48.2 −48.25 −48.3 −48.35 −48.4 −48.45 210.8 211 211.2 211.4 211.6 211.8 Delaunay Triangulation Given a set of coplanar data points, Delaunay triangulation is a set of lines connecting each point to its natural neighbors. The delaunay function returns a Delaunay triangulation as a set of triangles having the property that, for each triangle, the unique circle circumscribed about the triangle contains no data points. 2-20 Interpolation You can use triplot to print the resulting triangles in two-dimensional space. You can also add data for a third dimension to the output of delaunay and plot the result as a surface with trisurf, or as a mesh with trimesh. Plotting a Delaunay Triangulation. To try delaunay, load the seamount data set and view the longitude (x) and latitude (y) data as a scatter plot: load seamount plot(x,y,'.','markersize',12) xlabel('Longitude'), ylabel('Latitude') grid on −47.95 −48 −48.05 −48.1 Latitude −48.15 −48.2 −48.25 −48.3 −48.35 −48.4 −48.45 210.8 211 211.2 211.4 Longitude 211.6 211.8 Apply Delaunay triangulation and use triplot to overplot the resulting triangles on the scatter plot: tri = delaunay(x,y); hold on, triplot(tri,x,y), hold off 2-21 2 Polynomials and Interpolation −47.95 −48 −48.05 −48.1 Latitude −48.15 −48.2 −48.25 −48.3 −48.35 −48.4 −48.45 210.8 211 211.2 211.4 Longitude 211.6 211.8 Mesh and Surface Plots. Add the depth data (z) from seamount, to the Delaunay triangulation, and use trimesh to produce a mesh in three-dimensional space. Similarly, you can use trisurf to produce a surface: figure hidden on trimesh(tri,x,y,z) grid on xlabel('Longitude'); ylabel('Latitude'); zlabel('Depth in Feet') 2-22 Interpolation 0 Depth in Feet −1000 −2000 −3000 −4000 −5000 −48 −48.1 211.6 −48.2 211.4 211.2 −48.3 −48.4 Latitude 211 210.8 Longitude Contour Plots. This code uses meshgrid, griddata, and contour to produce a contour plot of the seamount data: figure [xi,yi] = meshgrid(210.8:.01:211.8,-48.5:.01:-47.9); zi = griddata(x,y,z,xi,yi,'cubic'); [c,h] = contour(xi,yi,zi,'b-'); clabel(c,h) xlabel('Longitude'), ylabel('Latitude') 2-23 2 Polynomials and Interpolation −47.95 −4000 −48 −48.05 0 −3 00 50 −4 −48.1 −2500 0 −48.15 −10 0 00 −2 −1500 00 0 0 −30 −2000 −48.2 −40 00 0 0 00 −48.3 0 −250 −3 −35 50 Latitude −400 0 −3 − −48.25 0 00 0 0 35 −3 00 0 −48.35 00 −40 00 −4 −48.4 −48.45 210.8 211 211.2 211.4 Longitude 211.6 211.8 The arguments for meshgrid encompass the largest and smallest x and y values in the original seamount data. To obtain these values, use min(x), max(x), min(y), and max(y). Closest-Point Searches. You can search through the Delaunay triangulation data with two functions: • dsearch finds the indices of the (x,y) points in a Delaunay triangulation closest to the points you specify. This code searches for the point closest to (211.32, -48.35) in the triangulation of the seamount data. xi = 211.32; yi = -48.35; p = dsearch(x,y,tri,xi,yi); [x(p), y(p)] ans = 211.3400 -48.3700 • tsearch finds the indices into the delaunay output that specify the enclosing triangles of the points you specify. This example uses the index of the 2-24 Interpolation enclosing triangle for the point (211.32, -48.35) to obtain the coordinates of the vertices of the triangle: xi = 211.32; yi = -48.35; t = tsearch(x,y,tri,xi,yi); r = tri(t,:); A = [x(r) y(r)] A = 211.3000 211.3400 211.2800 -48.3000 -48.3700 -48.3200 Voronoi Diagrams Voronoi diagrams are a closest-point plotting technique related to Delaunay triangulation. For each point in a set of coplanar points, you can draw a polygon that encloses all the intermediate points that are closer to that point than to any other point in the set. Such a polygon is called a Voronoi polygon, and the set of all Voronoi polygons for a given point set is called a Voronoi diagram. The voronoi function can plot the cells of the Voronoi diagram, or return the vertices of the edges of the diagram. This example loads the seamount data, then uses the voronoi function to produce the Voronoi diagram for the longitudinal (x) and latitudinal (y) dimensions. Note that voronoi plots only the bounded cells of the Voronoi diagram: load seamount voronoi(x,y) grid on xlabel('Longitude'), ylabel('Latitude') 2-25 2 Polynomials and Interpolation −48 −48.05 −48.1 Latitude −48.15 −48.2 −48.25 −48.3 −48.35 −48.4 210.9 211 211.1 211.2 211.3 Longitude 211.4 211.5 211.6 Note See the voronoi function for an example that uses the vertices of the edges to plot a Voronoi diagram. Tessellation and Interpolation of Scattered Data in Higher Dimensions Many applications in science, engineering, statistics, and mathematics require structures like convex hulls, Voronoi diagrams, and Delaunay tessellations. Using Qhull [1], MATLAB functions enable you to geometrically analyze data sets in any dimension. 2-26 Interpolation Functions for Multidimensional Geometrical Analysis Function Description convhulln N-dimensional convex hull. delaunayn N-dimensional Delaunay tessellation. dsearchn N-dimensional nearest point search. griddatan N-dimensional data gridding and hypersurface fitting. tsearchn N-dimensional closest simplex search. voronoin N-dimensional Voronoi diagrams. This section demonstrates these geometric analysis techniques: • Convex hulls • Delaunay triangulations • Voronoi diagrams • Interpolation of scattered multidimensional data Convex Hulls The convex hull of a data set in n-dimensional space is defined as the smallest convex region that contains the data set. Computing a Convex Hull. The convhulln function returns the indices of the points in a data set that comprise the facets of the convex hull for the set. For example, suppose X is an 8-by-3 matrix that consists of the 8 vertices of a cube. The convex hull of X then consists of 12 facets: d = [-1 1]; [x,y,z] = meshgrid(d,d,d); X = [x(:),y(:),z(:)]; C = convhulln(X) % 8 corner points of a cube C = 4 3 2 4 1 1 2-27 2 Polynomials and Interpolation 7 5 7 4 2 6 4 6 6 7 3 7 4 7 6 5 6 4 7 6 1 1 3 8 1 1 2 8 5 8 Because the data is three-dimensional, the facets that make up the convex hull are triangles. The 12 rows of C represent 12 triangles. The elements of C are indices of points in X. For example, the first row, 3 1 5, means that the first triangle has X(3,:), X(1,:), and X(5,:) as its vertices. For three-dimensional convex hulls, you can use trisurf to plot the output. However, using patch to plot the output gives you more control over the color of the facets. Note that you cannot plot convhulln output for n > 3. This code plots the convex hull by drawing the triangles as three-dimensional patches: figure, hold on d = [1 2 3 1]; % Index into C column. for i = 1:size(C,1) % Draw each triangle. j= C(i,d); % Get the ith C to make a patch. h(i)=patch(X(j,1),X(j,2),X(j,3),i,'FaceAlpha',0.9); end % 'FaceAlpha' is used to make it transparent. hold off view(3), axis equal, axis off camorbit(90,-5); % To view it from another angle title('Convex hull of a cube') 2-28 Interpolation Delaunay Tessellations A Delaunay tessellation is a set of simplices with the property that, for each simplex, the unique sphere circumscribed about the simplex contains no data points. In two-dimensional space, a simplex is a triangle. In three-dimensional space, a simplex is a tetrahedron. Computing a Delaunay Tessellation. The delaunayn function returns the indices of the points in a data set that comprise the simplices of an n-dimensional Delaunay tessellation of the data set. This example uses the same X as in the convex hull example, i.e. the 8 corner points of a cube, with the addition of a center point: d = [-1 1]; [x,y,z] = meshgrid(d,d,d); X = [x(:),y(:),z(:)]; % 8 corner points of a cube X(9,:) = [0 0 0]; % Add center to the vertex list. T = delaunayn(X) % Generate Delaunay tessellation. 2-29 2 Polynomials and Interpolation T = 4 4 7 7 7 7 6 6 6 6 6 6 9 9 9 9 4 4 9 9 4 4 7 7 3 2 3 5 9 8 2 5 9 8 9 8 1 1 1 1 3 9 1 1 2 9 5 9 The 12 rows of T represent the 12 simplices, in this case irregular tetrahedrons, that partition the cube. Each row represents one tetrahedron, and the row elements are indices of points in X. For three-dimensional tessellations, you can use tetramesh to plot the output. However, using patch to plot the output gives you more control over the color of the facets. Note that you cannot plot delaunayn output for n > 3. This code plots the tessellation T by drawing the tetrahedrons using three-dimensional patches: figure, hold on d = [1 1 1 2; 2 2 3 3; 3 4 4 4]; % Index into T for i = 1:size(T,1) % Draw each tetrahedron. y = T(i,d); % Get the ith T to make a patch. x1 = reshape(X(y,1),3,4); x2 = reshape(X(y,2),3,4); x3 = reshape(X(y,3),3,4); h(i)=patch(x1,x2,x3,(1:4)*i,'FaceAlpha',0.9); end hold off view(3), axis equal axis off camorbit(65,120) % To view it from another angle title('Delaunay tessellation of a cube with a center point') 2-30 Interpolation You can use cameramenu to rotate the figure in any direction. Voronoi Diagrams Given m data points in n-dimensional space, a Voronoi diagram is the partition of n-dimensional space into m polyhedral regions, one region for each data point. Such a region is called a Voronoi cell. A Voronoi cell satisfies the condition that it contains all points that are closer to its data point than any other data point in the set. Computing a Voronoi Diagram. The voronoin function returns two outputs: • V is an m-by-n matrix of m points in n-space. Each row of V represents a Voronoi vertex. • C is a cell array of vectors. Each vector in the cell array C represents a Voronoi cell. The vector contains indices of the points in V that are the vertices of the Voronoi cell. Each Voronoi cell may have a different number of points. 2-31 2 Polynomials and Interpolation Because a Voronoi cell can be unbounded, the first row of V is a point at infinity. Then any unbounded Voronoi cell in C includes the point at infinity, i.e., the first point in V. This example uses the same X as in the Delaunay example, i.e., the 8 corner points of a cube and its center. Random noise is added to make the cube less regular. The resulting Voronoi diagram has 9 Voronoi cells: d = [-1 1]; [x,y,z] = meshgrid(d,d,d); X = [x(:),y(:),z(:)]; % 8 corner points of a cube X(9,:) = [0 0 0]; % Add center to the vertex list. X = X+0.01*rand(size(X)); % Make the cube less regular. [V,C] = voronoin(X); V = Inf 0.0055 0.0037 0.0052 0.0030 0.0072 -1.7912 -1.4886 -1.4886 0.0101 1.5115 1.5115 0.0104 0.0026 Inf 1.5054 0.0101 0.0087 1.5054 0.0072 0.0000 0.0011 0.0002 0.0044 0.0074 0.0081 -1.4846 -1.4846 C = [1x8 double] [1x6 double] [1x4 double] [1x6 double] [1x6 double] [1x6 double] [1x6 double] [1x6 double] [1x12 double] 2-32 Inf 0.0004 -1.4990 -1.4990 0.0030 1.4971 0.0044 0.0036 0.0045 1.4971 0.0033 0.0040 -0.0007 0.0071 Interpolation In this example, V is a 13-by-3 matrix, the 13 rows are the coordinates of the 13 Voronoi vertices. The first row of V is a point at infinity. C is a 9-by-1 cell array, where each cell in the array contains an index vector into V corresponding to one of the 9 Voronoi cells. For example, the 9th cell of the Voronoi diagram is C{9} = 2 3 4 5 6 7 8 9 10 11 12 13 If any index in a cell of the cell array is 1, then the corresponding Voronoi cell contains the first point in V, a point at infinity. This means the Voronoi cell is unbounded. To view a bounded Voronoi cell, i.e., one that does not contain a point at infinity, use the convhulln function to compute the vertices of the facets that make up the Voronoi cell. Then use patch and other plot functions to generate the figure. For example, this code plots the Voronoi cell defined by the 9th cell in C: X = V(C{9},:); % View 9th Voronoi cell. K = convhulln(X); figure hold on d = [1 2 3 1]; % Index into K for i = 1:size(K,1) j = K(i,d); h(i)=patch(X(j,1),X(j,2),X(j,3),i,'FaceAlpha',0.9); end hold off view(3) axis equal title('One cell of a Voronoi diagram') 2-33 2 Polynomials and Interpolation Interpolating N-Dimensional Data Use the griddatan function to interpolate multidimensional data, particularly scattered data. griddatan uses the delaunayn function to tessellate the data, and then interpolates based on the tessellation. Suppose you want to visualize a function that you have evaluated at a set of n scattered points. In this example, X is an n-by-3 matrix of points, each row containing the (x,y,z) coordinates for one of the points. The vector v contains the n function values at these points. The function for this example is the squared distance from the origin, v = x.^2 + y.^2 + z.^2. Start by generating n = 5000 points at random in three-dimensional space, and computing the value of a function on those points: n = 5000; X = 2*rand(n,3)-1; v = sum(X.^2,2); 2-34 Interpolation The next step is to use interpolation to compute function values over a grid. Use meshgrid to create the grid, and griddatan to do the interpolation: delta = 0.05; d = -1:delta:1; [x0,y0,z0] = meshgrid(d,d,d); X0 = [x0(:), y0(:), z0(:)]; v0 = griddatan(X,v,X0); v0 = reshape(v0, size(x0)); Then use isosurface and related functions to visualize the surface that consists of the (x,y,z) values for which the function takes a constant value. You could pick any value, but the example uses the value 0.6. Since the function is the squared distance from the origin, the surface at a constant value is a sphere: p = patch(isosurface(x0,y0,z0,v0,0.6)); isonormals(x0,y0,z0,v0,p); set(p,'FaceColor','red','EdgeColor','none'); view(3); camlight; lighting phong axis equal title('Interpolated sphere from scattered data') Note A smaller delta produces a smoother sphere, but increases the compute time. 2-35 2 Polynomials and Interpolation 2-36 Selected Bibliography Selected Bibliography [1] National Science and Technology Research Center for Computation and Visualization of Geometric Structures (The Geometry Center), University of Minnesota. 1993. For information about qhull, see http://www.qhull.org. [2] Parker, Robert. L., Loren Shure, & John A. Hildebrand, “The Application of Inverse Theory to Seamount Magnetism.” Reviews of Geophysics. Vol. 25, No. 1, 1987. 2-37 2 Polynomials and Interpolation 2-38 3 Data Analysis and Statistics Column-Oriented Data Sets (p. 3-3) Organizing arrays for data analysis. Basic Data Analysis Functions (p. 3-7) Basic data analysis functions and an example that uses some of the functions. This section also discusses functions for the computation of correlation coefficients and covariance, and for finite difference calculations. Data Preprocessing (p. 3-13) Working with missing values, and outliers or misplaced data points in a data set. Regression and Curve Fitting (p. 3-16) Investigates the use of different regression methods to find functions that describe the relationship among observed variables. Case Study: Curve Fitting (p. 3-21) Uses a case study to look at some of the MATLAB basic data analysis capabilities. This section also provides information about the Basic Fitting interface. Difference Equations and Filtering (p. 3-39) Discusses MATLAB functions for working with difference equations and filters. Fourier Analysis and the Fast Fourier Discusses Fourier analysis in MATLAB. Transform (FFT) (p. 3-42) 3 Data Analysis and Statistics Data Analysis and Statistics Functions The data analysis and statistics functions are located in the MATLAB datafun directory. Use online help to get a complete list of functions. Related Toolboxes A number of related toolboxes provide advanced functionality for specialized data analysis applications. 3-2 Toolbox Data Analysis Application Optimization Nonlinear curve fitting and regression. Signal Processing Signal processing, filtering, and frequency analysis. Spline Curve fitting and regression. Statistics Advanced statistical analysis, nonlinear curve fitting, and regression. System Identification Parametric / ARMA modeling. Wavelet Wavelet analysis. Column-Oriented Data Sets Column-Oriented Data Sets Univariate statistical data is typically stored in individual vectors. The vectors can be either 1-by-n or n-by-1. For multivariate data, a matrix is the natural representation but there are, in principle, two possibilities for orientation. By MATLAB convention, however, the different variables are put into columns, allowing observations to vary down through the rows. Therefore, a data set consisting of twenty four samples of three variables is stored in a matrix of size 24-by-3. Vehicle Traffic Sample Data Set Consider a sample data set comprising vehicle traffic count observations at three locations over a 24-hour period. Vehicle Traffic Sample Data Set Time Location 1 Location 2 Location 3 01h00 11 11 9 02h00 7 13 11 03h00 14 17 20 04h00 11 13 9 05h00 43 51 69 06h00 38 46 76 07h00 61 132 186 08h00 75 135 180 09h00 38 88 115 10h00 28 36 55 11h00 12 12 14 12h00 18 27 30 13h00 18 19 29 3-3 3 Data Analysis and Statistics Vehicle Traffic Sample Data Set (Continued) Time Location 1 Location 2 Location 3 14h00 17 15 18 15h00 19 36 48 16h00 32 47 10 17h00 42 65 92 18h00 57 66 151 19h00 44 55 90 20h00 114 145 257 21h00 35 58 68 22h00 11 12 15 23h00 13 9 15 24h00 10 9 7 Loading and Plotting the Data The raw data is stored in the file, count.dat. 11 7 14 11 43 38 61 75 38 28 12 18 18 17 19 3-4 11 13 17 13 51 46 132 135 88 36 12 27 19 15 36 9 11 20 9 69 76 186 180 115 55 14 30 29 18 48 Column-Oriented Data Sets 32 42 57 44 114 35 11 13 10 47 65 66 55 145 58 12 9 9 10 92 151 90 257 68 15 15 7 Use the load command to import the data: load count.dat This creates the matrix count in the workspace. For this example, there are 24 observations of three variables. This is confirmed by [n,p] = size(count) n = 24 p = 3 Create a time vector, t, of integers from 1 to n: t = 1:n; Now plot the counts versus time and annotate the plot: set(0,'defaultaxeslinestyleorder','-|--|-.') set(0,'defaultaxescolororder',[0 0 0]) plot(t,count), legend('Location 1','Location 2','Location 3',2) xlabel('Time'), ylabel('Vehicle Count'), grid on 3-5 3 Data Analysis and Statistics The plot shows the vehicle counts at three locations over a 24-hour period. 300 Location 1 Location 2 Location 3 250 Vehicle Count 200 150 100 50 0 0 5 10 15 Time 3-6 20 25 Basic Data Analysis Functions Basic Data Analysis Functions This section introduces functions for • Basic column-oriented data analysis • Computation of correlation coefficients and covariance • Calculating finite differences Function Summary A collection of functions provides basic column-oriented data analysis capabilities. These functions are located in the MATLAB datafun directory. This section also gives you some hints about using row and column data, and provides some basic examples. This table lists the functions. Basic Data Analysis Function Summary Function Description cumprod Cumulative product of elements. cumsum Cumulative sum of elements. cumtrapz Cumulative trapezoidal numerical integration. diff Difference function and approximate derivative. max Largest component. mean Average or mean value. median Median value. min Smallest component. prod Product of elements. sort Sort array elements in ascending or descending order. sortrows Sort rows in ascending order. std Standard deviation. 3-7 3 Data Analysis and Statistics Basic Data Analysis Function Summary (Continued) Function Description sum Sum of elements. trapz Trapezoidal numerical integration. To use the Data Statistics Tool to calculate the maximum, minimum, mean, median, range, and standard deviation on plotted data, and create plots of these statistics, see “Using the Data Statistics Tool” in the MATLAB graphics documentation. Working with Row and Column Data For vector input arguments to these functions, it does not matter whether the vectors are oriented in row or column direction. For array arguments, however, the functions operate column by column on the data in the array. This means, for example, that if you apply max to an array, the result is a row vector containing the maximum values over each column. Note You can add more functions to this list using M-files, but when doing so, you must exercise care to handle the row-vector case. If you are writing your own column-oriented M-files, check other M-files; for example, mean.m and diff.m. Basic Examples Continuing with the vehicle traffic count example, the statements mx = max(count) mu = mean(count) sigma = std(count) result in mx = 114 145 257 32.0000 46.5417 65.5833 mu = 3-8 Basic Data Analysis Functions sigma = 25.3703 41.4057 68.0281 To locate the index at which the minimum or maximum occurs, a second output parameter can be specified. For example, [mx,indx] = min(count) mx = 7 9 7 indx = 2 23 24 shows that the lowest vehicle count is recorded at 02h00 for the first observation point (column one) and at 23h00 and 24h00 for the other observation points. You can subtract the mean from each column of the data using an outer product involving a vector of n ones: [n,p] = size(count) e = ones(n,1) x = count - e*mu Rearranging the data may help you evaluate a vector function over an entire data set. For example, to find the smallest value in the entire data set, use min(count(:)) which produces ans = 7 The syntax count(:) rearranges the 24-by-3 matrix into a 72-by-1 column vector. 3-9 3 Data Analysis and Statistics Covariance and Correlation Coefficients The MATLAB statistical capabilities include two functions for the computation of correlation coefficients and covariance. Covariance and Correlation Coefficient Function Summary Function Description cov Variance of vector – measure of spread or dispersion of sample variable. Covariance of matrix – measure of strength of linear relationships between variables. Correlation coefficient – normalized measure of linear relationship strength between variables. corrcoef Covariance cov returns the variance for a vector of data. The variance of the data in the first column of count is cov(count(:,1)) ans = 643.6522 For an array of data, cov calculates the covariance matrix. The variance values for the array columns are arranged along the diagonal of the covariance matrix. The remaining entries reflect the covariance between the columns of the original array. For an m-by-n matrix, the covariance matrix has size n-by-n. For example, the covariance matrix for count, cov(count), is arranged as 2 2 2 2 2 2 2 2 2 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 2 2 σ ij = σ ji 3-10 Basic Data Analysis Functions Correlation Coefficients corrcoef produces a matrix of correlation coefficients for an array of data where each row is an observation and each column is a variable. The correlation coefficient is a normalized measure of the strength of the linear relationship between two variables. Uncorrelated data results in a correlation coefficient of 0; equivalent data sets have a correlation coefficient of 1. For an m-by-n matrix, the correlation coefficient matrix has size n-by-n. The arrangement of the elements in the correlation coefficient matrix corresponds to the location of the elements in the covariance matrix described above. For the traffic count example corrcoef(count) results in ans = 1.0000 0.9331 0.9599 0.9331 1.0000 0.9553 0.9599 0.9553 1.0000 Clearly there is a strong linear correlation between the three traffic counts observed at the three locations, as the results are close to 1. Finite Differences MATLAB provides three functions for finite difference calculations. Function Description diff Difference between successive elements of a vector. Numerical partial derivatives of a vector. gradient Numerical partial derivatives a matrix. del2 Discrete Laplacian of a matrix. 3-11 3 Data Analysis and Statistics The diff function computes the difference between successive elements in a numeric vector. That is, diff(X) is [X(2)-X(1) X(3)-X(2)... X(n)-X(n-1)]. So, for a vector A, A = [9 -2 3 0 1 5 4]; diff(A) ans = -11 5 -3 1 4 -1 Besides computing the first difference, diff is useful for determining certain characteristics of vectors. For example, you can use diff to determine if a vector is monotonic (elements are always either increasing or decreasing), or if a vector has equally spaced elements. This table describes a few different ways to use diff with a vector x. 3-12 Test Description diff(x)==0 Tests for repeated elements. all(diff(x)>0) Tests for monotonicity. all(diff(diff(x))==0) Tests for equally spaced vector elements. Data Preprocessing Data Preprocessing This section tells you how to work with • Missing values • Outliers and misplaced data points Missing Values The special value, NaN, stands for Not-a-Number in MATLAB. IEEE floating-point arithmetic convention specifies NaN as the result of undefined expressions such as 0/0. The correct handling of missing data is a difficult problem and often varies in different situations. For data analysis purposes, it is often convenient to use NaNs to represent missing values or data that are not available. MATLAB treats NaNs in a uniform and rigorous way. They propagate naturally through to the final result in any calculation. Any mathematical calculation involving NaNs produces NaNs in the results. For example, consider a matrix containing the 3-by-3 magic square with its center element set to NaN: a = magic(3); a(2,2) = NaN a = 8 3 4 1 NaN 9 6 7 2 Compute a sum for each column in the matrix: sum(a) ans = 15 NaN 15 Any mathematical calculation involving NaNs propagates NaNs through to the final result as appropriate. 3-13 3 Data Analysis and Statistics You should remove NaNs from the data before performing statistical computations. Here are some ways to use isnan to remove NaNs from data. Code Description i = find(~isnan(x)); x = x(i) Find indices of elements in vector that are not NaNs, then keep only the non-NaN elements. x = x(find(~isnan(x))) Remove NaNs from vector. x = x(~isnan(x)); Remove NaNs from vector (faster). x(isnan(x)) = []; Remove NaNs from vector. X(any(isnan(X)'),:) = []; Remove any rows of matrix X containing NaNs. Note You must use the special function isnan to find NaNs because, by IEEE arithmetic convention, the logical comparison, NaN == NaN always produces 0. You cannot use x(x==NaN) = [] to remove NaNs from your data. If you frequently need to remove NaNs, write a short M-file function: function X = excise(X) X(any(isnan(X)'),:) = []; Now, typing X = excise(X); accomplishes the same thing. 3-14 Data Preprocessing Removing Outliers You can remove outliers or misplaced data points from a data set in much the same manner as NaNs. For the vehicle traffic count data, the mean and standard deviations of each column of the data are mu = mean(count) sigma = std(count) mu = 32.0000 46.5417 65.5833 sigma = 25.3703 41.4057 68.0281 The number of rows with outliers greater than three standard deviations is obtained with [n,p] = size(count) outliers = abs(count - mu(ones(n, 1),:)) > 3*sigma(ones(n, 1),:); nout = sum(outliers) nout = 1 0 0 There is one outlier in the first column. Remove this entire observation with count(any(outliers'),:) = []; 3-15 3 Data Analysis and Statistics Regression and Curve Fitting It is often useful to find functions that describe the relationship between some variables you have observed. Identification of the coefficients of the function often leads to the formulation of an overdetermined system of simultaneous linear equations. You can find these coefficients efficiently by using the MATLAB backslash operator. Suppose you measure a quantity y at several values of time t: t = [0 .3 .8 1.1 1.6 2.3]'; y = [0.5 0.82 1.14 1.25 1.35 1.40]'; plot(t,y,'o'), grid on 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0 0.5 1 1.5 2 2.5 The following sections look at three ways of modeling the data: • Polynomial regression • Linear-in-the-parameters regression • Multiple regression 3-16 Regression and Curve Fitting Polynomial Regression Based on the plot, it is possible that the data can be modeled by a polynomial function y = a0 + a1 t + a2 t 2 The unknown coefficients a0, a1, and a2 can be computed by doing a least squares fit, which minimizes the sum of the squares of the deviations of the data from the model. There are six equations in three unknowns, 2 1 t1 t1 y1 y2 y3 y4 y5 2 1 t2 t2 a0 2 1 t3 t3 = × a1 2 1 t4 t4 a2 y6 2 1 t5 t5 2 1 t6 t6 represented by the 6-by-3 matrix X = [ones(size(t)) t t.^2] X = 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0 0.3000 0.8000 1.1000 1.6000 2.3000 0 0.0900 0.6400 1.2100 2.5600 5.2900 The solution is found with the backslash operator: a = X\y a = 0.5318 0.9191 - 0.2387 3-17 3 Data Analysis and Statistics The second-order polynomial model of the data is therefore y = 0.5318 + 0.919 ( 1 )t – 0.2387t 2 Now evaluate the model at regularly spaced points and overlay the original data in a plot: T = (0:0.1:2.5)'; Y = [ones(size(T)) T T.^2]*a; plot(T,Y,'-',t,y,'o'), grid on 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0 0.5 1 1.5 2 2.5 Clearly this fit does not perfectly approximate the data. You could either increase the order of the polynomial fit, or explore some other functional form to get a better approximation. Linear-in-the-Parameters Regression Instead of a polynomial function, you could try using a function that is linear-in-the-parameters. In this case, consider the exponential function y = a0 + a1 e –t + a 2 te –t The unknown coefficients a 0 , a 1 , and a 2 , are computed by performing a least squares fit. Construct and solve the set of simultaneous equations by forming 3-18 Regression and Curve Fitting the regression matrix, X, and solving for the coefficients using the backslash operator: X = [ones(size(t)) a = X\y exp(-t) t.*exp(-t)]; a = 1.3974 - 0.8988 0.4097 The fitted model of the data is, therefore, y = 1.3974 – 0.8988 e –t + 0.4097 te –t Now evaluate the model at regularly spaced points and overlay the original data in a plot: T = (0:0.1:2.5)'; Y = [ones(size(T)) exp(-T) T.*exp(-T)]*a; plot(T,Y,'-',t,y,'o'), grid on 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0 0.5 1 1.5 2 2.5 This is a much better fit than the second-order polynomial function. 3-19 3 Data Analysis and Statistics Multiple Regression If y is a function of more than one independent variable, the matrix equations that express the relationships among the variables can be expanded to accommodate the additional data. Suppose you measure a quantity y for several values of parameters x 1 and x 2 . The observations are entered as x1 = [.2 .5 .6 .8 1.0 1.1]'; x2 = [.1 .3 .4 .9 1.1 1.4]'; y = [.17 .26 .28 .23 .27 .24]'; A multivariate model of the data is y = a0 + a1 x1 + a2 x2 Multiple regression solves for unknown coefficients a 0 , a 1 , and a 2 , by performing a least squares fit. Construct and solve the set of simultaneous equations by forming the regression matrix, X, and solving for the coefficients using the backslash operator: X = [ones(size(x1)) a = X\y x1 x2]; a = 0.1018 0.4844 -0.2847 The least squares fit model of the data is y = 0.1018 + 0.4844 x 1 – 0.2847 x 2 To validate the model, find the maximum of the absolute value of the deviation of the data from the model: Y = X*a; MaxErr = max(abs(Y - y)) MaxErr = 0.0038 This is sufficiently small to be confident the model reasonably fits the data. 3-20 Case Study: Curve Fitting Case Study: Curve Fitting This section provides an overview of some of the MATLAB basic data analysis capabilities in the form of a case study. The examples that follow work with a collection of census data, using MATLAB functions to experiment with fitting curves to the data: • Polynomial fit • Analyzing residuals • Exponential fit • Error bounds This section also tells you how to use the Basic Fitting interface to perform curve fitting tasks. Loading the Data The file census.mat contains U.S. population data for the years 1790 through 1990. Load it into MATLAB: load census Your workspace now contains two new variables, cdate and pop: • cdate is a column vector containing the years from 1790 to 1990 in increments of 10. • pop is a column vector with the U.S. population figures that correspond to the years in cdate. Polynomial Fit A first try in fitting the census data might be a simple polynomial fit. Two MATLAB functions help with this process. Curve Fitting Function Summary Function Description polyfit Polynomial curve fit. polyval Evaluation of polynomial fit. 3-21 3 Data Analysis and Statistics The MATLAB polyfit function generates a “best fit” polynomial (in the least squares sense) of a specified order for a given set of data. For a polynomial fit of the fourth-order, p = polyfit(cdate,pop,4) Warning: Polynomial is badly conditioned. Remove repeated data points or try centering and scaling as described in HELP POLYFIT. p = 1.0e+05 ∗ 0.0000 -0.0000 0.0000 -0.0126 6.0020 The warning arises because the polyfit function uses the cdate values as the basis for a matrix with very large values (it creates a Vandermonde matrix in its calculations – see the polyfit M-file for details). The spread of the cdate values results in scaling problems. One way to deal with this is to normalize the cdate data. Preprocessing: Normalizing the Data Normalization is a process of scaling the numbers in a data set to improve the accuracy of the subsequent numeric computations. A way to normalize cdate is to center it at zero mean and scale it to unit standard deviation: sdate = (cdate - mean(cdate))./std(cdate) Now try the fourth-degree polynomial model using the normalized data: p = polyfit(sdate,pop,4) p = 0.7047 0.9210 23.4706 73.8598 62.2285 Evaluate the fitted polynomial at the normalized year values, and plot the fit against the observed data points: pop4 = polyval(p,sdate); plot(cdate,pop4,'-',cdate,pop,'+'), grid on 3-22 Case Study: Curve Fitting 300 250 200 150 100 50 0 1750 1800 1850 1900 1950 2000 Another way to normalize data is to use some knowledge of the solution and units. For example, with this data set, choosing 1790 to be year zero would also have produced satisfactory results. Analyzing Residuals A measure of the “goodness” of fit is the residual, the difference between the observed and predicted data. Compare the residuals for the various fits, using normalized cdate values. It’s evident from studying the fit plots and residuals that it should be possible to do better than a simple polynomial fit with this data set. 3-23 3 Data Analysis and Statistics Comparison Plots of Fit and Residual Fit Residuals p1 = polyfit(sdate,pop,1); pop1 = polyval(p1,sdate); plot(cdate,pop1,'b-',cdate,pop,'g+') res1 = pop - pop1; figure, plot(cdate,res1,'g+') 250 200 50 Linear fit appears unsatisfactory – note negative population values at lower end of scale. Residuals of linear fit show strongly patterned behavior. 40 30 150 20 100 10 0 50 −10 0 −20 −50 1750 1800 1850 1900 1950 2000 p = polyfit(sdate,pop,2); pop2 = polyval(p,sdate); plot(cdate,pop2,'b-',cdate,pop,'g+') 250 200 −30 1750 1800 Quadratic polynomial provides better fit to data points. 1950 2000 res2 = pop - pop2; figure, plot(cdate,res2,'g+') 4 2 0 −2 100 −4 Residuals still appear strongly patterned. 50 −6 3-24 1900 6 150 0 1750 1850 1800 1850 1900 1950 2000 −8 1750 1800 1850 1900 1950 2000 Case Study: Curve Fitting Comparison Plots of Fit and Residual (Continued) Fit Residuals p = polyfit(sdate,pop,4); pop4 = polyval(p,sdate); plot(cdate,pop4,'b-',cdate,pop,'g+') res4 = pop - pop4; figure, plot(cdate,res4,'g+') 300 6 250 200 Fourth-degree model provides little improvement – note that curve still begins to turn upward at lower end of plot. 4 2 0 150 −2 100 −4 50 0 1750 Residuals still appear strongly patterned. −6 1800 1850 1900 1950 2000 −8 1750 1800 1850 1900 1950 2000 Exponential Fit By looking at the population data plots on the previous pages, the population data curve is somewhat exponential in appearance. To take advantage of this, try to fit the logarithm of the population values, again working with normalized year values: logp1 = polyfit(sdate,log10(pop),1); logpred1 = 10.^polyval(logp1,sdate); semilogy(cdate,logpred1,'-',cdate,pop,'+'); grid on 3-25 3 Data Analysis and Statistics 3 10 2 10 1 10 0 10 1750 1800 1850 1900 1950 2000 Now try the logarithm analysis with a second-order model: logp2 = polyfit(sdate,log10(pop),2); logpred2 = 10.^polyval(logp2,sdate); semilogy(cdate,logpred2,'-',cdate,pop,'+'); grid on 3 10 2 10 1 10 0 10 1750 3-26 1800 1850 1900 1950 2000 Case Study: Curve Fitting This is a more accurate model. The upper end of the plot appears to taper off, while the polynomial fits in the previous section continue, concave up, to infinity. Compare the residuals for the second-order logarithmic model. Residuals in Log Population Scale Residuals in Population Scale logres2 = log10(pop)polyval(logp2,sdate); plot(cdate,logres2,'+') r = pop - 10.^(polyval(logp2,sdate)); plot(cdate,r,'+') 0.03 10 0.02 5 0.01 0 0 −0.01 −5 −0.02 −10 −0.03 −0.04 1750 1800 1850 1900 1950 2000 −15 1750 1800 1850 1900 1950 2000 The residuals are more random than for the simple polynomial fit. As might be expected, the residuals tend to get larger in magnitude as the population increases. But overall, the logarithmic model provides a more accurate fit to the population data. Error Bounds Error bounds are useful for determining if your data is reasonably modeled by the fit. You can obtain the error bounds by passing an optional second output parameter from polyfit as an input parameter to polyval. This example uses the census demo data and normalizes the data by centering it at zero mean and scaling it to unit standard deviation. The example then 3-27 3 Data Analysis and Statistics uses polyfit and polyval to produce error bounds for a second-order polynomial model. Year values are normalized. This code uses an interval of ±2∆, corresponding to a 95% confidence interval: load census sdate = (cdate - mean(cdate))./std(cdate) [p2,S2] = polyfit(sdate,pop,2); [pop2,del2] = polyval(p2,sdate,S2); plot(cdate,pop,'+',cdate,pop2,'g-',cdate,pop2+2*del2,'r:',... cdate,pop2-2∗del2,'r:'), grid on 300 250 200 150 100 50 0 −50 1750 1800 1850 1900 1950 2000 The Basic Fitting Interface MATLAB supports curve fitting through the Basic Fitting interface. Using this interface, you can quickly perform many curve fitting tasks within the same easy-to-use environment. The interface is designed so that you can • Fit data using a spline interpolant, a shape-preserving interpolant, or a polynomial up to degree 10 • Plot multiple fits simultaneously for a given data set 3-28 Case Study: Curve Fitting • Plot the fit residuals • Examine the numerical results of a fit • Evaluate (interpolate or extrapolate) a fit • Annotate the plot with the numerical fit results and the norm of residuals • Save the fit and evaluated results to the MATLAB workspace Depending on your specific curve fitting application, you can use the Basic Fitting interface, the command line functionality, or both. You can use the Basic Fitting interface only with 2-D data. However, if you plot multiple data sets as a subplot, and at least one data set is 2-D, then the interface is enabled. Improving Fitting Efficiency MATLAB does not sort your data before fitting it with the Basic Fitting interface, although it did so in prior releases. This change in behavior was made to improve handling of residual plots. However, as sorted data is faster to fit and plot, you can improve performance of the Basic Fitting interface with large data sets by presorting your data so that the x values are in ascending order. If your original data consists of the vectors x and y, you can create sorted vectors x_sorted and y_sorted as follows: [x_sorted, i] = sort(x); y_sorted = y(i); Overview of the Basic Fitting Interface The full Basic Fitting interface is shown below. To reproduce this state, follow these three steps: 1 Plot some data. 2 Select Basic Fitting from the Tools menu. 3 Click the button twice. 3-29 3 Data Analysis and Statistics Select data – This parameter list is populated with the names of all the data sets you display in the figure window associated with the Basic Fitting interface. Use this list to select the current data set — the data set that you want to fit. You can fit only one data set at a time. However, you can perform multiple fits for the current data set. Use the Plot Editor to change the name of a data set. Center and scale X data – If checked, the data is centered at zero mean and scaled to unit standard deviation. You may need to center and scale your data to improve the accuracy of the subsequent numerical computations. MATLAB displays a warning is displayed if a fit produces results that might be inaccurate. 3-30 Case Study: Curve Fitting Plot fits – This panel enables you to visually explore one or more fits to the current data set: • Check to display fits on figure – Select the fits you want to display for the current data set. There are two types of fits to choose from: interpolants and polynomials. The spline interpolant uses the spline function, while the shape-preserving interpolant uses the pchip function. Refer to the pchip online help for a comparison of these two functions. The polynomial fits use the polyfit function. You can choose as many fits for a given data set as you want. If your data set has N points, then you should use polynomials with, at most, N coefficients. If your fit uses polynomials with more than N coefficients, the interface automatically sets a sufficient number of coefficients to 0 during the calculation so that the system is not underdetermined. • Show equations – If checked, the fit equation is displayed on the plot. - Significant digits – Select the significant digits associated with the equation display. • Plot residuals – If checked, the fit residuals are displayed. The fit residuals are defined as the difference between the ordinate data point and the resulting fit for each abscissa data point. You can display the residuals as a bar plot, as a scatter plot, or as a line plot in the same figure window as the data or in a separate figure window. If you use subplots to plot multiple data sets, then residuals can be plotted only in a separate figure window. - Show norm of residuals – If checked, the norm of residuals are displayed. The norm of residuals is a measure of the goodness of fit, where a smaller value indicates a better fit than a larger value. It is calculated using the norm function, norm(V,2), where V is the vector of residuals. Numerical results – This panel allows you to explore the numerical results of a single fit to the current data set without plotting the fit: • Fit – Select the equation to fit to the current data set. The fit results are displayed in the list box below the menu. Note that selecting an equation in this menu does not affect the state of the Plot fits panel. Therefore, if you want to display the fit in the data plot, you may need to select the associated check box in Plot fits. 3-31 3 Data Analysis and Statistics • Coefficients and norm of residuals – Display the numerical results for the equation selected in Fit. Note that when you first open the Numerical Results panel, the results of the last fit you selected in Plot fits are displayed. • Save to workspace – Launch a dialog box that allows you to save the fit results to workspace variables. • Find Y = f(X) – Interpolate or extrapolate the current fit. - Enter value(s) – Enter a MATLAB expression to evaluate for the current fit. The expression is evaluated after you press the Evaluate button, and the results are displayed in the associated table. The current fit is displayed in the Fit menu. - Save to workspace – Launch a dialog box that allows you to save the evaluated results to workspace variables. - Plot results – If checked, the evaluated results are displayed on the data plot. Example: Using the Basic Fitting Interface This example illustrates the features of the Basic Fitting interface by fitting a cubic polynomial to the census data. You may want to repeat this example using different equations and compare results. To launch the interface, 1 Plot some data: plot(cdate,pop,'ro') 3-32 Case Study: Curve Fitting 2 Select Basic Fitting from the Tools menu in the figure. Configure the Basic Fitting interface to • Fit a cubic polynomial to the data. • Display the equation in the data plot. • Plot the fit residuals as a bar plot, and display the residuals as a subplot of the data figure window. • Display the norm of the residuals. 3-33 3 Data Analysis and Statistics This configuration is shown below. Current data set Fit a cubic polynomial to the data Show the equation Plot the residuals as a bar plot in the data figure window Show the norm of the residuals The Plot fits panel enables you to visually explore multiple fits to the current data set. For comparison, try fitting additional equations to the census data by selecting the appropriate check boxes. If an equation produces results that might be numerically inaccurate, MATLAB displays a warning. In this case, you should select the Center and scale X data check box to improve the numerical accuracy. 3-34 Case Study: Curve Fitting The resulting fit and the residuals are shown in the following plot. Click and drag legend if it covers part of the plot The plot legend indicates the name of the data set and the equation. If the legend covers part of the plot, you can click and drag it to another location. The legend is automatically updated as you add or remove data sets or fits. Additionally, fits are displayed using a default set of line styles and colors. You can change any of the default plot settings using the Plot Editor. However, any changes you make are undone if you subsequently perform another fit. To retain changes, you should wait until after you have finished fitting your data. Note If you change the name of a data set in the legend, then the name is automatically changed in the Select data menu. 3-35 3 Data Analysis and Statistics By selecting the of the residuals. button, you can examine the fit coefficients and the norm The Fit menu enables you to explore numerical fit results for the current data set without plotting the fit. For comparison, you can display the numerical results for other fits by selecting the desired equation. Note that if you want to display a fit in the data plot, you have to select the associated check box in Plot fits. You can save the fit results to the MATLAB workspace by selecting the Save to workspace button. 3-36 Case Study: Curve Fitting The fit structure is fit1 fit1 = type: 'polynomial degree 3' coeff: [3.8555e-006 -0.0153 17.7815 -4.8519e+003] You may want to use this structure for subsequent display or analysis. For example, you can use the saved coefficients and the polyval function to evaluate the cubic polynomial at the command line. By selecting the button again, you can specify a vector of x-values at which to evaluate the current fit. Enter the vector in the field next to the Evaluate button, and then click Evaluate. For example, if you enter the vector 2000:10:2050, the population for the years 2000 to 2050 is evaluated in increments of 10. The x-values and the corresponding values for f(x), evaluated from the fit, are displayed in the pane below Evaluate, as shown in the following figure. 3-37 3 Data Analysis and Statistics Select the Plot evaluated results check box to display the evaluated points along with the current data set in the data plot, as shown in the following figure. You can save the evaluated data to the MATLAB workspace by selecting the Save to workspace button. 3-38 Difference Equations and Filtering Difference Equations and Filtering MATLAB has functions for working with difference equations and filters. These functions operate primarily on vectors. Vectors are used to hold sampled-data signals, or sequences, for signal processing and data analysis. For multi-input systems, each row of a matrix corresponds to a sample point with each input appearing as columns of the matrix. The function y = filter(b,a,x) processes the data in vector x with the filter described by vectors a and b, creating filtered data y. The filter command can be thought of as an efficient implementation of the difference equation. The filter structure is the general tapped delay-line filter described by the difference equation below, where n is the index of the current sample, na is the order of the polynomial described by vector a and nb is the order of the polynomial described by vector b. The output y(n), is a linear combination of current and previous inputs, x(n) x(n-1) ..., and previous outputs, y(n-1) y(n-2) ... a ( 1 )y ( n ) = b ( 1 )x ( n ) + b ( 2 )x ( n – 1 ) + … + b ( nb )x ( n – nb + 1 ) – a ( 2 )y ( n – 1 ) – … – a ( na )y ( n – na + 1 ) Suppose, for example, you want to smooth the traffic count data with a moving average filter to see the average traffic flow over a 4-hour window. This process is represented by the difference equation 1 1 1 1 y ( n ) = --- x ( n ) + --- x ( n – 1 ) + --- x ( n – 2 ) + --- x ( n – 3 ) 4 4 4 4 The corresponding vectors are a = 1; b = [1/4 1/4 1/4 1/4]; 3-39 3 Data Analysis and Statistics Note Enter the format command, format rat, to display and enter data using the rational format. Executing the command load count.dat creates the matrix count in the workspace. For this example, extract the first column of traffic counts and assign it to the vector x: x = count(:,1); The 4-hour moving-average of the data is efficiently calculated with y = filter(b,a,x); Compare the original data and the smoothed data with an overlaid plot of the two curves: t = 1:length(x); plot(t,x,'-.',t,y,'-'), grid on legend('Original Data','Smoothed Data',2) 3-40 Difference Equations and Filtering 120 Original Data Smoothed Data 100 80 60 40 20 0 0 5 10 15 20 25 The filtered data represented by the solid line is the 4-hour moving average of the observed traffic count data represented by the dashed line. For practical filtering applications, the Signal Processing Toolbox includes numerous functions for designing and analyzing filters. 3-41 3 Data Analysis and Statistics Fourier Analysis and the Fast Fourier Transform (FFT) Fourier analysis is extremely useful for data analysis, as it breaks down a signal into constituent sinusoids of different frequencies. For sampled vector data, Fourier analysis is performed using the discrete Fourier transform (DFT). The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT of a sequence; it is not a separate transform. It is particularly useful in areas such as signal and image processing, where its uses range from filtering, convolution, and frequency analysis to power spectrum estimation. This section • Summarizes the Fourier transform functions • Introduces Fourier transform analysis with an example about sunspot activity • Calculates magnitude and phase of transformed data • Discusses the dependence of execution time on length of the transform Function Summary MATLAB provides a collection of functions for computing and working with Fourier transforms. FFT Function Summary 3-42 Function Description fft Discrete Fourier transform. fft2 Two-dimensional discrete Fourier transform. fftn N-dimensional discrete Fourier transform. ifft Inverse discrete Fourier transform. ifft2 Two-dimensional inverse discrete Fourier transform. ifftn N-dimensional inverse discrete Fourier transform. abs Magnitude. Fourier Analysis and the Fast Fourier Transform (FFT) FFT Function Summary (Continued) Function Description angle Phase angle. unwrap Unwrap phase angle in radians. fftshift Move zeroth lag to center of spectrum. cplxpair Sort numbers into complex conjugate pairs. nextpow2 Next higher power of two. Introduction For length N input sequence x, the DFT is a length N vector, X. fft and ifft implement the relationships N X(k)= ∑ x ( n )e n–1 – j2π ( k – 1 ) ⎛ -------------⎞ ⎝ N ⎠ 1≤k≤N n=1 N 1 x ( n ) = ---N ∑ X ( k )e n–1 j2π ( k – 1 ) ⎛ -------------⎞ ⎝ N ⎠ 1≤n≤N k=1 Note Since the first element of a MATLAB vector has an index 1, the summations in the equations above are from 1 to N. These produce identical results as traditional Fourier equations with summations from 0 to N-1. If x(n) is real, you can rewrite the above equation in terms of a summation of sine and cosine functions with real coefficients: N 1 x ( n ) = ---N 2π ( k – 1 ) ( n – 1 ) 2π ( k – 1 ) ( n – 1 ) -⎞ + b ( k ) sin ⎛ -------------------------------------------⎞ ∑ a ( k ) cos ⎛⎝ -----------------------------------------⎠ ⎝ ⎠ N N k=1 3-43 3 Data Analysis and Statistics where a ( k ) = real ( X ( k ) ), b ( k ) = – imag ( X ( k ) ), 1 ≤ n ≤ N Finding an FFT The FFT of a column vector x x = [4 3 7 -9 1 0 0 0]' ; is found with y = fft(x) which results in y = 6.0000 11.4853 -2.0000 -5.4853 18.0000 -5.4853 -2.0000 11.4853 - 2.7574i -12.0000i +11.2426i -11.2426i +12.0000i + 2.7574i Notice that although the sequence x is real, y is complex. The first component of the transformed data is the constant contribution and the fifth element corresponds to the Nyquist frequency. The last three values of y correspond to negative frequencies and, for the real sequence x, they are complex conjugates of three components in the first half of y. Example: Using FFT to Calculate Sunspot Periodicity Suppose, you want to analyze the variations in sunspot activity over the last 300 years. You are probably aware that sunspot activity is cyclical, reaching a maximum about every 11 years. This example confirms that. Astronomers have tabulated a quantity called the Wolfer number for almost 300 years. This quantity measures both number and size of sunspots. Load and plot the sunspot data: load sunspot.dat year = sunspot(:,1); 3-44 Fourier Analysis and the Fast Fourier Transform (FFT) wolfer = sunspot(:,2); plot(year,wolfer) title('Sunspot Data') Sunspot Data 200 180 160 140 120 100 80 60 40 20 0 1700 1750 1800 1850 1900 1950 2000 Now take the FFT of the sunspot data: Y = fft(wolfer); The result of this transform is the complex vector, Y. The magnitude of Y squared is called the power and a plot of power versus frequency is a “periodogram.” Remove the first component of Y, which is simply the sum of the data, and plot the results: N = length(Y); Y(1) = []; power = abs(Y(1:N/2)).^2; nyquist = 1/2; freq = (1:N/2)/(N/2)*nyquist; plot(freq,power), grid on xlabel('cycles/year') title('Periodogram') 3-45 3 Data Analysis and Statistics Periodogram 7 2 x 10 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 cycles/year 0.35 0.4 0.45 0.5 The scale in cycles/year is somewhat inconvenient. You can plot in years/cycle and estimate what one cycle is. For convenience, plot the power versus period (where period = 1./freq) from 0 to 40 years/cycle: period = 1./freq; plot(period,power), axis([0 40 0 2e7]), grid on ylabel('Power') xlabel('Period(Years/Cycle)') 3-46 Fourier Analysis and the Fast Fourier Transform (FFT) 7 2 x 10 1.8 1.6 1.4 Power 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Period(Years/Cycle) 30 35 40 In order to determine the cycle more precisely, [mp,index] = max(power); period(index) ans = 11.0769 Magnitude and Phase of Transformed Data Important information about a transformed sequence includes its magnitude and phase. The MATLAB functions abs and angle calculate this information. To try this, create a time vector t, and use this vector to create a sequence x consisting of two sinusoids at different frequencies: t = 0:1/100:10-1/100; x = sin(2*pi*15*t) + sin(2*pi*40*t); Now use the fft function to compute the DFT of the sequence. The code below calculates the magnitude and phase of the transformed sequence. It uses the abs function to obtain the magnitude of the data, the angle function to obtain 3-47 3 Data Analysis and Statistics the phase information, and unwrap to remove phase jumps greater than pi to their 2*pi complement: y = fft(x); m = abs(y); p = unwrap(angle(y)); Now create a frequency vector for the x-axis and plot the magnitude and phase: f = (0:length(y)-1)'*100/length(y); subplot(2,1,1), plot(f,m), ylabel('Abs. Magnitude'), grid on subplot(2,1,2), plot(f,p*180/pi) ylabel('Phase [Degrees]'), grid on xlabel('Frequency [Hertz]') The magnitude plot is perfectly symmetrical about the Nyquist frequency of 50 hertz. The useful information in the signal is found in the range 0 to 50 hertz. 500 Abs. Magnitude 400 300 200 100 0 0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 Frequency [Hertz] 70 80 90 100 4 2.5 x 10 Phase [Degrees] 2 1.5 1 0.5 0 −0.5 3-48 0 Fourier Analysis and the Fast Fourier Transform (FFT) FFT Length Versus Speed You can add a second argument to fft to specify a number of points n for the transform: y = fft(x,n) With this syntax, fft pads x with zeros if it is shorter than n, or truncates it if it is longer than n. If you do not specify n, fft defaults to the length of the input sequence. The execution time for fft depends on the length of the transform. It is fastest for powers of two. It is almost as fast for lengths that have only small prime factors. It is typically several times slower for lengths that are prime or which have large prime factors. The inverse FFT function ifft also accepts a transform length argument. For practical application of the FFT, the Signal Processing Toolbox includes numerous functions for spectral analysis. 3-49 3 Data Analysis and Statistics 3-50 4 Function Functions Function Summary (p. 4-2) A summary of some function functions Representing Functions in MATLAB (p. 4-3) Some guidelines for representing functions in MATLAB Plotting Mathematical Functions (p. 4-5) A discussion about using fplot to plot mathematical functions Minimizing Functions and Finding Zeros (p. 4-8) A discussion of high-level function functions that perform optimization-related tasks Numerical Integration (Quadrature) (p. 4-27) A discussion of the MATLAB quadrature functions Parameterizing Functions Called by Function Functions (p. 4-30) Explains how to pass additional arguments to user-defined functions that are called by a function function. See the “Differential Equations” and “Sparse Matrices” chapters for information about the use of other function functions. For information about function handles, see the function_handle (@), func2str, and str2func reference pages, and the “Function Handles” section of “Programming and Data Types” in the MATLAB documentation. 4 Function Functions Function Summary Function functions are functions that call other functions as input arguments. An example of a function function is fplot, which plots the graphs of functions. You can call the function fplot with the syntax fplot(@fun, [-pi pi]) where the input argument @fun is a handle to the function you want to plot. The function fun is referred to as the called function. The function functions are located in the MATLAB funfun directory. This table provides a brief description of the functions discussed in this chapter. Related functions are grouped by category. Function Summary Category Function Description Plotting fplot Plot function Optimization and zero finding fminbnd Minimize function of one variable with bound constraints. fminsearch Minimize function of several variables. fzero Find zero of function of one variable. quad Numerically evaluate integral, adaptive Simpson quadrature. quadl Numerically evaluate integral, adaptive Lobatto quadrature. quadv Vectorized quadrature dblquad Numerically evaluate double integral. triplequad Numerically evaluate triple integral. Numerical integration 4-2 Representing Functions in MATLAB Representing Functions in MATLAB MATLAB can represent mathematical functions by expressing them as MATLAB functions in M-files or as anonymous functions. For example, consider the function 1 1 f ( x ) = ------------------------------------------- + ------------------------------------------ –6 2 2 ( x – 0.3 ) + 0.01 ( x – 0.9 ) + 0.04 This function can be used as input to any of the function functions. MATLAB Functions You can find the function above in the M-file named humps.m. function y = humps(x) y = 1./((x - 0.3).^2 + 0.01) + 1./((x - 0.9).^2 + 0.04) - 6; To evaluate the function humps at 2.0, use @ to obtain a function handle for humps, and then use the function handle in the same way you would use a function name to call the function: fh = @humps; fh(2.0) ans = -4.8552 Anonymous Functions A second way to represent a mathematical function at the command line is by creating an anonymous function from a string expression. For example, you can create an anonymous function of the humps function. The value returned, fh, is a function handle: fh = @(x)1./((x-0.3).^2 + 0.01) + 1./((x-0.9).^2 + 0.04)-6; You can then evaluate fh at 2.0 in the same way that you can with a function handle for a MATLAB function: fh(2.0) ans = -4.8552 4-3 4 Function Functions You can also create anonymous functions of more than one argument. The following function has two input arguments x and y. fh = @(x,y)y*sin(x)+x*cos(y); fh(pi,2*pi) ans = 3.1416 4-4 Plotting Mathematical Functions Plotting Mathematical Functions The fplot function plots a mathematical function between a given set of axes limits. You can control the x-axis limits only, or both the x- and y-axis limits. For example, to plot the humps function over the x-axis range [-5 5], use fplot(@humps,[-5 5]) grid on 100 80 60 40 20 0 −20 −5 −4 −3 −2 −1 0 1 2 3 4 5 You can zoom in on the function by selecting y-axis limits of -10 and 25, using fplot(@humps,[-5 5 -10 25]) grid on 4-5 4 Function Functions 25 20 15 10 5 0 −5 −10 −5 −4 −3 −2 −1 0 1 2 3 4 5 You can also pass the function handle for an anonymous function for fplot to graph, as in fplot(@(x)2*sin(x+3),[-1 1]); You can plot more than one function on the same graph with one call to fplot. If you use this with a function, then the function must take a column vector x and return a matrix where each column corresponds to each function, evaluated at each value of x. If you pass an anonymous function consisting of several functions to fplot, the anonymous function also must return a matrix where each column corresponds to each function evaluated at each value of x, as in fplot(@(x)[2*sin(x+3), humps(x)],[-5 5]) which plots the first and second functions on the same graph. 4-6 Plotting Mathematical Functions 100 80 60 40 20 0 −20 −5 0 5 Note that the anonymous function fh = @(x)[2*sin(x+3), humps(x)]; evaluates to a matrix of two columns, one for each function, when x is a column vector. fh([1;2;3]) returns -1.5136 -1.9178 -0.5588 16.0000 -4.8552 -5.6383 4-7 4 Function Functions Minimizing Functions and Finding Zeros MATLAB provides a number of high-level function functions that perform optimization-related tasks. This section describes the following topics: • “Minimizing Functions of One Variable” on page 4-8 • “Minimizing Functions of Several Variables” on page 4-9 • “Fitting a Curve to Data” on page 4-10 • “Setting Minimization Options” on page 4-13 • “Output Functions” on page 4-14 • “Finding Zeros of Functions” on page 4-21 The MATLAB optimization functions are: fminbnd Minimize a function of one variable on a fixed interval fminsearch Minimize a function of several variables fzero Find zero of a function of one variable lsqnonneg Linear least squares with nonnegativity constraints optimget Get optimization options structure parameter values optimset Create or edit optimization options parameter structure For more optimization capabilities, see the Optimization Toolbox. Minimizing Functions of One Variable Given a mathematical function of a single variable coded in an M-file, you can use the fminbnd function to find a local minimizer of the function in a given interval. For example, to find a minimum of the humps function in the range (0.3, 1), use x = fminbnd(@humps,0.3,1) which returns x = 0.6370 4-8 Minimizing Functions and Finding Zeros You can ask for a tabular display of output by passing a fourth argument created by the optimset command to fminbnd x = fminbnd(@humps,0.3,1,optimset('Display','iter')) which gives the output Func-count 3 4 5 6 7 8 9 10 11 x 0.567376 0.732624 0.465248 0.644416 0.6413 0.637618 0.636985 0.637019 0.637052 f(x) 12.9098 13.7746 25.1714 11.2693 11.2583 11.2529 11.2528 11.2528 11.2528 Procedure initial golden golden parabolic parabolic parabolic parabolic parabolic parabolic Optimization terminated: the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-004 x = 0.6370 This shows the current value of x and the function value at f(x) each time a function evaluation occurs. For fminbnd, one function evaluation corresponds to one iteration of the algorithm. The last column shows what procedure is being used at each iteration, either a golden section search or a parabolic interpolation. Minimizing Functions of Several Variables The fminsearch function is similar to fminbnd except that it handles functions of many variables, and you specify a starting vector x0 rather than a starting interval. fminsearch attempts to return a vector x that is a local minimizer of the mathematical function near this starting vector. 4-9 4 Function Functions To try fminsearch, create a function three_var of three variables, x, y, and z. function b = three_var(v) x = v(1); y = v(2); z = v(3); b = x.^2 + 2.5*sin(y) - z^2*x^2*y^2; Now find a minimum for this function using x = -0.6, y = -1.2, and z = 0.135 as the starting values. v = [-0.6 -1.2 0.135]; a = fminsearch(@three_var,v) a = 0.0000 -1.5708 0.1803 Fitting a Curve to Data This section gives an example that shows how to fit an exponential function of the form Ae – λ t to some data. The example uses the function fminsearch to minimize the sum of squares of errors between the data and an exponential function Ae – λ t for varying parameters A and λ. This section covers the following topics. • “Creating an M-file for the Example” on page 4-10 • “Running the Example” on page 4-11 • “Plotting the Results” on page 4-12 Creating an M-file for the Example To run the example, first create an M-file that • Accepts vectors corresponding to the x- and y-coordinates of the data • Returns the parameters of the exponential function that best fits the data To do so, copy and paste the following code into an M-file and save it as fitcurvedemo in a directory on the MATLAB path. function [estimates, model] = fitcurvedemo(xdata, ydata) % Call fminsearch with a random starting point. start_point = rand(1, 2); 4-10 Minimizing Functions and Finding Zeros model = @expfun; estimates = fminsearch(model, start_point); % expfun accepts curve parameters as inputs, and outputs sse, % the sum of squares error for A * exp(-lambda * xdata) - ydata, % and the FittedCurve. FMINSEARCH only needs sse, but we want to % plot the FittedCurve at the end. function [sse, FittedCurve] = expfun(params) A = params(1); lambda = params(2); FittedCurve = A .* exp(-lambda * xdata); ErrorVector = FittedCurve - ydata; sse = sum(ErrorVector .^ 2); end end The M-file calls the function fminsearch, which find parameters A and lambda that minimize the sum of squares of the differences between the data and the exponential function A*exp(-lambda*t). The nested function expfun computes the sum of squares. Running the Example To run the example, first create some random data to fit. The following commands create random data that is approximately exponential with parameters A = 40 and lambda = .5. xdata = (0:.1:10)'; ydata = 40 * exp(-.5 * xdata) + randn(size(xdata)); To fit an exponential function to the data, enter [estimates, model] = fitcurvedemo(xdata,ydata) This returns estimates for the parameters A and lambda, estimates = 40.1334 0.5025 and a function handle, model, to the function that computes the exponential function A*exp(-lambda*t). 4-11 4 Function Functions Plotting the Results To plot the fit and the data, enter the following commands. plot(xdata, ydata, '*') hold on [sse, FittedCurve] = model(estimates); plot(xdata, FittedCurve, 'r') xlabel('xdata') ylabel('f(estimates,xdata)') title(['Fitting to function ', func2str(model)]); legend('data', ['fit using ', func2str(model)]) hold off The resulting plot displays the data points and the exponential fit. Fitting to function fitcurvedemo/expfun 45 data fit using fitcurvedemo/expfun 40 35 f(estimates,xdata) 30 25 20 15 10 5 0 −5 0 2 4 6 xdata 4-12 8 10 Minimizing Functions and Finding Zeros Setting Minimization Options You can specify control options that set some minimization parameters using an options structure that you create using the function optimset. You then pass options as in input to the optimization function, for example, by calling fminbnd with the syntax x = fminbnd(fun,x1,x2,options) or fminsearch with the syntax x = fminsearch(fun,x0,options) Use optimset to set the values of the options structure. For example, to set the 'Display' option to 'iter', in order to display output from the algorithm at each iteration, enter options = optimset('Display','iter'); fminbnd and fminsearch use only the options parameters shown in the following table. options.Display A flag that determines if intermediate steps in the minimization appear on the screen. If set to 'iter', intermediate steps are displayed; if set to 'off', no intermediate solutions are displayed, if set to final, displays just the final output. options.TolX The termination tolerance for x. Its default value is 1.e-4. options.TolFun The termination tolerance for the function value. The default value is 1.e-4. This parameter is used by fminsearch, but not fminbnd. options.MaxIter Maximum number of iterations allowed. options.MaxFunEvals The maximum number of function evaluations allowed. The default value is 500 for fminbnd and 200*length(x0) for fminsearch. 4-13 4 Function Functions The number of function evaluations, the number of iterations, and the algorithm are returned in the structure output when you provide fminbnd or fminsearch with a fourth output argument, as in [x,fval,exitflag,output] = fminbnd(@humps,0.3,1); or [x,fval,exitflag,output] = fminsearch(@three_var,v); Output Functions An output function is a function that an optimization function calls at each iteration of its algorithm. Typically, you might use an output function to generate graphical output, record the history of the data the algorithm generates, or halt the algorithm based on the data at the current iteration. You can create an output function as an M-file function, a subfunction, or a nested function. You can use the OutputFcn option with the following MATLAB optimization functions: • fminbnd • fminsearch • fzero This section covers the following topics: • “Creating and Using an Output Function” on page 4-15 • “Structure of the Output Function” on page 4-16 • “Example of a Nested Output Function” on page 4-17 • “Fields in optimValues” on page 4-19 • “States of the Algorithm” on page 4-20 • “Stop Flag” on page 4-20 4-14 Minimizing Functions and Finding Zeros Creating and Using an Output Function The following is a simple example of an output function that plots the points generated by an optimization function. function stop = outfun(x, optimValues, state) stop = false; hold on; plot(x(1),x(2),'.'); drawnow You can use this output function to plot the points generated by fminsearch in solving the optimization problem x 2 2 minimize f ( x ) = e 1 ( 4x 1 + 2x 2 + 4x 1 x 2 + 2x 2 + 1 ) x To do so, 1 Create an M-file containing the preceding code and save it as outfun.m in a directory on the MATLAB path. 2 Enter the command options = optimset('OutputFcn', @outfun); to set the value of the Outputfcn field of the options structure to a function handle to outfun. 3 Enter the following commands: hold on [email protected](x) exp(x(1))*(4*x(1)^2+2*x(2)^2+x(1)*x(2)+2*x(2)); [x fval] = fminsearch(objfun, [-1 1], options) hold off This returns the solution x = 0.1290 -0.5323 fval = -0.5689 4-15 4 Function Functions and displays the following plot of the points generated by fminsearch: Structure of the Output Function The function definition line of the output function has the following form: stop = outfun(x, optimValues, state) where • stop is a flag that is true or false depending on whether the optimization routine should quit or continue. See “Stop Flag” on page 4-20. • x is the point computed by the algorithm at the current iteration. • optimValues is a structure containing data from the current iteration. “Fields in optimValues” on page 4-19 describes the structure in detail. • state is the current state of the algorithm. “States of the Algorithm” on page 4-20 lists the possible values. The optimization function passes the values of the input arguments to outfun at each iteration. 4-16 Minimizing Functions and Finding Zeros Example of a Nested Output Function The example in “Creating and Using an Output Function” on page 4-15 does not require the output function to preserve data from one iteration to the next. When this is the case, you can write the output function as an M-file and call the optimization function directly from the command line. However, if you want your output function to record data from one iteration to the next, you should write a single M-file that does the following: • Contains the output function as a nested function — see Nested Functions in the online MATLAB documentation for more information. • Calls the optimization function. In the following example, the M-file also contains the objective function as a subfunction, although you could also write the objective function as a separate M-file or as an anonymous function. Since the nested function has access to variables in the M-file function that contains it, this method enables the output function to preserve variables from one iteration to the next. The following example uses an output function to record the points generated by fminsearch in solving the optimization problem x 2 2 minimize f ( x ) = e 1 ( 4x 1 + 2x 2 + 4x 1 x 2 + 2x 2 + 1 ) x and returns the sequence of points as a matrix called history. To run the example, do the following steps: 1 Open a new M-file in the MATLAB editor. 2 Copy and paste the following code into the M-file. function [x fval history] = myproblem(x0) history = []; options = optimset('OutputFcn', @myoutput); [x fval] = fminsearch(@objfun, x0,options); 4-17 4 Function Functions function stop = myoutput(x,optimvalues,state); stop = false; if state == 'iter' history = [history; x]; end end function z = objfun(x) z = exp(x(1))*(4*x(1)^2+2*x(2)^2+x(1)*x(2)+2*x(2)); end end 3 Save the file as myproblem.m in a directory on the MATLAB path. 4 At the MATLAB prompt, enter [x fval history] = myproblem([-1 1]) The function fminsearch returns x, the optimal point, and fval, the value of the objective function at x. x = 0.1290 -0.5323 fval = -0.5689 In addition, the output function myoutput returns the matrix history, which contains the points generated by the algorithm at each iteration, to the MATLAB workspace. The first four rows of history are history(1:4,:) ans = -1.0000 -1.0000 -1.0750 -1.0125 4-18 1.0000 1.0500 0.9000 0.8500 Minimizing Functions and Finding Zeros The final row of points is the same as the optimal point, x. history(end,:) ans = 0.1290 -0.5323 objfun(history(end,:)) ans = -0.5689 Fields in optimValues The following table lists the fields of the optimValues structure that are provided by all three optimization functions, fminbnd, fminsearch, and fzero. The function fzero also provides additional fields that are described in its reference page. The “Command-Line Display Headings” column of the table lists the headings, corresponding to the optimValues fields that are displayed at the command line when you set the Display parameter of options to 'iter'. optimValues Field (optimValues.field) Description Command-Line Display Heading funcount Cumulative number of function evaluations. Func-count fval Function value at current point. min f(x) iteration Iteration number — starts at 0. Iteration procedure Procedure messages Procedure 4-19 4 Function Functions States of the Algorithm The following table lists the possible values for state: State Description 'init' The algorithm is in the initial state before the first iteration. 'interrupt' The algorithm is performing an iteration. In this state, the output function can interrupt the current iteration of the optimization. You might want the output function to do this to improve the efficiency of the computations. When state is set to 'interrupt', the values of x and optimValues are the same as at the last call to the output function, in which state is set to 'iter'. 'iter' The algorithm is at the end of an iteration. 'done' The algorithm is in the final state after the last iteration. The following code illustrates how the output function might use the value of state to decide which tasks to perform at the current iteration. switch state case 'init' % Setup for plots or guis case 'iter' % Make updates to plot or guis as needed. case 'interrupt' % Check conditions to see whether optimization % should quit. case 'done' % Cleanup of plots, guis, or final plot otherwise end Stop Flag The output argument stop is a flag that is true or false. The flag tells the optimization function whether the optimization should quit or continue. The following examples show typical ways to use the stop flag. 4-20 Minimizing Functions and Finding Zeros Stopping an Optimization Based on Data in optimValues. The output function can stop an optimization at any iteration based on the current data in optimValues. For example, the following code sets stop to true if the objective function value is less than 5: function stop = myoutput(x, optimValues, state) stop = false; % Check if objective function is less than 5. if optimValues.fval < 5 stop = true; end Stopping an Optimization Based on GUI Input. If you design a GUI to perform optimizations, you can make the output function stop an optimization when a user clicks a Stop button on the GUI. The following code shows how to do this, assuming that the Stop button callback stores the value true in the optimstop field of a handles structure called hObject stored in appdata. function stop = myoutput(x, optimValues, state) stop = false; % Check if user has requested to stop the optimization. stop = getappdata(hObject,'optimstop'); Finding Zeros of Functions The fzero function attempts to find a zero of one equation with one variable. You can call this function with either a one-element starting point or a two-element vector that designates a starting interval. If you give fzero a starting point x0, fzero first searches for an interval around this point where the function changes sign. If the interval is found, fzero returns a value near where the function changes sign. If no such interval is found, fzero returns NaN. Alternatively, if you know two points where the function value differs in sign, you can specify this starting interval using a two-element vector; fzero is guaranteed to narrow down the interval and return a value near a sign change. The following sections contain two examples that illustrate how to find a zero of a function using a starting interval and a starting point. The examples use the function humps, which is provided with MATLAB. The following figure shows the graph of humps. 4-21 4 Function Functions 100 80 60 40 20 0 −20 −1 −0.5 0 0.5 1 1.5 2 Using a Starting Interval The graph of humps indicates that the function is negative at x = -1 and positive at x = 1. You can confirm this by calculating humps at these two points. humps(1) ans = 16 humps(-1) ans = -5.1378 Consequently, you can use [-1 1] as a starting interval for fzero. 4-22 Minimizing Functions and Finding Zeros The iterative algorithm for fzero finds smaller and smaller subintervals of [-1 1]. For each subinterval, the sign of humps differs at the two endpoints. As the endpoints of the subintervals get closer and closer, they converge to a zero for humps. To show the progress of fzero at each iteration, set the Display option to iter using the function optimset. options = optimset('Display','iter'); Then call fzero as follows: a = fzero(@humps,[-1 1],options) This returns the following iterative output: a = fzero(@humps,[-1 1],options) Func-count 2 3 4 5 6 7 8 9 10 11 12 13 x -1 -0.513876 -0.513876 -0.473635 -0.115287 -0.115287 -0.132562 -0.131666 -0.131618 -0.131618 -0.131618 -0.131618 f(x) -5.13779 -4.02235 -4.02235 -3.83767 0.414441 0.414441 -0.0226907 -0.0011492 1.88371e-007 -2.7935e-011 8.88178e-016 8.88178e-016 Procedure initial interpolation bisection interpolation bisection interpolation interpolation interpolation interpolation interpolation interpolation interpolation Zero found in the interval [-1, 1] a = -0.1316 Each value x represents the best endpoint so far. The Procedure column tells you whether each step of the algorithm uses bisection or interpolation. 4-23 4 Function Functions You can verify that the function value at a is close to zero by entering humps(a) ans = 8.8818e-016 Using a Starting Point Suppose you do not know two points at which the function values of humps differ in sign. In that case, you can choose a scalar x0 as the starting point for fzero. fzero first searches for an interval around this point on which the function changes sign. If fzero finds such an interval, it proceeds with the algorithm described in the previous section. If no such interval is found, fzero returns NaN. For example, if you set the starting point to -0.2, the Display option to Iter, and call fzero by a = fzero(@humps,-0.2,options) fzero returns the following output: Search for an interval around -0.2 containing a sign change: Func-count a f(a) b f(b) 1 -0.2 -1.35385 -0.2 -1.35385 3 -0.194343 -1.26077 -0.205657 -1.44411 5 -0.192 -1.22137 -0.208 -1.4807 7 -0.188686 -1.16477 -0.211314 -1.53167 9 -0.184 -1.08293 -0.216 -1.60224 11 -0.177373 -0.963455 -0.222627 -1.69911 13 -0.168 -0.786636 -0.232 -1.83055 15 -0.154745 -0.51962 -0.245255 -2.00602 17 -0.136 -0.104165 -0.264 -2.23521 18 -0.10949 0.572246 -0.264 -2.23521 Search for a zero in the interval [-0.10949, -0.264]: Func-count x f(x) Procedure 18 -0.10949 0.572246 initial 19 -0.140984 -0.219277 interpolation 20 -0.132259 -0.0154224 interpolation 21 -0.131617 3.40729e-005 interpolation 22 -0.131618 -6.79505e-008 interpolation 23 -0.131618 -2.98428e-013 interpolation 24 -0.131618 8.88178e-016 interpolation 25 -0.131618 8.88178e-016 interpolation Zero found in the interval [-0.10949, -0.264] 4-24 Procedure initial interval search search search search search search search search search Minimizing Functions and Finding Zeros a = -0.1316 The endpoints of the current subinterval at each iteration are listed under the headings a and b, while the corresponding values of humps at the endpoints are listed under f(a) and f(b), respectively. Note The endpoints a and b are not listed in any specific order: a can be greater than b or less than b. For the first nine steps, the sign of humps is negative at both endpoints of the current subinterval, which are listed under in the output. At the tenth step, the sign of humps is positive at the endpoint, -0.10949, but negative at the endpoint, -0.264. From this point on, the algorithm continues to narrow down the interval [-0.10949 -0.264], as described in the previous section, until it reaches the value -0.1316. Tips Optimization problems may take many iterations to converge. Most optimization problems benefit from good starting guesses. Providing good starting guesses improves the execution efficiency and may help locate the global minimum instead of a local minimum. Sophisticated problems are best solved by an evolutionary approach, whereby a problem with a smaller number of independent variables is solved first. Solutions from lower order problems can generally be used as starting points for higher order problems by using an appropriate mapping. The use of simpler cost functions and less stringent termination criteria in the early stages of an optimization problem can also reduce computation time. Such an approach often produces superior results by avoiding local minima. Troubleshooting Below is a list of typical problems and recommendations for dealing with them. 4-25 4 Function Functions Problem Recommendation The solution found by fminbnd or fminsearch does not appear to be a global minimum. There is no guarantee that you have a global minimum unless your problem is continuous and has only one minimum. Starting the optimization from a number of different starting points (or intervals in the case of fminbnd) may help to locate the global minimum or verify that there is only one minimum. Use different methods, where possible, to verify results. Sometimes an optimization problem has values of x for which it is impossible to evaluate f. Modify your function to include a penalty function to give a large positive value to f when infeasibility is encountered. The minimization routine appears to enter an infinite loop or returns a solution that is not a minimum (or not a zero in the case of fzero). Your objective function (fun) may be returning NaN or complex values. The optimization routines expect only real numbers to be returned. Any other values may cause unexpected results. To determine whether this is the case, set options = optimset('FunValCheck', 'on') and call the optimization function with options as an input argument. This displays a warning when the objective function returns NaN or complex values. 4-26 Numerical Integration (Quadrature) Numerical Integration (Quadrature) The area beneath a section of a function F(x) can be determined by numerically integrating F(x), a process referred to as quadrature. The MATLAB quadrature functions are: quad Use adaptive Simpson quadrature quadl Use adaptive Lobatto quadrature quadv Vectorized quadrature dblquad Numerically evaluate double integral triplequad Numerically evaluate triple integral To integrate the function defined by humps.m from 0 to 1, use q = quad(@humps,0,1) q = 29.8583 Both quad and quadl operate recursively. If either method detects a possible singularity, it prints a warning. You can include a fourth argument for quad or quadl that specifies a relative error tolerance for the integration. If a nonzero fifth argument is passed to quad or quadl, the function evaluations are traced. Two examples illustrate use of these functions: • Computing the length of a curve • Double integration Example: Computing the Length of a Curve You can use quad or quadl to compute the length of a curve. Consider the curve parameterized by the equations x ( t ) = sin ( 2t ), y ( t ) = cos ( t ), z(t) = t where t ∈ [ 0, 3π ] . 4-27 4 Function Functions A three-dimensional plot of this curve is t = 0:0.1:3*pi; plot3(sin(2*t),cos(t),t) The arc length formula says the length of the curve is the integral of the norm of the derivatives of the parameterized equations 3π ∫ 2 2 4 cos ( 2t ) + sin ( t ) + 1 dt 0 The function hcurve computes the integrand function f = hcurve(t) f = sqrt(4*cos(2*t).^2 + sin(t).^2 + 1); Integrate this function with a call to quad len = quad(@hcurve,0,3*pi) len = 1.7222e+01 The length of this curve is about 17.2. Example: Double Integration Consider the numerical solution of ymax xmax ∫ ∫ ymin xmin f ( x , y ) dx dy For this example f ( x, y ) = y sin ( x ) + x cos ( y ) . The first step is to build the function to be evaluated. The function must be capable of returning a vector output when given a vector input. You must also consider which variable is in the inner integral, and which goes in the outer integral. In this example, the inner variable is x and the outer variable is y (the order in the integral is dxdy). In this case, the integrand function is function out = integrnd(x,y) out = y*sin(x) + x*cos(y); 4-28 Numerical Integration (Quadrature) To perform the integration, two functions are available in the funfun directory. The first, dblquad, is called directly from the command line. This M-file evaluates the outer loop using quad. At each iteration, quad calls the second helper function that evaluates the inner loop. To evaluate the double integral, use result = dblquad(@integrnd,xmin,xmax,ymin,ymax); The first argument is a string with the name of the integrand function. The second to fifth arguments are xmin Lower limit of inner integral xmax Upper limit of the inner integral ymin Lower limit of outer integral ymax Upper limit of the outer integral Here is a numeric example that illustrates the use of dblquad. xmin = xmax = ymin = ymax = result pi; 2*pi; 0; pi; = dblquad(@integrnd,xmin,xmax,ymin,ymax) The result is -9.8698. By default, dblquad calls quad. To integrate the previous example using quadl (with the default values for the tolerance argument), use result = dblquad(@integrnd,xmin,xmax,ymin,ymax,[],@quadl); Alternatively, you can pass any user-defined quadrature function name to dblquad as long as the quadrature function has the same calling and return arguments as quad. 4-29 4 Function Functions Parameterizing Functions Called by Function Functions At times, you might want use a function function that calls a function with several parameters. For example, if you want to use fzero to find zeros of the cubic polynomial x3 + bx + c for different values of the coefficients b and c, you would like the function that computes the polynomial to accept the additional parameters b and c. When you invoke fzero, you must also provide values for these additional parameters to the polynomial function. This section describes two ways to do this: • “Providing Parameter Values Using Nested Functions” on page 4-30 • “Providing Parameter Values to Anonymous Functions” on page 4-31 Providing Parameter Values Using Nested Functions One way to provide parameters to the polynomial is to write a single M-file that • Accepts the additional parameters as inputs • Invokes the function function • Contains the function called by the function function as a nested function The following example illustrates how to find a zero of the cubic polynomial x3 + bx + c, for different values of the coefficients b and c, using this method. To do so, write an M-file with the following code. function y = findzero(b, c, x0) options = optimset('Display', 'off'); % Turn off Display y = fzero(@poly, x0, options); function y = poly(x) % Compute the polynomial. y = x^3 + b*x + c; end end The main function, findzero, does two things: • Invokes the function fzero to find a zero of the polynomial • Computes the polynomial in a nested function, poly, which is called by fzero 4-30 Parameterizing Functions Called by Function Functions You can call findzero with any values of the coefficients b and c, which are seen by poly because it is a nested function. As an example, to find a zero of the polynomial with b = 2 and c = 3.5, using the starting point x0 = 0, call findzero as follows. x = findzero(2, 3.5, 0) This returns the zero x = -1.0945 Providing Parameter Values to Anonymous Functions Suppose you have already written a standalone M-file for the function poly containing the following code, which computes the polynomial for any coefficients b and c, function y = poly(x, b, c) % Compute the polynomial. y = x^3 + b*x + c; You then want to find a zero for the coefficient values b = 2 and c = 3.5. You cannot simply apply fzero to poly, which has three input arguments, because fzero only accepts functions with a single input argument. As an alternative to rewriting poly as a nested function, as described in “Providing Parameter Values Using Nested Functions” on page 4-30, you can pass poly to fzero as a function handle to an anonymous function that has the form @(x) poly(x, b, c). The function handle has just one input argument x, so fzero accepts it. b = 2; c = 3.5; x = fzero(@(x) poly(x, b, c), 0) This returns the zero x = -1.0945 4-31 4 Function Functions “Anonymous Functions” on page 4-3 explains how to create anonymous functions. If you later decide to find a zero for different values of b and c, you must redefine the anonymous function using the new values. For example, b = 4; c = -1; fzero(@(x) poly(x, b, c), 0) ans = 0.2463 For more complicated objective functions, it is usually preferable to write the function as a nested function, as described in “Providing Parameter Values Using Nested Functions” on page 4-30. 4-32 5 Differential Equations Initial Value Problems for ODEs and DAEs (p. 5-2) Describes the solution of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs), where the solution of interest satisfies initial conditions at a given initial value of the independent variable. Initial Value Problems for DDEs (p. 5-45) Describes the solution of delay differential equations (DDEs) where the solution of interest is determined by a history function. Boundary Value Problems for ODEs (p. 5-57) Describes the solution of ODEs, where the solution of interest satisfies certain boundary conditions. The boundary conditions specify a relationship between the values of the solution at the initial and final values of the independent variable. Partial Differential Equations (p. 5-81) Describes the solution of initial-boundary value problems for systems of parabolic and elliptic partial differential equations (PDEs) in one spatial variable and time. Selected Bibliography (p. 5-98) Lists published materials that support concepts described in this chapter. Note In function tables, commonly used functions are listed first, followed by more advanced functions. The same is true of property tables. 5 Differential Equations Initial Value Problems for ODEs and DAEs This section describes how to use MATLAB to solve initial value problems (IVPs) of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). This section covers the following topics: • “ODE Function Summary” on page 5-2 • “Introduction to Initial Value ODE Problems” on page 5-4 • “Solvers for Explicit and Linearly Implicit ODEs” on page 5-5 • “Examples: Solving Explicit ODE Problems” on page 5-9 • “Solver for Fully Implicit ODEs” on page 5-15 • “Example: Solving a Fully Implicit ODE Problem” on page 5-16 • “Changing ODE Integration Properties” on page 5-17 • “Examples: Applying the ODE Initial Value Problem Solvers” on page 5-18 • “Questions and Answers, and Troubleshooting” on page 5-39 ODE Function Summary ODE Initial Value Problem Solvers The following table lists the initial value problem solvers, the kind of problem you can solve with each solver, and the method each solver uses. 5-2 Solver Solves These Kinds of Problems Method ode45 Nonstiff differential equations Runge-Kutta ode23 Nonstiff differential equations Runge-Kutta ode113 Nonstiff differential equations Adams ode15s Stiff differential equations and DAEs NDFs (BDFs) ode23s Stiff differential equations Rosenbrock ode23t Moderately stiff differential equations and DAEs Trapezoidal rule Initial Value Problems for ODEs and DAEs ode23tb Stiff differential equations TR-BDF2 ode15i Fully implicit differential equations BDFs ODE Solution Evaluation and Extension You can use the following functions to evaluate and extend solutions to ODEs. Function Description deval Evaluate the numerical solution using the output of dde23. odextend Extend the solution of an initial value problem for an ODE ODE Solvers Properties Handling An options structure contains named properties whose values are passed to ODE solvers, and which affect problem solution. Use these functions to create, alter, or access an options structure. Function Description odeset Create or alter options structure for input to ODE solver. odeget Extract properties from options structure created with odeset. ODE Solver Output Functions If an output function is specified, the solver calls the specified function after every successful integration step. You can use odeset to specify one of these sample functions as the OutputFcn property, or you can modify them to create your own functions. Function Description odeplot Time-series plot odephas2 Two-dimensional phase plane plot 5-3 5 Differential Equations odephas3 Three-dimensional phase plane plot odeprint Print to command window Introduction to Initial Value ODE Problems What Is an Ordinary Differential Equation? The ODE solvers are designed to handle ordinary differential equations. An ordinary differential equation contains one or more derivatives of a dependent variable y with respect to a single independent variable t , usually referred to as time. The derivative of y with respect to t is denoted as y′ , the second derivative as y′′ , and so on. Often y(t) is a vector, having elements y 1, y 2, … , y n . Types of Problems Handled by the ODE Solvers The ODE solvers handle the following types of first-order ODEs: • Explicit ODEs of the form y′ = f ( t, y ) • Linearly implicit ODEs of the form M(t, y) ⋅ y′ = f ( t, y ) , where M(t,y) is a matrix • Fully implicit ODEs of the form f ( t, y, y′ ) = 0 (ode15i only) Using Initial Conditions to Specify the Solution of Interest Generally there are many functions y(t) that satisfy a given ODE, and additional information is necessary to specify the solution of interest. In an initial value problem, the solution of interest satisfies a specific initial condition, that is, y is equal to y 0 at a given initial time t 0 . An initial value problem for an ODE is then y′ = f ( t, y ) y ( t0 ) = y0 (5-1) If the function f(t, y) is sufficiently smooth, this problem has one and only one solution. Generally there is no analytic expression for the solution, so it is necessary to approximate y(t) by numerical means, such as using one of the ODE solvers. 5-4 Initial Value Problems for ODEs and DAEs Working with Higher Order ODEs The ODE solvers accept only first-order differential equations. However, ODEs often involve a number of dependent variables, as well as derivatives of order higher than one. To use the ODE solvers, you must rewrite such equations as an equivalent system of first-order differential equations of the form y′ = f ( t, y ) You can write any ordinary differential equation y (n) = f ( t, y, y′, …, y (n – 1) ) as a system of first-order equations by making the substitutions y 1 = y, y 2 = y′, …, y n = y (n – 1) The result is an equivalent system of n first-order ODEs. y′1 = y 2 y 2′ = y 3 … y n′ = f ( t, y 1, y 2, ..., y n ) “Example: Solving an IVP ODE (van der Pol Equation, Nonstiff)” on page 5-9 rewrites the second-order van der Pol equation 2 y′′ 1 – µ ( 1 – y 1 ) y′1 + y 1 = 0 as a system of first-order ODEs. Solvers for Explicit and Linearly Implicit ODEs This section describes the ODE solver functions for explicit or linearly implicit ODEs, as described in “Types of Problems Handled by the ODE Solvers” on page 5-4. The solver functions implement numerical integration methods for solving initial value problems for ODEs. Beginning at the initial time with initial conditions, they step through the time interval, computing a solution at each time step. If the solution for a time step satisfies the solver’s error tolerance criteria, it is a successful step. Otherwise, it is a failed attempt; the solver shrinks the step size and tries again. 5-5 5 Differential Equations This section describes: • Solvers for nonstiff ODE problems • Solvers for stiff ODE problems • Basic ODE solver syntax “Mass Matrix and DAE Properties,” in the reference page for odeset, explains how to set options to solve more general linearly implicit problems. The function ode15i, which solves implicit ODEs, is described in “Solver for Fully Implicit ODEs” on page 5-15. Solvers for Nonstiff Problems There are three solvers designed for nonstiff problems: ode45 Based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. It is a one-step solver – in computing y(t n) , it needs only the solution at the immediately preceding time point, y(t n – 1) . In general, ode45 is the best function to apply as a “first try” for most problems. ode23 Based on an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of mild stiffness. Like ode45, ode23 is a one-step solver. ode113 Variable order Adams-Bashforth-Moulton PECE solver. It may be more efficient than ode45 at stringent tolerances and when the ODE function is particularly expensive to evaluate. ode113 is a multistep solver – it normally needs the solutions at several preceding time points to compute the current solution. Solvers for Stiff Problems Not all difficult problems are stiff, but all stiff problems are difficult for solvers not specifically designed for them. Solvers for stiff problems can be used exactly like the other solvers. However, you can often significantly improve the efficiency of these solvers by providing them with additional information about the problem. (See “Changing ODE Integration Properties” on page 5-17.) 5-6 Initial Value Problems for ODEs and DAEs There are four solvers designed for stiff problems: ode15s Variable-order solver based on the numerical differentiation formulas (NDFs). Optionally it uses the backward differentiation formulas, BDFs, (also known as Gear’s method). Like ode113, ode15s is a multistep solver. If you suspect that a problem is stiff or if ode45 failed or was very inefficient, try ode15s. ode23s Based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective. ode23t An implementation of the trapezoidal rule using a “free” interpolant. Use this solver if the problem is only moderately stiff and you need a solution without numerical damping. ode23tb An implementation of TR-BDF2, an implicit Runge-Kutta formula with a first stage that is a trapezoidal rule step and a second stage that is a backward differentiation formula of order 2. Like ode23s, this solver may be more efficient than ode15s at crude tolerances. Basic ODE Solver Syntax All of the ODE solver functions, except for ode15i, share a syntax that makes it easy to try any of the different numerical methods, if it is not apparent which is the most appropriate. To apply a different method to the same problem, simply change the ODE solver function name. The simplest syntax, common to all the solver functions, is [t,y] = solver(odefun,tspan,y0,options) where solver is one of the ODE solver functions listed previously. 5-7 5 Differential Equations The basic input arguments are odefun Handle to a function that evaluates the system of ODEs. The function has the form dydt = odefun(t,y) where t is a scalar, and dydt and y are column vectors. See “Function Handles” in the MATLAB Programming documentation for more information. tspan Vector specifying the interval of integration. The solver imposes the initial conditions at tspan(1), and integrates from tspan(1) to tspan(end). y0 Vector of initial conditions for the problem See also “Introduction to Initial Value ODE Problems” on page 5-4. options Structure of optional parameters that change the default integration properties. “Changing ODE Integration Properties” on page 5-17 tells you how to create the structure and describes the properties you can specify. The output arguments are t Column vector of time points y Solution array. Each row in y corresponds to the solution at a time returned in the corresponding row of t. See the reference page for the ODE solvers for more information about these arguments. 5-8 Initial Value Problems for ODEs and DAEs Examples: Solving Explicit ODE Problems This section uses the van der Pol equation 2 y′′ 1 – µ ( 1 – y 1 ) y′1 + y 1 = 0 to describe the process for solving initial value ODE problems using the ODE solvers. • “Example: Solving an IVP ODE (van der Pol Equation, Nonstiff)” on page 5-9 describes each step of the process. Because the van der Pol equation is a second-order equation, the example must first rewrite it as a system of first order equations. • “Example: The van der Pol Equation, µ = 1000 (Stiff)” on page 5-12 demonstrates the solution of a stiff problem. • “Evaluating the Solution at Specific Points” on page 5-15 tells you how to evaluate the solution at specific points. Note See “Basic ODE Solver Syntax” on page 5-7 for more information. Example: Solving an IVP ODE (van der Pol Equation, Nonstiff) This example explains and illustrates the steps you need to solve an initial value ODE problem: 1 Rewrite the problem as a system of first-order ODEs. Rewrite the van der Pol equation (second-order) 2 y′′ 1 – µ ( 1 – y 1 ) y′1 + y 1 = 0 where µ > 0 is a scalar parameter, by making the substitution y′ 1 = y 2 . The resulting system of first-order ODEs is y′ 1 = y 2 2 y′ 2 = µ ( 1 – y 1 )y 2 – y 1 See “Working with Higher Order ODEs” on page 5-5 for more information. 5-9 5 Differential Equations 2 Code the system of first-order ODEs. Once you represent the equation as a system of first-order ODEs, you can code it as a function that an ODE solver can use. The function must be of the form dydt = odefun(t,y) Although t and y must be the function’s two arguments, the function does not need to use them. The output dydt, a column vector, is the derivative of y. The code below represents the van der Pol system in the function, vdp1. The vdp1 function assumes that µ = 1 . The variables y 1 and y 2 are the entries y(1) and y(2) of a two-element vector. function dydt = vdp1(t,y) dydt = [y(2); (1-y(1)^2)∗y(2)-y(1)]; Note that, although vdp1 must accept the arguments t and y, it does not use t in its computations. 3 Apply a solver to the problem. Decide which solver you want to use to solve the problem. Then call the solver and pass it the function you created to describe the first-order system of ODEs, the time interval on which you want to solve the problem, and an initial condition vector. See “Examples: Solving Explicit ODE Problems” on page 5-9 and the ODE solver reference page for descriptions of the ODE solvers. For the van der Pol system, you can use ode45 on time interval [0 20] with initial values y(1) = 2 and y(2) = 0. [t,y] = ode45(@vdp1,[0 20],[2; 0]); This example uses @ to pass vdp1 as a function handle to ode45. The resulting output is a column vector of time points t and a solution array y. Each row in y corresponds to a time returned in the corresponding row of t. The first column of y corresponds to y 1 , and the second column to y 2 . 5-10 Initial Value Problems for ODEs and DAEs Note For information on function handles, see the function_handle (@), func2str, and str2func reference pages, and the Function Handles chapter of “Programming and Data Types” in the MATLAB documentation. 4 View the solver output. You can simply use the plot command to view the solver output. plot(t,y(:,1),'-',t,y(:,2),'--') title('Solution of van der Pol Equation, \mu = 1'); xlabel('time t'); ylabel('solution y'); legend('y_1','y_2') Solution of van der Pol Equation, µ = 1 3 y1 y2 2 solution y 1 0 −1 −2 −3 0 2 4 6 8 10 time t 12 14 16 18 20 As an alternative, you can use a solver output function to process the output. The solver calls the function specified in the integration property OutputFcn after each successful time step. Use odeset to set OutputFcn to the desired 5-11 5 Differential Equations function. See “Solver Output Properties,” in the reference page for odeset, for more information about OutputFcn. Example: The van der Pol Equation, µ = 1000 (Stiff) This example presents a stiff problem. For a stiff problem, solutions can change on a time scale that is very short compared to the interval of integration, but the solution of interest changes on a much longer time scale. Methods not designed for stiff problems are ineffective on intervals where the solution changes slowly because they use time steps small enough to resolve the fastest possible change. When µ is increased to 1000, the solution to the van der Pol equation changes dramatically and exhibits oscillation on a much longer time scale. Approximating the solution of the initial value problem becomes a more difficult task. Because this particular problem is stiff, a solver intended for nonstiff problems, such as ode45, is too inefficient to be practical. A solver such as ode15s is intended for such stiff problems. The vdp1000 function evaluates the van der Pol system from the previous example, but with µ = 1000. function dydt = vdp1000(t,y) dydt = [y(2); 1000∗(1-y(1)^2)∗y(2)-y(1)]; Note This example hardcodes µ in the ODE function. The vdpode example solves the same problem, but passes a user-specified µ as an additional argument to the ODE function. Now use the ode15s function to solve the problem with the initial condition vector of [2; 0], but a time interval of [0 3000]. For scaling reasons, plot just the first component of y(t). [t,y] = ode15s(@vdp1000,[0 3000],[2; 0]); plot(t,y(:,1),'-'); title('Solution of van der Pol Equation, \mu = 1000'); xlabel('time t'); ylabel('solution y_1'); 5-12 Initial Value Problems for ODEs and DAEs Solution of van der Pol Equation, µ = 1000 2.5 2 1.5 1 solution y1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0 500 1000 1500 time t 2000 2500 3000 Note For detailed instructions for solving an initial value ODE problem, see “Example: Solving an IVP ODE (van der Pol Equation, Nonstiff)” on page 5-9. Parameterizing an ODE Function The preceding sections showed how to solve the van der Pol equation for two different values of the parameter µ. In those examples, the values µ = 1 and µ = 1000 are hard-coded in the ODE functions. If you are solving an ODE for several different parameter values, it might be more convenient to include the parameter in the ODE function and assign a value to the parameter each time you run the ODE solver. This section explains how to do this for the van der Pol equation. 5-13 5 Differential Equations One way to provide parameter values to the ODE function is to write an M-file that • Accepts the parameters as inputs. • Contains ODE function as a nested function, internally using the input parameters. • Calls the ODE solver. The following code illustrates this: function [t,y] = run_vdp(mu) tspan = [0 max(20, 3*mu)]; y0 = [2; 0]; % Call the ODE solver ode15s. [t,y] = ode15s(@vdp,tspan,y0); % Plot the results. plot(t,y(:,1),'-'); title(strcat('Solution of van der Pol Equation, \mu =',... num2str(mu))); xlabel('time t'); ylabel('solution y_1'); % Define the ODE function as nested function, % using the parameter mu. function dydt = vdp(t,y) dydt = [y(2); mu*(1-y(1)^2)*y(2)-y(1)]; end end Because the ODE function vdp is a nested function, the value of the parameter mu is available to it. To run the M-file for mu = 1, as in “Example: Solving an IVP ODE (van der Pol Equation, Nonstiff)” on page 5-9, enter [t,y] = run_vdp(1); To run the code for µ = 1000, as in “Example: The van der Pol Equation, µ = 1000 (Stiff)” on page 5-12, enter [t,y] = run_vdp(1000); 5-14 Initial Value Problems for ODEs and DAEs See the vdpode code for a complete example based on these functions. Evaluating the Solution at Specific Points The numerical methods implemented in the ODE solvers produce a continuous solution over the interval of integration [ a, b ] . You can evaluate the approximate solution, S ( x ) , at any point in [ a, b ] using the function deval and the structure sol returned by the solver. For example, if you solve the problem described in “Example: Solving an IVP ODE (van der Pol Equation, Nonstiff)” on page 5-9 by calling ode45 with a single output argument sol, sol = ode45(@vdp1,[0 20],[2; 0]); ode45 returns the solution as a structure. You can then evaluate the approximate solution at points in the vector xint = 1:5 as follows: xint = 1:5; Sxint = deval(sol,xint) Sxint = 1.5081 -0.7803 0.3235 -1.8320 -1.8686 -1.0220 -1.7407 0.6260 -0.8344 1.3095 The deval function is vectorized. For a vector xint, the ith column of Sxint approximates the solution y ( xint(i) ) . Solver for Fully Implicit ODEs The solver ode15i solves fully implicit differential equations of the form f ( t, y, y′ ) = 0 using the variable order BDF method. The basic syntax for ode15i is [t,y] = ode15i(odefun,tspan,y0,yp0,options) 5-15 5 Differential Equations The input arguments are odefun A function that evaluates the left side of the differential equation of the form f(t, y, y′) = 0 . tspan A vector specifying the interval of integration, [t0,tf]. To obtain solutions at specific times (all increasing or all decreasing), use tspan = [t0,t1,...,tf]. y0, yp0 Vectors of initial conditions for y(t 0) and y′(t 0) , respectively. The specified values must be consistent; that is, they must satisfy f(t 0, y(t 0), y′(t 0)) = 0 . “Example: Solving a Fully Implicit ODE Problem” on page 5-16 shows how to use the function decic to compute consistent initial conditions. options Optional integration argument created using the odeset function. See the odeset reference page for details. The output arguments are t Column vector of time points y Solution array. Each row in y corresponds to the solution at a time returned in the corresponding row of t. See the ode15i reference page for more information about these arguments. Example: Solving a Fully Implicit ODE Problem The following example shows how to use the function ode15i to solve the implicit ODE problem defined by Weissinger’s equation 2 3 3 2 2 2 ty ( y′ ) – y ( y′ ) + t ( t + 1 )y′ – t y = 0 with the initial value y(1) = y(t) = 3 ⁄ 2 . The exact solution of the ODE is 2 t + 0.5 The example uses the function weissinger, which is provided with MATLAB, to compute the left-hand side of the equation. 5-16 Initial Value Problems for ODEs and DAEs Before calling ode15i, the example uses a helper function decic to compute a consistent initial value for y′(t 0) . In the following call, the given initial value y(1) = 3 ⁄ 2 is held fixed and a guess of 0 is specified for y′(1) . See the reference page for decic for more information. t0 = 1; y0 = sqrt(3/2); yp0 = 0; [y0,yp0] = decic(@weissinger,t0,y0,1,yp0,0); You can now call ode15i to solve the ODE and then plot the numerical solution against the analytical solution with the following commands. [t,y] = ode15i(@weissinger,[1 10],y0,yp0); ytrue = sqrt(t.^2 + 0.5); plot(t,y,t,ytrue,'o'); 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Changing ODE Integration Properties The default integration properties in the ODE solvers are selected to handle common problems. In some cases, you can improve ODE solver performance by overriding these defaults. You do this by supplying the solvers with an options structure that specifies one or more property values. 5-17 5 Differential Equations For example, to change the value of the relative error tolerance of the solver from the default value of 1e-3 to 1e-4, 1 Create an options structure using the function odeset by entering options = odeset('RelTol', 1e-4); 2 Pass the options structure to the solver as follows: - For all solvers except ode15i, use the syntax [t,y] = solver(odefun,tspan,y0,options) - For ode15i, use the syntax [t,y] = ode15i(odefun,tspan,y0,yp0,options) For an example that uses the options structure, see “Example: Stiff Problem (van der Pol Equation)” on page 5-20. For a complete description of the available options, see the reference page for odeset. Examples: Applying the ODE Initial Value Problem Solvers This section contains several examples that illustrate the kinds of problems you can solve. For each example, there is a corresponding M-file, included in MATLAB. You can • View the M-file code in an editor by entering edit followed by the name of the M-file at the MATLAB prompt. For example, to view the code for the simple nonstiff problem example, enter edit rigidode Alternatively, if you are reading this in the MATLAB Help Browser, you can click the name of the M-file in the list below. • Run the example by entering the name of the M-file at the MATLAB prompt. This section presents the following examples: • Simple nonstiff problem (rigidode) • Stiff problem (vdpode) • Finite element discretization (fem1ode) 5-18 Initial Value Problems for ODEs and DAEs • Large, stiff, sparse problem (brussode) • Simple event location (ballode) • Advanced event location (orbitode) • Differential-algebraic problem (hb1dae) • “Summary of Code Examples” on page 5-37 Example: Simple Nonstiff Problem rigidode illustrates the solution of a standard test problem proposed by Krogh for solvers intended for nonstiff problems [8]. The ODEs are the Euler equations of a rigid body without external forces. y′1 = y2 y 3 y′2 = – y 1 y 3 y′3 = – 0.51 y 1 y 2 For your convenience, the entire problem is defined and solved in a single M-file. The differential equations are coded as a subfunction f. Because the example calls the ode45 solver without output arguments, the solver uses the default output function odeplot to plot the solution components. To run this example, click on the example name, or type rigidode at the command line. function rigidode %RIGIDODE Euler equations of a rigid body without external forces tspan = [0 12]; y0 = [0; 1; 1]; % Solve the problem using ode45 ode45(@f,tspan,y0); % -----------------------------------------------------------function dydt = f(t,y) dydt = [ y(2)*y(3) -y(1)*y(3) -0.51*y(1)*y(2) ]; 5-19 5 Differential Equations 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 2 4 6 8 10 12 Example: Stiff Problem (van der Pol Equation) vdpode illustrates the solution of the van der Pol problem described in “Example: The van der Pol Equation, µ = 1000 (Stiff)” on page 5-12. The differential equations y′1 = y2 2 y′2 = µ(1 – y 1 )y 2 – y 1 involve a constant parameter µ . As µ increases, the problem becomes more stiff, and the period of oscillation becomes larger. When µ is 1000 the equation is in relaxation oscillation and the problem is very stiff. The limit cycle has portions where the solution components change slowly and the problem is quite stiff, alternating with regions of very sharp change where it is not stiff (quasi-discontinuities). By default, the solvers in the ODE suite that are intended for stiff problems approximate Jacobian matrices numerically. However, this example provides a nested function J(t,y) to evaluate the Jacobian matrix ∂f ⁄ ∂y analytically at 5-20 Initial Value Problems for ODEs and DAEs (t,y) for µ = MU. The use of an analytic Jacobian can improve the reliability and efficiency of integration. To run this example, click on the example name, or type vdpode at the command line. From the command line, you can specify a value of µ as an argument to vdpode. The default is µ = 1000. function vdpode(MU) %VDPODE Parameterizable van der Pol equation (stiff for large MU) if nargin < 1 MU = 1000; % default end tspan = [0; max(20,3*MU)]; y0 = [2; 0]; options = odeset('Jacobian',@J); % Several periods [t,y] = ode15s(@f,tspan,y0,options); plot(t,y(:,1)); title(['Solution of van der Pol Equation, \mu = ' num2str(MU)]); xlabel('time t'); ylabel('solution y_1'); axis([tspan(1) tspan(end) -2.5 2.5]); --------------------------------------------------------------function dydt = f(t,y) dydt = [ y(2) MU*(1-y(1)^2)*y(2)-y(1) ]; end % End nested function f --------------------------------------------------------------function dfdy = J(t,y) dfdy = [ 0 1 -2*MU*y(1)*y(2)-1 MU*(1-y(1)^2) ]; end % End nested function J end 5-21 5 Differential Equations Solution of van der Pol Equation, µ = 1000 2.5 2 1.5 1 solution y1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0 500 1000 1500 time t 2000 2500 3000 Example: Finite Element Discretization fem1ode illustrates the solution of ODEs that result from a finite element discretization of a partial differential equation. The value of N in the call fem1ode(N) controls the discretization, and the resulting system consists of N equations. By default, N is 19. This example involves a mass matrix. The system of ODEs comes from a method of lines solution of the partial differential equation 2 – t ∂u u e ------ = ∂--------∂t ∂x 2 with initial condition u ( 0, x ) = sin ( x ) and boundary conditions u ( t, 0 ) = u ( t, π ) = 0 . An integer N is chosen, h is defined as π ⁄ ( N + 1 ) , and 5-22 Initial Value Problems for ODEs and DAEs the solution of the partial differential equation is approximated at x k = kh for k = 0, 1, ..., N+1 by N u (t,x k) ≈ ∑ ck ( t ) φk ( x ) k=1 Here φ k(x) is a piecewise linear function that is 1 at x k and 0 at all the other x j . A Galerkin discretization leads to the system of ODEs M ( t )c′ = Jc where c ( t ) = c1 ( t ) cN ( t ) and the tridiagonal matrices M ( t ) and J are given by ⎧ 2h ⁄ 3 exp ( – t ) ⎪ M ij = ⎨ h ⁄ 6 exp ( – t ) ⎪ ⎩0 if i = j if i = j ± 1 otherwise and ⎧ –2 ⁄ h ⎪ J ij = ⎨ 1 ⁄ h ⎪ ⎩ 0 if i = j if i = j ± 1 otherwise The initial values c ( 0 ) are taken from the initial condition for the partial differential equation. The problem is solved on the time interval [ 0, π ] . In the fem1ode example, the properties options = odeset('Mass',@mass,'MStateDep','none','Jacobian',J) indicate that the problem is of the form M(t)y′ = Jy . The nested function mass(t) evaluates the time-dependent mass matrix M ( t ) and J is the constant Jacobian. 5-23 5 Differential Equations To run this example, click on the example name, or type fem1ode at the command line. From the command line, you can specify a value of N as an argument to fem1ode. The default is N = 19. function fem1ode(N) %FEM1ODE Stiff problem with a time-dependent mass matrix if nargin < 1 N = 19; end h = pi/(N+1); y0 = sin(h*(1:N)'); tspan = [0; pi]; % e d J The Jacobian is constant. = repmat(1/h,N,1); % e=[(1/h) ... (1/h)]; = repmat(-2/h,N,1); % d=[(-2/h) ... (-2/h)]; = spdiags([e d e], -1:1, N, N); options = odeset('Mass',@mass,'MStateDependence','none', ... 'Jacobian',J); [t,y] = ode15s(@f,tspan,y0,options); surf((1:N)/(N+1),t,y); set(gca,'ZLim',[0 1]); view(142.5,30); title(['Finite element problem with time-dependent mass ' ... 'matrix, solved by ODE15S']); xlabel('space ( x/\pi )'); ylabel('time'); zlabel('solution'); %--------------------------------------------------------------function out = f(t,y) h = pi/(N+1); e = repmat(1/h,N,1); % e=[(1/h) ... (1/h)]; d = repmat(-2/h,N,1); % d=[(-2/h) ... (-2/h)]; J = spdiags([e d e], -1:1, N, N); out = J*y; end % End nested function f 5-24 Initial Value Problems for ODEs and DAEs %--------------------------------------------------------------function M = mass(t) h = pi/(N+1); e = repmat(exp(-t)*h/6,N,1); % e(i)=exp(-t)*h/6 e4 = repmat(4*exp(-t)*h/6,N,1); M = spdiags([e e4 e], -1:1, N, N); end % End nested function mass end Finite element problem with time−dependent mass matrix, solved by ODE15S 1 0.8 solution 0.6 0.4 0.2 0 0 0 1 0.2 2 0.4 0.6 3 0.8 4 1 space ( x/π ) time Example: Large, Stiff, Sparse Problem brussode illustrates the solution of a (potentially) large stiff sparse problem. The problem is the classic “Brusselator” system [3] that models diffusion in a chemical reaction 2 2 u′i = 1 + u i v i – 4u i + α ( N + 1 ) ( u i – 1 – 2u i + u i + 1 ) 2 2 v′i = 3u i – u i v i + α ( N + 1 ) ( v i – 1 – 2v i + v i + 1 ) 5-25 5 Differential Equations and is solved on the time interval [0,10] with α = 1/50 and u i ( 0 ) = 1 + sin ( 2πx i ) ⎫ ⎬ vi ( 0 ) = 3 ⎭ with x i = i ⁄ ( N + 1 ), for i = 1, ..., N There are 2N equations in the system, but the Jacobian is banded with a constant width 5 if the equations are ordered as u 1, v 1, u 2, v 2, … In the call brussode(N), where N corresponds to N , the parameter N ≥ 2 specifies the number of grid points. The resulting system consists of 2N equations. By default, N is 20. The problem becomes increasingly stiff and the Jacobian increasingly sparse as N increases. The nested function f(t,y) returns the derivatives vector for the Brusselator problem. The subfunction jpattern(N) returns a sparse matrix of 1s and 0s showing the locations of nonzeros in the Jacobian ∂f ⁄ ∂y . The example assigns this matrix to the property JPattern, and the solver uses the sparsity pattern to generate the Jacobian numerically as a sparse matrix. Providing a sparsity pattern can significantly reduce the number of function evaluations required to generate the Jacobian and can accelerate integration. For the Brusselator problem, if the sparsity pattern is not supplied, 2N evaluations of the function are needed to compute the 2N-by-2N Jacobian matrix. If the sparsity pattern is supplied, only four evaluations are needed, regardless of the value of N. To run this example, click on the example name, or type brussode at the command line. From the command line, you can specify a value of N as an argument to brussode. The default is N = 20. function brussode(N) %BRUSSODE Stiff problem modeling a chemical reaction if nargin < 1 N = 20; end tspan = [0; 10]; y0 = [1+sin((2*pi/(N+1))*(1:N)); repmat(3,1,N)]; options = odeset('Vectorized','on','JPattern',jpattern(N)); 5-26 Initial Value Problems for ODEs and DAEs [t,y] = ode15s(@f,tspan,y0,options); u = y(:,1:2:end); x = (1:N)/(N+1); surf(x,t,u); view(-40,30); xlabel('space'); ylabel('time'); zlabel('solution u'); title(['The Brusselator for N = ' num2str(N)]); % -------------------------------------------------------------function dydt = f(t,y) c = 0.02 * (N+1)^2; dydt = zeros(2*N,size(y,2)); % preallocate dy/dt % Evaluate the two components of the function at one edge of % the grid (with edge conditions). i = 1; dydt(i,:) = 1 + y(i+1,:).*y(i,:).^2 - 4*y(i,:) + ... c*(1-2*y(i,:)+y(i+2,:)); dydt(i+1,:) = 3*y(i,:) - y(i+1,:).*y(i,:).^2 + ... c*(3-2*y(i+1,:)+y(i+3,:)); % Evaluate the two components of the function at all interior % grid points. i = 3:2:2*N-3; dydt(i,:) = 1 + y(i+1,:).*y(i,:).^2 - 4*y(i,:) + ... c*(y(i-2,:)-2*y(i,:)+y(i+2,:)); dydt(i+1,:) = 3*y(i,:) - y(i+1,:).*y(i,:).^2 + ... c*(y(i-1,:)-2*y(i+1,:)+y(i+3,:)); % Evaluate the two components of the function at the other edge % of the grid (with edge conditions). i = 2*N-1; dydt(i,:) = 1 + y(i+1,:).*y(i,:).^2 - 4*y(i,:) + ... c*(y(i-2,:)-2*y(i,:)+1); dydt(i+1,:) = 3*y(i,:) - y(i+1,:).*y(i,:).^2 + ... c*(y(i-1,:)-2*y(i+1,:)+3); end % End nested function f end % End function brussode % -------------------------------------------------------------function S = jpattern(N) 5-27 5 Differential Equations B = ones(2*N,5); B(2:2:2*N,2) = zeros(N,1); B(1:2:2*N-1,4) = zeros(N,1); S = spdiags(B,-2:2,2*N,2*N); end; The Brusselator for N = 20 3 2.5 solution u 2 1.5 1 0.5 0 10 1 8 0.8 6 0.6 4 0.4 2 time 0.2 0 0 space Example: Simple Event Location ballode models the motion of a bouncing ball. This example illustrates the event location capabilities of the ODE solvers. The equations for the bouncing ball are y′1 = y2 y′2 = – 9.8 In this example, the event function is coded in a subfunction events [value,isterminal,direction] = events(t,y) 5-28 Initial Value Problems for ODEs and DAEs which returns • A value of the event function • The information whether or not the integration should stop when value = 0 (isterminal = 1 or 0, respectively) • The desired directionality of the zero crossings: -1 Detect zero crossings in the negative direction only 0 Detect all zero crossings 1 Detect zero crossings in the positive direction only The length of value, isterminal, and direction is the same as the number of event functions. The ith element of each vector, corresponds to the ith event function. For an example of more advanced event location, see orbitode (“Example: Advanced Event Location” on page 5-31). In ballode, setting the Events property to @events causes the solver to stop the integration (isterminal = 1) when the ball hits the ground (the height y(1) is 0) during its fall (direction = -1). The example then restarts the integration with initial conditions corresponding to a ball that bounced. To run this example, click on the example name, or type ballode at the command line. function ballode %BALLODE Run a demo of a bouncing ball. tstart = 0; tfinal = 30; y0 = [0; 20]; refine = 4; options = odeset('Events',@events,'OutputFcn', @odeplot,... 'OutputSel',1,'Refine',refine); set(gca,'xlim',[0 30],'ylim',[0 25]); box on hold on; tout = tstart; 5-29 5 Differential Equations yout = y0.'; teout = []; yeout = []; ieout = []; for i = 1:10 % Solve until the first terminal event. [t,y,te,ye,ie] = ode23(@f,[tstart tfinal],y0,options); if ~ishold hold on end % Accumulate output. nt = length(t); tout = [tout; t(2:nt)]; yout = [yout; y(2:nt,:)]; teout = [teout; te]; % Events at tstart are never reported. yeout = [yeout; ye]; ieout = [ieout; ie]; ud = get(gcf,'UserData'); if ud.stop break; end % Set the new initial conditions, with .9 attenuation. y0(1) = 0; y0(2) = -.9*y(nt,2); % A good guess of a valid first time step is the length of % the last valid time step, so use it for faster computation. options = odeset(options,'InitialStep',t(nt)-t(nt-refine),... 'MaxStep',t(nt)-t(1)); tstart = t(nt); end plot(teout,yeout(:,1),'ro') xlabel('time'); ylabel('height'); title('Ball trajectory and the events'); hold off odeplot([],[],'done'); 5-30 Initial Value Problems for ODEs and DAEs % -------------------------------------------------------------function dydt = f(t,y) dydt = [y(2); -9.8]; % -------------------------------------------------------------function [value,isterminal,direction] = events(t,y) % Locate the time when height passes through zero in a % decreasing direction and stop integration. value = y(1); % Detect height = 0 isterminal = 1; % Stop the integration direction = -1; % Negative direction only Ball trajectory and the events 25 20 height 15 10 5 0 0 5 10 15 time 20 25 30 Example: Advanced Event Location orbitode illustrates the solution of a standard test problem for those solvers that are intended for nonstiff problems. It traces the path of a spaceship traveling around the moon and returning to the earth. (Shampine and Gordon [8], p.246). 5-31 5 Differential Equations The orbitode problem is a system of the following four equations shown: y′1 = y 3 y′2 = y 4 µ∗ ( y 1 + µ ) µ ( y 1 – µ∗ ) y′3 = 2y 4 + y 1 – -------------------------- – --------------------------3 3 r1 r2 µy µ∗ y y′4 = – 2y 3 + y 2 – -----------2- – --------23 r1 r 23 where µ = 1 ⁄ 82.45 µ∗ = 1 – µ r1 = ( y 1 + µ ) 2 + y 22 r2 = ( y 1 – µ∗ ) + y 22 2 The first two solution components are coordinates of the body of infinitesimal mass, so plotting one against the other gives the orbit of the body. The initial conditions have been chosen to make the orbit periodic. The value of µ corresponds to a spaceship traveling around the moon and the earth. Moderately stringent tolerances are necessary to reproduce the qualitative behavior of the orbit. Suitable values are 1e-5 for RelTol and 1e-4 for AbsTol. The nested events function includes event functions that locate the point of maximum distance from the starting point and the time the spaceship returns to the starting point. Note that the events are located accurately, even though the step sizes used by the integrator are not determined by the location of the events. In this example, the ability to specify the direction of the zero crossing is critical. Both the point of return to the initial point and the point of maximum distance have the same event function value, and the direction of the crossing is used to distinguish them. 5-32 Initial Value Problems for ODEs and DAEs To run this example, click on the example name, or type orbitode at the command line. The example uses the output function odephas2 to produce the two-dimensional phase plane plot and let you to see the progress of the integration. function orbitode %ORBITODE Restricted three-body problem mu = 1 / 82.45; mustar = 1 - mu; y0 = [1.2; 0; 0; -1.04935750983031990726]; tspan = [0 7]; options = odeset('RelTol',1e-5,'AbsTol',1e-4,... 'OutputFcn',@odephas2,'Events',@events); [t,y,te,ye,ie] = ode45(@f,tspan,y0,options); plot(y(:,1),y(:,2),ye(:,1),ye(:,2),'o'); title ('Restricted three body problem') ylabel ('y(t)') xlabel ('x(t)') % -------------------------------------------------------------function dydt = f(t,y) r13 = ((y(1) + mu)^2 + y(2)^2) ^ 1.5; r23 = ((y(1) - mustar)^2 + y(2)^2) ^ 1.5; dydt = [ y(3) y(4) 2*y(4) + y(1) - mustar*((y(1)+mu)/r13) - ... mu*((y(1)-mustar)/r23) -2*y(3) + y(2) - mustar*(y(2)/r13) - mu*(y(2)/r23) ]; end % End nested function f % -------------------------------------------------------------function [value,isterminal,direction] = events(t,y) % Locate the time when the object returns closest to the % initial point y0 and starts to move away, and stop integration. % Also locate the time when the object is farthest from the % initial point y0 and starts to move closer. % 5-33 5 Differential Equations % The current distance of the body is % % DSQ = (y(1)-y0(1))^2 + (y(2)-y0(2))^2 % = <y(1:2)-y0(1:2),y(1:2)-y0(1:2)> % % A local minimum of DSQ occurs when d/dt DSQ crosses zero % heading in the positive direction. We can compute d(DSQ)/dt as % % d(DSQ)/dt = 2*(y(1:2)-y0(1:2))'*dy(1:2)/dt = ... % 2*(y(1:2)-y0(1:2))'*y(3:4) % dDSQdt = 2 * ((y(1:2)-y0(1:2))' * y(3:4)); value = [dDSQdt; dDSQdt]; isterminal = [1; 0]; % Stop at local minimum direction = [1; -1]; % [local minimum, local maximum] end % End nested function events end Restricted three body problem 0.8 0.6 0.4 y(t) 0.2 0 −0.2 −0.4 −0.6 −0.8 −1.5 5-34 −1 −0.5 0 x(t) 0.5 1 1.5 Initial Value Problems for ODEs and DAEs Example: Differential-Algebraic Problem hb1dae reformulates the hb1ode example as a differential-algebraic equation (DAE) problem. The Robertson problem coded in hb1ode is a classic test problem for codes that solve stiff ODEs. 4 y′1 = – 0.04 y 1 + 10 y 2 y 3 7 2 4 y′2 = 0.04y 1 – 10 y 2 y 3 – 3 ⋅ 10 y 2 7 2 y′3 = 3 ⋅ 10 y 2 Note The Robertson problem appears as an example in the prolog to LSODI [4]. In hb1ode, the problem is solved with initial conditions y 1(0) = 1 , y 2(0) = 0 , y 3(0) = 0 to steady state. These differential equations satisfy a linear conservation law that is used to reformulate the problem as the DAE 4 y′1 = – 0.04 y 1 + 10 y 2 y 3 4 7 2 y′2 = 0.04y 1 – 10 y 2 y 3 – 3 ⋅ 10 y 2 0 = y1 + y2 + y3 – 1 Obviously these equations do not have a solution for y ( 0 ) with components that do not sum to 1. The problem has the form of My′ = f ( t, y ) with M = 1 0 0 0 1 0 0 0 0 M is obviously singular, but hb1dae does not inform the solver of this. The solver must recognize that the problem is a DAE, not an ODE. Similarly, although consistent initial conditions are obvious, the example uses an –3 inconsistent value y 3(0) = 10 to illustrate computation of consistent initial conditions. 5-35 5 Differential Equations To run this example, click on the example name, or type hb1dae at the command line. Note that hb1dae: • Imposes a much smaller absolute error tolerance on y 2 than on the other components. This is because y 2 is much smaller than the other components and its major change takes place in a relatively short time. • Specifies additional points at which the solution is computed to more clearly show the behavior of y 2 . • Multiplies y 2 by 104 to make y 2 visible when plotting it with the rest of the solution. • Uses a logarithmic scale to plot the solution on the long time interval. function hb1dae %HB1DAE Stiff differential-algebraic equation (DAE) % A constant, singular mass matrix M = [1 0 0 0 1 0 0 0 0]; % Use an inconsistent initial condition to test initialization. y0 = [1; 0; 1e-3]; tspan = [0 4*logspace(-6,6)]; % Use the LSODI example tolerances. The 'MassSingular' property % is left at its default 'maybe' to test the automatic detection % of a DAE. options = odeset('Mass',M,'RelTol',1e-4,... 'AbsTol',[1e-6 1e-10 1e-6],'Vectorized','on'); [t,y] = ode15s(@f,tspan,y0,options); y(:,2) = 1e4*y(:,2); semilogx(t,y); ylabel('1e4 * y(:,2)'); title(['Robertson DAE problem with a Conservation Law, '... 'solved by ODE15S']); xlabel('This is equivalent to the stiff ODEs coded in HB1ODE.'); 5-36 Initial Value Problems for ODEs and DAEs % -------------------------------------------------------------function out = f(t,y) out = [ -0.04*y(1,:) + 1e4*y(2,:).*y(3,:) 0.04*y(1,:) - 1e4*y(2,:).*y(3,:) - 3e7*y(2,:).^2 y(1,:) + y(2,:) + y(3,:) - 1 ]; Robertson DAE problem with a Conservation Law, solved by ODE15S 1 0.9 0.8 0.7 1e4 * y(:,2) 0.6 0.5 0.4 0.3 0.2 0.1 0 −6 10 −4 10 −2 0 2 4 10 10 10 10 This is equivalent to the stiff ODEs coded in HB1ODE. 6 10 8 10 Summary of Code Examples The following table lists the M-files for all the ODE initial value problem examples. Click the example name to see the code in an editor. Type the example name at the command line to run it. Note The Differential Equations Examples browser enables you to view the code for the ODE examples and DAE examples. You can also run the examples from the browser. Click on these links to invoke the browser, or type odeexamples('ode')or odeexamples('dae')at the command line. 5-37 5 Differential Equations 5-38 Example Description amp1dae Stiff DAE — electrical circuit ballode Simple event location — bouncing ball batonode ODE with time- and state-dependent mass matrix — motion of a baton brussode Stiff large problem — diffusion in a chemical reaction (the Brusselator) burgersode ODE with strongly state-dependent mass matrix — Burger’s equation solved using a moving mesh technique fem1ode Stiff problem with a time-dependent mass matrix — finite element method fem2ode Stiff problem with a constant mass matrix — finite element method hb1ode Stiff ODE problem solved on a very long interval — Robertson chemical reaction hb1dae Robertson problem — stiff, linearly implicit DAE from a conservation law ihb1dae Robertson problem — stiff, fully implicit DAE iburgersode Burgers' equation solved as implicit ODE system orbitode Advanced event location — restricted three body problem rigidode Nonstiff problem — Euler equations of a rigid body without external forces vdpode Parameterizable van der Pol equation (stiff for large µ ) Initial Value Problems for ODEs and DAEs Questions and Answers, and Troubleshooting This section contains a number of tables that answer questions about the use and operation of the ODE solvers: • General ODE solver questions • Problem size, memory use, and computation speed • Time steps for integration • Error tolerance and other options • Solving different kinds of problems • Troubleshooting General ODE Solver Questions Question Answer How do the ODE solvers differ from quad or quadl? quad and quadl solve problems of the form y′ = f ( t ) . The ODE Can I solve ODE systems in which there are more equations than unknowns, or vice versa? solvers handle more general problems y′ = f ( t, y ) , linearly implicit problems that involve a mass matrix M(t, y) y′ = f(t, y) , and fully implicit problems f(t, y, y′) = 0 . No. 5-39 5 Differential Equations Problem Size, Memory Use, and Computation Speed Question Answer How large a problem can I solve with the ODE suite? The primary constraints are memory and time. At each time step, the solvers for nonstiff problems allocate vectors of length n, where n is the number of equations in the system. The solvers for stiff problems but also allocate an n-by-n Jacobian matrix. For these solvers it may be advantageous to use the sparse option. If the problem is nonstiff, or if you are using the sparse option, it may be possible to solve a problem with thousands of unknowns. In this case, however, storage of the result can be problematic. Try asking the solver to evaluate the solution at specific points only, or call the solver with no output arguments and use an output function to monitor the solution. 5-40 I'm solving a very large system, but only care about a couple of the components of y. Is there any way to avoid storing all of the elements? Yes. The user-installable output function capability is designed specifically for this purpose. When you call the solver with no output arguments, the solver does not allocate storage to hold the entire solution history. Instead, the solver calls OutputFcn(t,y,flag) at each time step. To keep the history of specific elements, write an output function that stores or plots only the elements you care about. What is the startup cost of the integration and how can I reduce it? The biggest startup cost occurs as the solver attempts to find a step size appropriate to the scale of the problem. If you happen to know an appropriate step size, use the InitialStep property. For example, if you repeatedly call the integrator in an event location loop, the last step that was taken before the event is probably on scale for the next integration. See ballode for an example. Initial Value Problems for ODEs and DAEs Time Steps for Integration Question Answer The first step size that the integrator takes is too large, and it misses important behavior. You can specify the first step size with the InitialStep property. The integrator tries this value, then reduces it if necessary. Can I integrate with fixed step sizes? No. Error Tolerance and Other Options Question Answer How do I choose RelTol and AbsTol? RelTol, the relative accuracy tolerance, controls the number of correct digits in the answer. AbsTol, the absolute error tolerance, controls the difference between the answer and the solution. At each step, the error e in component i of the solution satisfies |e(i)| <= max(RelTol*abs(y(i)),AbsTol(i)) Roughly speaking, this means that you want RelTol correct digits in all solution components except those smaller than thresholds AbsTol(i). Even if you are not interested in a component y(i) when it is small, you may have to specify AbsTol(i) small enough to get some correct digits in y(i) so that you can accurately compute more interesting components. I want answers that are correct to the precision of the computer. Why can’t I simply set RelTol to eps? You can get close to machine precision, but not that close. The solvers do not allow RelTol near eps because they try to approximate a continuous function. At tolerances comparable to eps, the machine arithmetic causes all functions to look discontinuous. 5-41 5 Differential Equations Error Tolerance and Other Options (Continued) Question Answer How do I tell the solver that I don’t care about getting an accurate answer for one of the solution components? You can increase the absolute error tolerance corresponding to this solution component. If the tolerance is bigger than the component, this specifies no correct digits for the component. The solver may have to get some correct digits in this component to compute other components accurately, but it generally handles this automatically. Solving Different Kinds of Problems Question Answer Can the solvers handle partial differential equations (PDEs) that have been discretized by the method of lines? Yes, because the discretization produces a system of ODEs. Depending on the discretization, you might have a form involving mass matrices – the ODE solvers provide for this. Often the system is stiff. This is to be expected when the PDE is parabolic and when there are phenomena that happen on very different time scales such as a chemical reaction in a fluid flow. In such cases, use one of the four solvers: ode15s, ode23s, ode23t, ode23tb. If there are many equations, set the JPattern property. This might make the difference between success and failure due to the computation being too expensive. For an example that uses JPattern, see “Example: Large, Stiff, Sparse Problem” on page 5-25. When the system is not stiff, or not very stiff, ode23 or ode45 is more efficient than ode15s, ode23s, ode23t, or ode23tb. Parabolic-elliptic partial differential equations in 1-D can be solved directly with the MATLAB PDE solver, pdepe. For more information, see “Partial Differential Equations” on page 5-81. Can I solve differential-algebraic equation (DAE) systems? 5-42 Yes. The solvers ode15s and ode23t can solve some DAEs of the form M ( t, y )y′ = f ( t, y ) where M ( t, y ) is singular. The DAEs must be of index 1. ode15i can solve fully implicit DAEs of index 1, f(t, y, y′) = 0 . For examples, see amp1dae, hb1dae, or ihb1dae. Initial Value Problems for ODEs and DAEs Solving Different Kinds of Problems (Continued) Question Answer Can I integrate a set of sampled data? Not directly. You have to represent the data as a function by interpolation or some other scheme for fitting data. The smoothness of this function is critical. A piecewise polynomial fit like a spline can look smooth to the eye, but rough to a solver; the solver takes small steps where the derivatives of the fit have jumps. Either use a smooth function to represent the data or use one of the lower order solvers (ode23, ode23s, ode23t, ode23tb) that is less sensitive to this. What do I do when I have the final and not the initial value? All the solvers of the ODE suite allow you to solve backwards or forwards in time. The syntax for the solvers is [t,y] = ode45(odefun,[t0 tf],y0); and the syntax accepts t0 > tf. Troubleshooting Question Answer The solution doesn’t look like what I expected. If you’re right about its appearance, you need to reduce the error tolerances from their default values. A smaller relative error tolerance is needed to compute accurately the solution of problems integrated over “long” intervals, as well as solutions of problems that are moderately unstable. You should check whether there are solution components that stay smaller than their absolute error tolerance for some time. If so, you are not asking for any correct digits in these components. This may be acceptable for these components, but failing to compute them accurately may degrade the accuracy of other components that depend on them. My plots aren’t smooth enough. Increase the value of Refine from its default of 4 in ode45 and 1 in the other solvers. The bigger the value of Refine, the more output points. Execution speed is not affected much by the value of Refine. 5-43 5 Differential Equations Troubleshooting (Continued) Question Answer I’m plotting the solution as it is computed and it looks fine, but the code gets stuck at some point. First verify that the ODE function is smooth near the point where the code gets stuck. If it isn’t, the solver must take small steps to deal with this. It may help to break tspan into pieces on which the ODE function is smooth. If the function is smooth and the code is taking extremely small steps, you are probably trying to solve a stiff problem with a solver not intended for this purpose. Switch to ode15s, ode23s, ode23t, or ode23tb. My integration proceeds very slowly, using too many time steps. First, check that your tspan is not too long. Remember that the solver uses as many time points as necessary to produce a smooth solution. If the ODE function changes on a time scale that is very short compared to the tspan, the solver uses a lot of time steps. Long-time integration is a hard problem. Break tspan into smaller pieces. If the ODE function does not change noticeably on the tspan interval, it could be that your problem is stiff. Try using ode15s, ode23s, ode23t, or ode23tb. Finally, make sure that the ODE function is written in an efficient way. The solvers evaluate the derivatives in the ODE function many times. The cost of numerical integration depends critically on the expense of evaluating the ODE function. Rather than recompute complicated constant parameters at each evaluation, store them in globals or calculate them once and pass them to nested functions. I know that the solution undergoes a radical change at time t where t0 ≤ t ≤ tf but the integrator steps past without “seeing” it. 5-44 If you know there is a sharp change at time t, it might help to break the tspan interval into two pieces, [t0 t] and [t tf], and call the integrator twice. If the differential equation has periodic coefficients or solution, you might restrict the maximum step size to the length of the period so the integrator won’t step over periods. Initial Value Problems for DDEs Initial Value Problems for DDEs This section describes how to use MATLAB to solve initial value problems (IVPs) for delay differential equations (DDEs). It provides: • A summary of the DDE functions and examples • An introduction to DDEs • A description of the DDE solver and its syntax • General instructions for representing a DDE • A discussion and example about discontinuities and restarting • A discussion about changing default integration properties DDE Function Summary DDE Initial Value Problem Solver Solver Description dde23 Solve initial value problems for delay differential equations with constant delays. DDE Helper Functions Function Description deval Evaluate the numerical solution using the output of dde23. DDE Solver Properties Handling An options structure contains named properties whose values are passed to dde23, and which affect problem solution. Use these functions to create, alter, or access an options structure. 5-45 5 Differential Equations Function Description ddeset Create/alter the DDE options structure. ddeget Extract properties from options structure created with ddeset. DDE Initial Value Problem Examples These examples illustrate the kind of problems you can solve using dde23. Click the example name to see the code in an editor. Type the example name at the command line to run it. Note The Differential Equations Examples browser enables you to view the code for the DDE examples, and also run them. Click on the link to invoke the browser, or type odeexamples('dde')at the command line. Example Description ddex1 Straightforward example ddex2 Cardiovascular model with discontinuities Additional examples are provided by “Tutorial on Solving DDEs with DDE23,” available at http://www.mathworks.com/dde_tutorial. Introduction to Initial Value DDE Problems The DDE solver can solve systems of ordinary differential equations y′ ( t ) = f ( t, y ( t ), y ( t – τ 1 ), …, y ( t – τ k ) ) where t is the independent variable, y is the dependent variable, and y′ represents dy ⁄ dt . The delays (lags) τ 1, …, τ k are positive constants. Using a History to Specify the Solution of Interest In an initial value problem, we seek the solution on an interval [ t 0, t f ] . with t 0 < t f . The DDE shows that y′ ( t ) depends on values of the solution at times 5-46 Initial Value Problems for DDEs prior to t . In particular, y′ ( t 0 ) depends on y ( t 0 – τ 1 ), …, y ( t 0 – τ k ) . Because of this, a solution on [ t 0, t f ] depends on its values for t ≤ t 0 , i.e., its history S ( t ) . Propagation of Discontinuities Generally, the solution y ( t ) of an IVP for a system of DDEs has a jump in its first derivative at the initial point t 0 because the first derivative of the history function does not satisfy the DDE there. – + S′ ( t 0 ) ≠ y′ ( t 0 ) = f ( t 0, y ( t 0 ), S ( t 0 – τ 1 ), …, S ( t 0 – τ k ) ) A discontinuity in any derivative propagates into the future at spacings of τ 1, τ 2 , … , τ k . For reliable and efficient integration of DDEs, a solver must track discontinuities in low order derivatives and deal with them. For DDEs with constant lags, the solution gets smoother as the integration progresses, so after a while the solver can stop tracking a discontinuity. See “Discontinuities” on page 5-53 for more information. DDE Solver This section describes: • The DDE solver, dde23 • DDE solver basic syntax The DDE Solver The function dde23 solves initial value problems for delay differential equations (DDEs) with constant delays. It integrates a system of first-order differential equations y′ ( t ) = f ( t, y ( t ), y ( t – τ 1 ), …, y ( t – τ k ) ) on the interval [ t 0, t f ] , with t 0 < t f and given history y ( t ) = S ( t ) for t ≤ t 0 . dde23 produces a solution that is continuous on [ t 0, t f ] . You can use the function deval and the output of dde23 to evaluate the solution at specific points on the interval of integration. dde23 tracks discontinuities and integrates the differential equations with the explicit Runge-Kutta (2,3) pair and interpolant used by ode23. The 5-47 5 Differential Equations Runge-Kutta formulas are implicit for step sizes longer than the delays. When the solution is smooth enough that steps this big are justified, the implicit formulas are evaluated by a predictor-corrector iteration. DDE Solver Basic Syntax The basic syntax of the DDE solver is sol = dde23(ddefun,lags,history,tspan,options) The input arguments are ddefun Handle to a function that evaluates the right side of the differential equations. The function must have the form dydt = ddefun(t,y,Z) where the scalar t is the independent variable, the column vector y is the dependent variable, and Z(:,j) is y ( t – τ j ) for τ j = lags(j). See “Function Handles” in the MATLAB Programming documentation for more information. lags history A vector of constant positive delays τ 1, …, τ k . Handle to a function of t that evaluates the solution y ( t ) for t ≤ t 0 . The function must be of the form S = history(t) where S is a column vector. Alternatively, if y ( t ) is constant, you can specify history as this constant vector. If the current call to dde23 continues a previous integration to t0, use the solution sol from that call as the history. tspan 5-48 The interval of integration as a two-element vector [t0,tf] with t0 < tf. Initial Value Problems for DDEs For more advanced applications, you can specify solver options by passing an input argument options. options Structure of optional parameters that change the default integration properties. You can create the structure options using odeset. The odeset reference page describes the properties you can specify. The output argument sol is a structure created by the solver. It has fields: sol.x Nodes of the mesh selected by dde23 sol.y Approximation to y ( t ) at the mesh points of sol.x sol.yp Approximation to y′ ( t ) at the mesh points of sol.x sol.solver 'dde23' To evaluate the numerical solution at any point from [t0,tf], use deval with the output structure sol as its input. Solving DDE Problems This section uses an example to describe: • Using dde23 to solve initial value problems (IVPs) for delay differential equations (DDEs) • Evaluating the solution at specific points Example: A Straightforward Problem This example illustrates the straightforward formulation, computation, and display of the solution of a system of DDEs with constant delays. The history is constant, which is often the case. The differential equations are y1 ′ ( t ) = y1 ( t – 1 ) y 2 ′ ( t ) = y 1 ( t – 1 ) + y 2 ( t – 0.2 ) y3 ′ ( t ) = y2 ( t ) 5-49 5 Differential Equations The example solves the equations on [0,5] with history y1 ( t ) = 1 y2 ( t ) = 1 y3 ( t ) = 1 for t ≤ 0 . Note The demo ddex1 contains the complete code for this example. To see the code in an editor, click the example name, or type edit ddex1 at the command line. To run the example type ddex1 at the command line. See “DDE Solver Basic Syntax” on page 5-48 for more information. 1 Rewrite the problem as a first-order system. To use dde23, you must rewrite the equations as an equivalent system of first-order differential equations. Do this just as you would when solving IVPs and BVPs for ODEs (see “Examples: Solving Explicit ODE Problems” on page 5-9). However, this example needs no such preparation because it already has the form of a first-order system of equations. 2 Identify the lags. The delays (lags) τ 1, …, τ k are supplied to dde23 as a vector. For the example we could use lags = [1,0.2]; In coding the differential equations, τ j = lags(j). 3 Code the system of first-order DDEs. Once you represent the equations as a first-order system, and specify lags, you can code the equations as a function that dde23 can use. This code represents the system in the function, ddex1de. function dydt = ddex1de(t,y,Z) ylag1 = Z(:,1); ylag2 = Z(:,2); dydt = [ylag1(1) ylag1(1) + ylag2(2) 5-50 Initial Value Problems for DDEs y(2) ]; 4 Code the history function. The history function for this example is function S = ddex1hist(t) S = ones(3,1); 5 Apply the DDE solver. The example now calls dde23 with the functions ddex1de and ddex1hist. sol = dde23(@ddex1de,lags,@ddex1hist,[0,5]); Here the example supplies the interval of integration [0,5] directly. Because the history is constant, we could also call dde23 by sol = dde23(@ddex1de,lags,ones(3,1),[0,5]); 6 View the results. Complete the example by displaying the results. dde23 returns the mesh it selects and the solution there as fields in the solution structure sol. Often, these provide a smooth graph. plot(sol.x,sol.y); title('An example of Wille'' and Baker'); xlabel('time t'); ylabel('solution y'); legend('y_1','y_2','y_3',2) 5-51 5 Differential Equations An example of Wille’ and Baker 200 180 160 y 1 y 2 y3 solution y 140 120 100 80 60 40 20 0 0 1 2 3 4 5 time t Evaluating the Solution at Specific Points The method implemented in dde23 produces a continuous solution over the whole interval of integration [ t 0, t f ] . You can evaluate the approximate solution, S ( t ) , at any point in [ t 0, t f ] using the helper function deval and the structure sol returned by dde23. Sint = deval(sol,tint) The deval function is vectorized. For a vector tint, the ith column of Sint approximates the solution y ( tint(i) ) . Using the output sol from the previous example, this code evaluates the numerical solution at 100 equally spaced points in the interval [0,5] and plots the result. tint = linspace(0,5); Sint = deval(sol,tint); plot(tint,Sint); 5-52 Initial Value Problems for DDEs Discontinuities dde23 can solve problems with discontinuities in the history or discontinuities in coefficients of the equations. It provides properties that enable you to supply locations of known discontinuities and a different initial value. Discontinuity Property Comments At the initial value t = t0 InitialY Generally the initial value y ( t 0 ) is the value S ( t 0 ) returned by the history function, which is to say that the solution is continuous at the initial point. However, if this is not the case, supply a different initial value using the InitialY property. In the history, i.e., the solution at t < t 0 , or in the equation coefficients for t > t0 Jumps Provide the known locations t of the discontinuities in a vector as the value of the Jumps property. State-dependent Events dde23 uses the events function you supply to locate these discontinuities. When dde23 finds such a discontinuity, restart the integration to continue. Specify the solution structure for the current integration as the history for the new integration. dde23 extends each element of the solution structure after each restart so that the final structure provides the solution for the whole interval of integration. If the new problem involves a change in the solution, use the InitialY property to specify the initial value for the new integration. 5-53 5 Differential Equations Example: Cardiovascular Model This example solves a cardiovascular model due to J. T. Ottesen [6]. The equations are integrated over the interval [0,1000]. The situation of interest is when the peripheral pressure R is reduced exponentially from its value of 1.05 to 0.84 beginning at t = 600. This is a problem with one delay, a constant history, and three differential equations with fourteen physical parameters. It has a discontinuity in a low order derivative at t = 600. Note The demo ddex2 contains the complete code for this example. To see the code in an editor, click the example name, or type edit ddex2 at the command line. To run the example type ddex2 at the command line. See “DDE Solver Basic Syntax” on page 5-48 for more information. In ddex2, the fourteen physical parameters are set as fields in a structure p that is shared with nested functions. The function ddex2de for evaluating the equations begins with function dydt = ddex2de(t,y,Z) if t <= 600 p.R = 1.05; else p.R = 0.21 * exp(600-t) + 0.84; end . . . Solve Using the Jumps Property. The peripheral pressure R is a continuous function of t , but it does not have a continuous derivative at t = 600. The example uses the Jumps property to inform dde23 about the location of this discontinuity. opts = ddeset('Jumps',600); After defining the delay tau and the constant history, the call is sol = dde23(@ddex2de,tau,history,[0, 1000],opts); 5-54 Initial Value Problems for DDEs The demo ddex2 plots only the third component, the heart rate, which shows a sharp change at t = 600. Solve by Restarting. The example could have solved this problem by breaking it into two pieces sol = dde23(@ddex2de,tau,history,[0, 600]); sol = dde23(@ddex2de,tau,sol,[600, 1000]); The solution structure sol on the interval [0,600] serves as history for restarting the integration at t = 600. In the second call, dde23 extends sol so that on return the solution is available on the whole interval [0,1000]. That is, after this second return, Sint = deval(sol,[300,900]); evaluates the solution obtained in the first integration at t = 300, and the solution obtained in the second integration at t = 900. When discontinuities occur in low order derivatives at points known in advance, it is better to use the Jumps property. When you use event functions to locate such discontinuities, you must restart the integration at discontinuities. Heart Rate for Baroflex−Feedback Mechanism. 1.7 1.6 1.5 H(t) 1.4 1.3 1.2 1.1 1 0 200 400 600 800 1000 time t 5-55 5 Differential Equations Changing DDE Integration Properties The default integration properties in the DDE solver dde23 are selected to handle common problems. In some cases, you can improve solver performance by overriding these defaults. You do this by supplying dde23 with an options structure that specifies one or more property values. For example, to change the relative error tolerance of dde23 from the default value of 1e-3 to 1e-4, 1 Create an options structure using the function ddeset by entering options = ddeset('RelTol', 1e-4); 2 Pass the options structure to dde23 as follows: sol = dde23(ddefun,lags,history,tspan,options) For a complete description of the available options, see the reference page for ddeset. 5-56 Boundary Value Problems for ODEs Boundary Value Problems for ODEs This section describes how to use MATLAB to solve boundary value problems (BVPs) of ordinary differential equations (ODEs). It provides: • A summary of the BVP functions and examples • An introduction to BVPs • A description of the BVP solver and its syntax • General instructions for solving a BVP • A discussion and examples about using continuation to solve a difficult problem • Instructions for solving singular BVPs • A discussion about changing default integration properties 5-57 5 Differential Equations BVP Function Summary ODE Boundary Value Problem Solver Solver Description bvp4c Solve boundary value problems for ordinary differential equations. BVP Helper Functions Function Description bvpinit Form the initial guess for bvp4c. deval Evaluate the numerical solution using the output of bvp4c. BVP Solver Properties Handling An options structure contains named properties whose values are passed to bvp4c, and which affect problem solution. Use these functions to create, alter, or access an options structure. Function Description bvpset Create/alter the BVP options structure. bvpget Extract properties from options structure created with bvpset. ODE Boundary Value Problem Examples These examples illustrate the kind of problems you can solve using the BVP solver. Click the example name to see the code in an editor. Type the example name at the command line to run it. 5-58 Boundary Value Problems for ODEs Note The Differential Equations Examples browser enables you to view the code for the BVP examples, and also run them. Click on the link to invoke the browser, or type odeexamples('bvp')at the command line. Example Description emdenbvp Emden's equation, a singular BVP fsbvp Falkner-Skan BVP on an infinite interval mat4bvp Fourth eigenfunction of Mathieu’s equation shockbvp Solution with a shock layer near x = 0 twobvp BVP with exactly two solutions threebvp Three-point boundary value problem Additional examples are provided by “Tutorial on Solving BVPs with BVP4C,” available at http://www.mathworks.com/bvp_tutorial. Introduction to Boundary Value ODE Problems The BVP solver is designed to handle systems of ordinary differential equations y′ = f ( x, y ) where x is the independent variable, y is the dependent variable, and y′ represents dy ⁄ dx . See “What Is an Ordinary Differential Equation?” on page 5-4 for general information about ODEs. Using Boundary Conditions to Specify the Solution of Interest In a boundary value problem, the solution of interest satisfies certain boundary conditions. These conditions specify a relationship between the values of the solution at more than one x . In its basic syntax, bvp4c is designed to solve 5-59 5 Differential Equations two-point BVPs, i.e., problems where the solution sought on an interval [ a, b ] must satisfy the boundary conditions g(y(a), y(b)) = 0 Unlike initial value problems, a boundary value problem may not have a solution, may have a finite number of solutions, or may have infinitely many solutions. As an integral part of the process of solving a BVP, you need to provide a guess for the required solution. The quality of this guess can be critical for the solver performance and even for a successful computation. There may be other difficulties when solving BVPs, such as problems imposed on infinite intervals or problems that involve singular coefficients. Often BVPs involve unknown parameters p that have to be determined as part of solving the problem y′ = f ( x, y, p ) g(y(a), y(b), p) = 0 In this case, the boundary conditions must suffice to determine the value of p . Boundary Value Problem Solver This section describes: • The BVP solver, bvp4c • BVP solver basic syntax • BVP solver options The BVP Solver The function bvp4c solves two-point boundary value problems for ordinary differential equations (ODEs). It integrates a system of first-order ordinary differential equations y′ = f ( x, y ) on the interval [ a, b ] , subject to general two-point boundary conditions bc ( y ( a ), y ( b ) ) = 0 5-60 Boundary Value Problems for ODEs It can also accommodate other types of BVP problems, such as those that have any of the following: • Unknown parameters • Singularities in the solutions • Multipoint conditions In this case, the number of boundary conditions must be sufficient to determine the solution and the unknown parameters. For more information, see “Finding Unknown Parameters” on page 5-67. bvp4c produces a solution that is continuous on [ a, b ] and has a continuous first derivative there. You can use the function deval and the output of bvp4c to evaluate the solution at specific points on the interval of integration. bvp4c is a finite difference code that implements the 3-stage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fourth-order accurate uniformly in the interval of integration. Mesh selection and error control are based on the residual of the continuous solution. The collocation technique uses a mesh of points to divide the interval of integration into subintervals. The solver determines a numerical solution by solving a global system of algebraic equations resulting from the boundary conditions, and the collocation conditions imposed on all the subintervals. The solver then estimates the error of the numerical solution on each subinterval. If the solution does not satisfy the tolerance criteria, the solver adapts the mesh and repeats the process. The user must provide the points of the initial mesh as well as an initial approximation of the solution at the mesh points. BVP Solver Basic Syntax The basic syntax of the BVP solver is sol = bvp4c(odefun,bcfun,solinit) 5-61 5 Differential Equations The input arguments are: odefun Handle to a function that evaluates the differential equations. It has the basic form dydx = odefun(x,y) where x is a scalar, and dydx and y are column vectors. See “Function Handles” in the MATLAB Programming documentation for more information. odefun can also accept a vector of unknown parameters and a variable number of known parameters. bcfun Handle to a function that evaluates the residual in the boundary conditions. It has the basic form res = bcfun(ya,yb) where ya and yb are column vectors representing y(a) and y(b), and res is a column vector of the residual in satisfying the boundary conditions. bcfun can also accept a vector of unknown parameters and a variable number of known parameters. solinit Structure with fields x and y: x Ordered nodes of the initial mesh. Boundary conditions are imposed at a = solinit.x(1) and b = solinit.x(end). y Initial guess for the solution with solinit.y(:,i) a guess for the solution at the node solinit.x(i). The structure can have any name, but the fields must be named x and y. It can also contain a vector that provides an initial guess for unknown parameters. You can form solinit with the helper function bvpinit. See the bvpinit reference page for details. The output argument sol is a structure created by the solver. In the basic case the structure has fields x, y, yp, and solver. 5-62 sol.x Nodes of the mesh selected by bvp4c sol.y Approximation to y ( x ) at the mesh points of sol.x Boundary Value Problems for ODEs sol.yp Approximation to y′ ( x ) at the mesh points of sol.x sol.solver 'bvp4c' The structure sol returned by bvp4c contains an additional field if the problem involves unknown parameters: sol.parameters Value of unknown parameters, if present, found by the solver. The function deval uses the output structure sol to evaluate the numerical solution at any point from [a,b]. For information about using deval, see “Evaluating the Solution at Specific Points” on page 5-52. BVP Solver Options For more advanced applications, you can specify solver options by passing an input argument options. options Structure of optional parameters that change the default integration properties. This is the fourth input argument. sol = bvp4c(odefun,bcfun,solinit,options) You can create the structure options using the function bvpset. The bvpset reference page describes the properties you can specify. Solving BVP Problems This section describes: • The process for solving boundary value problems using bvp4c • Finding unknown parameters • Evaluating the solution at specific points 5-63 5 Differential Equations Example: Mathieu’s Equation This example determines the fourth eigenvalue of Mathieu's Equation. It illustrates how to write second-order differential equations as a system of two first-order ODEs and how to use bvp4c to determine an unknown parameter λ . The task is to compute the fourth ( q = 5 ) eigenvalue λ of Mathieu's equation y′′ + ( λ – 2 q cos 2x ) y = 0 Because the unknown parameter λ is present, this second-order differential equation is subject to three boundary conditions y(0) = 1 y′ ( 0 ) = 0 y′ ( π ) = 0 Note The demo mat4bvp contains the complete code for this example. The demo uses nested functions to place all functions required by bvp4c in a single M-file. To run this example type mat4bvp at the command line. See “BVP Solver Basic Syntax” on page 5-61 for more information. 1 Rewrite the problem as a first-order system. To use bvp4c, you must rewrite the equations as an equivalent system of first-order differential equations. Using a substitution y 1 = y and y 2 = y′ , the differential equation is written as a system of two first-order equations y1 ′ = y2 y 2 ′ = – ( λ – 2 q cos 2x ) y 1 Note that the differential equations depend on the unknown parameter λ . The boundary conditions become y1 ( 0 ) – 1 = 0 y2 ( 0 ) = 0 y2( π ) = 0 5-64 Boundary Value Problems for ODEs 2 Code the system of first-order ODEs. Once you represent the equation as a first-order system, you can code it as a function that bvp4c can use. Because there is an unknown parameter, the function must be of the form dydx = odefun(x,y,parameters) The following code represents the system in the function, mat4ode. Variable q is shared with the outer function: function dydx = mat4ode(x,y,lambda) dydx = [ y(2) -(lambda - 2*q*cos(2*x))*y(1) ]; end % End nested function mat4ode See “Finding Unknown Parameters” on page 5-67 for more information about using unknown parameters with bvp4c. 3 Code the boundary conditions function. You must also code the boundary conditions in a function. Because there is an unknown parameter, the function must be of the form res = bcfun(ya,yb,parameters) The code below represents the boundary conditions in the function, mat4bc. function res = mat4bc(ya,yb,lambda) res = [ ya(2) yb(2) ya(1)-1 ]; 4 Create an initial guess. To form the guess structure solinit with bvpinit, you need to provide initial guesses for both the solution and the unknown parameter. The function mat4init provides an initial guess for the solution. mat4init uses y = cos 4x because this function satisfies the boundary conditions and has the correct qualitative behavior (the correct number of sign changes). 5-65 5 Differential Equations function yinit = mat4init(x) yinit = [ cos(4*x) -4*sin(4*x) ]; In the call to bvpinit, the third argument, lambda, provides an initial guess for the unknown parameter λ . lambda = 15; solinit = bvpinit(linspace(0,pi,10),@mat4init,lambda); This example uses @ to pass mat4init as a function handle to bvpinit. Note See the function_handle (@), func2str, and str2func reference pages, and the “Function Handles” chapter of “Programming and Data Types” in the MATLAB documentation for information about function handles. 5 Apply the BVP solver. The mat4bvp example calls bvp4c with the functions mat4ode and mat4bc and the structure solinit created with bvpinit. sol = bvp4c(@mat4ode,@mat4bc,solinit); 6 View the results. Complete the example by displaying the results: a Print the value of the unknown parameter λ found by bvp4c. fprintf('The fourth eigenvalue is approximately %7.3f.\n',... sol.parameters) b Use deval to evaluate the numerical solution at 100 equally spaced points in the interval [ 0, π ] , and plot its first component. This component approximates y ( x ) . xint = linspace(0,pi); Sxint = deval(sol,xint); plot(xint,Sxint(1,:)) axis([0 pi -1 1.1]) title('Eigenfunction of Mathieu''s equation.') xlabel('x') 5-66 Boundary Value Problems for ODEs ylabel('solution y') See “Evaluating the Solution at Specific Points” on page 5-68 for information about using deval. The following plot shows the eigenfunction associated with the final eigenvalue λ = 17.097. Eigenfunction of Mathieu’s equation. 1 0.8 0.6 solution y 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.5 1 1.5 x 2 2.5 3 Finding Unknown Parameters The bvp4c solver can find unknown parameters p for problems of the form y′ = f ( x, y, p ) bc ( y ( a ), y ( b ), p ) = 0 You must provide bvp4c an initial guess for any unknown parameters in the vector solinit.parameters. When you call bvpinit to create the structure solinit, specify the initial guess as a vector in the additional argument parameters. solinit = bvpinit(x,v,parameters) 5-67 5 Differential Equations The bvp4c function arguments odefun and bcfun must each have a third argument. dydx = odefun(x,y,parameters) res = bcfun(ya,yb,parameters) While solving the differential equations, bvp4c adjusts the value of unknown parameters to satisfy the boundary conditions. The solver returns the final values of these unknown parameters in sol.parameters. See “Example: Mathieu’s Equation” on page 5-64. Evaluating the Solution at Specific Points The collocation method implemented in bvp4c produces a C1-continuous solution over the whole interval of integration [ a, b ] . You can evaluate the approximate solution, S ( x ) , at any point in [ a, b ] using the helper function deval and the structure sol returned by bvp4c. Sxint = deval(sol,xint) The deval function is vectorized. For a vector xint, the ith column of Sxint approximates the solution y ( xint(i) ) . Using Continuation to Make a Good Initial Guess To solve a boundary value problem, you need to provide an initial guess for the solution. The quality of your initial guess can be critical to the solver performance, and to being able to solve the problem at all. However, coming up with a sufficiently good guess can be the most challenging part of solving a boundary value problem. Certainly, you should apply the knowledge of the problem's physical origin. Often a problem can be solved as a sequence of relatively simpler problems, i.e., a continuation. This section provides examples that illustrate how to use continuation to: • Solve a difficult BVP. • Verify a solution’s consistent behavior. Example: Using Continuation to Solve a Difficult BVP This example solves the differential equation 2 εy″ + xy′ = επ cos ( πx ) – πx sin ( πx ) 5-68 Boundary Value Problems for ODEs –4 for ε = 10 , on the interval [-1 1], with boundary conditions y(– 1) = – 2 and y(1) = 0 . For 0 < ε < 1 , the solution has a transition layer at x = 0 . Because of this rapid change in the solution for small values of ε , the problem becomes difficult to solve numerically. The example solves the problem as a sequence of relatively simpler problems, i.e., a continuation. The solution of one problem is used as the initial guess for solving the next problem. Note The demo shockbvp contains the complete code for this example. The demo uses nested functions to place all required functions in a single M-file. To run this example type shockbvp at the command line. See “BVP Solver Basic Syntax” on page 5-61 and “Solving BVP Problems” on page 5-63 for more information. Note This problem appears in [1] to illustrate the mesh selection capability of a well established BVP code COLSYS. 1 Code the ODE and boundary condition functions. Code the differential equation and the boundary conditions as functions that bvp4c can use: The code below represents the differential equation and the boundary conditions in the functions shockODE and shockBC. Note that shockODE is vectorized to improve solver performance. The additional parameter ε is represented by e and is shared with the outer function. function dydx = shockODE(x,y) pix = pi*x; dydx = [ y(2,:) -x/e.*y(2,:) - pi^2*cos(pix) - pix/e.*sin(pix) ]; end % End nested function shockODE function res = shockBC(ya,yb) res = [ ya(1)+2 yb(1) ]; end % End nested function shockBC 5-69 5 Differential Equations 2 Provide analytical partial derivatives. For this problem, the solver benefits from using analytical partial derivatives. The code below represents the derivatives in functions shockJac and shockBCJac. function jac = shockJac(x,y) jac = [ 0 1 0 -x/e ]; end % End nested function shockJac function [dBCdya,dBCdyb] = shockBCJac(ya,yb) dBCdya = [ 1 0 0 0 ]; dBCdyb = [ 0 0 1 0 ]; end % End nested function shockBCJac shockJac shares e with the outer function. Tell bvp4c to use these functions to evaluate the partial derivatives by setting the options FJacobian and BCJacobian. Also set 'Vectorized' to 'on' to indicate that the differential equation function shockODE is vectorized. options = bvpset('FJacobian',@shockJac,... 'BCJacobian',@shockBCJac,... 'Vectorized','on'); 3 Create an initial guess. You must provide bvp4c with a guess structure that contains an initial mesh and a guess for values of the solution at the mesh points. A constant guess of y ( x ) ≡ 1 and y′ ( x ) ≡ 0 , and a mesh of five –2 equally spaced points on [-1 1] suffice to solve the problem for ε = 10 . Use bvpinit to form the guess structure. sol = bvpinit([-1 -0.5 0 0.5 1],[1 0]); 4 Use continuation to solve the problem. To obtain the solution for the –4 parameter ε = 10 , the example uses continuation by solving a sequence –2 –3 –4 of problems for ε = 10 , 10 , 10 . The solver bvp4c does not perform continuation automatically, but the code's user interface has been designed to make continuation easy. The code uses the output sol that bvp4c produces for one value of e as the guess in the next iteration. 5-70 Boundary Value Problems for ODEs e = 0.1; for i=2:4 e = e/10; sol = bvp4c(@shockODE,@shockBC,sol,options); end 5 View the results. Complete the example by displaying the final solution plot(sol.x,sol.y(1,:)) axis([-1 1 -2.2 2.2]) title(['There is a shock at x = 0 when \epsilon = '... sprintf('%.e',e) '.']) xlabel('x') ylabel('solution y') There is a shock at x = 0 when ε =1e−004. 2 1.5 1 solution y 0.5 0 −0.5 −1 −1.5 −2 −1 −0.8 −0.6 −0.4 −0.2 0 x 0.2 0.4 0.6 0.8 1 Example: Using Continuation to Verify a Solution’s Consistent Behavior Falkner-Skan BVPs arise from similarity solutions of viscous, incompressible, laminar flow over a flat plate. An example is 5-71 5 Differential Equations 2 f′′′ + ff′′ + β ( 1 – ( f′ ) ) = 0 for β = 0.5 on the interval [0, ∞) with boundary conditions f(0) = 0 , f′(0) = 0 , and f′(∞) = 1 . The BVP cannot be solved on an infinite interval, and it would be impractical to solve it for even a very large finite interval. So, the example tries to solve a sequence of problems posed on increasingly larger intervals to verify the solution’s consistent behavior as the boundary approaches ∞ . The example imposes the infinite boundary condition at a finite point called infinity. The example then uses continuation in this end point to get convergence for increasingly larger values of infinity. It uses bvpinit to extrapolate the solution sol for one value of infinity as an initial guess for the new value of infinity. The plot of each successive solution is superimposed over those of previous solutions so they can easily be compared for consistency. Note The demo fsbvp contains the complete code for this example. The demo uses nested functions to place all required functions in a single M-file. To run this example type fsbvp at the command line. See “BVP Solver Basic Syntax” on page 5-61 and “Solving BVP Problems” on page 5-63 for more information. 1 Code the ODE and boundary condition functions. Code the differential equation and the boundary conditions as functions that bvp4c can use. The problem parameter beta is shared with the outer function. function dfdeta = fsode(eta,f) dfdeta = [ f(2) f(3) -f(1)*f(3) - beta*(1 - f(2)^2) ]; end % End nested function fsode function res = fsbc(f0,finf) res = [f0(1) f0(2) finf(2) - 1]; end % End nested function fsbc 5-72 Boundary Value Problems for ODEs 2 Create an initial guess. You must provide bvp4c with a guess structure that contains an initial mesh and a guess for values of the solution at the mesh points. A crude mesh of five points and a constant guess that satisfies the boundary conditions are good enough to get convergence when infinity = 3. infinity = 3; maxinfinity = 6; solinit = bvpinit(linspace(0,infinity,5),[0 0 1]); 3 Solve on the initial interval. The example obtains the solution for infinity = 3. It then prints the computed value of f′′(0) for comparison with the value reported by Cebeci and Keller [2]. sol = bvp4c(@fsode,@fsbc,solinit); eta = sol.x; f = sol.y; fprintf('\n'); fprintf('Cebeci & Keller report that f''''(0) = 0.92768.\n') fprintf('Value computed using infinity = %g is %7.5f.\n', ... infinity,f(3,1)) The example prints Cebeci & Keller report that f''(0) = 0.92768. Value computed using infinity = 3 is 0.92915. 4 Setup the figure and plot the initial solution. figure plot(eta,f(2,:),eta(end),f(2,end),'o'); axis([0 maxinfinity 0 1.4]); title('Falkner-Skan equation, positive wall shear, \beta = 0.5.') xlabel('\eta') ylabel('df/d\eta') hold on drawnow shg 5-73 5 Differential Equations Falkner−Skan equation, positive wall shear, β = 0.5. 1.2 1 df/dη 0.8 0.6 0.4 0.2 0 0 1 2 3 η 4 5 6 5 Use continuation to solve the problem and plot subsequent solutions. The example then solves the problem for infinity = 4, 5, 6. It uses bvpinit to extrapolate the solution sol for one value of infinity as an initial guess for the next value of infinity. For each iteration, the example prints the computed value of f′′(0) and superimposes a plot of the solution in the existing figure. for Bnew = infinity+1:maxinfinity solinit = bvpinit(sol,[0 Bnew]); % Extend solution to Bnew. sol = bvp4c(@fsode,@fsbc,solinit); eta = sol.x; f = sol.y; fprintf('Value computed using infinity = %g is %7.5f.\n', ... Bnew,f(3,1)) plot(eta,f(2,:),eta(end),f(2,end),'o'); drawnow end 5-74 Boundary Value Problems for ODEs hold off The example prints Value computed using infinity = 4 is 0.92774. Value computed using infinity = 5 is 0.92770. Value computed using infinity = 6 is 0.92770. Note that the values approach 0.92768 as reported by Cebeci and Keller. The superimposed plots confirm the consistency of the solution’s behavior. Falkner−Skan equation, positive wall shear, β = 0.5. 1.2 1 df/dη 0.8 0.6 0.4 0.2 0 0 1 2 3 η 4 5 6 Solving Singular BVPs The function bvp4c solves a class of singular BVPs of the form 1 y′ = --- Sy + f(x, y) x (5-2) 0 = g(y(0), y(b)) 5-75 5 Differential Equations It can also accommodate unknown parameters for problems of the form 1 y′ = --- Sy + f(x, y, p) x 0 = g(y(0), y(b) ,p) Singular problems must be posed on an interval [ 0, b ] with b > 0 . Use bvpset to pass the constant matrix S to bvp4c as the value of the 'SingularTerm' integration property. Boundary conditions at x = 0 must be consistent with the necessary condition for a smooth solution, Sy ( 0 ) = 0 . An initial guess should also satisfy this necessary condition. When you solve a singular BVP using sol = bvp4c(@odefun,@bcfun,solinit,options) bvp4c requires that your function odefun(x,y) return only the value of the f ( x, y ) term in Equation 5-2. Example: Solving a BVP that Has a Singular Term Emden's equation arises in modeling a spherical body of gas. The PDE of the model is reduced by symmetry to the ODE 5 2 y′′ + --- y′ + y = 0 x on an interval [ 0, 1 ] . The coefficient 2 ⁄ x is singular at x = 0 , but symmetry implies the boundary condition y′ ( 0 ) = 0 . With this boundary condition, the term 2 --- y′ ( 0 ) x is well-defined as x approaches 0. For the boundary condition y ( 1 ) = this BVP has the analytical solution 2 –1 ⁄ 2 x y ( x ) = ⎛ 1 + ----- ⎞ ⎝ 3 ⎠ 5-76 3⁄2, Boundary Value Problems for ODEs Note The demo emdenbvp contains the complete code for this example. The demo uses subfunctions to place all required functions in a single M-file. To run this example type emdenbvp at the command line. See “BVP Solver Basic Syntax” on page 5-61 and “Solving BVP Problems” on page 5-63 for more information. 1 Rewrite the problem as a first-order system and identify the singular term. Using a substitution y 1 = y and y 2 = y′ , write the differential equation as a system of two first-order equations y1 ′ = y2 5 2 y 2 ′ = – --- y 2 – y 1 x The boundary conditions become y2( 0 ) = 0 y1 ( 1 ) = 3⁄2 Writing the ODE system in a vector-matrix form y2 1 0 0 y1 = --+ 5 x 0 –2 y y2 ′ –y1 2 y1 ′ the terms of Equation 5-2 are identified as S = 0 0 0 –2 and f ( x, y ) = y2 5 –y1 5-77 5 Differential Equations 2 Code the ODE and boundary condition functions. Code the differential equation and the boundary conditions as functions that bvp4c can use. function dydx = emdenode(x,y) dydx = [ y(2) -y(1)^5 ]; function res = emdenbc(ya,yb) res = [ ya(2) yb(1) - sqrt(3)/2 ]; 3 Setup integration properties. Use the matrix as the value of the 'SingularTerm' integration property. S = [0,0;0,-2]; options = bvpset('SingularTerm',S); 4 Create an initial guess. This example starts with a mesh of five points and a constant guess for the solution. y1 ( x ) ≡ 3 ⁄ 2 y2 ( x ) ≡ 0 Use bvpinit to form the guess structure guess = [sqrt(3)/2;0]; solinit = bvpinit(linspace(0,1,5),guess); 5 Solve the problem. Use the standard bvp4c syntax to solve the problem. sol = bvp4c(@emdenode,@emdenbc,solinit,options); 6 View the results. This problem has an analytical solution 2 –1 ⁄ 2 x y ( x ) = ⎛ 1 + ----- ⎞ ⎝ 3 ⎠ The example evaluates the analytical solution at 100 equally-spaced points and plots it along with the numerical solution computed using bvp4c. x = linspace(0,1); truy = 1 ./ sqrt(1 + (x.^2)/3); plot(x,truy,sol.x,sol.y(1,:),'ro'); title('Emden problem -- BVP with singular term.') 5-78 Boundary Value Problems for ODEs legend('Analytical','Computed'); xlabel('x'); ylabel('solution y'); Emden problem −− BVP with singular term. Analytical Computed solution y 1 0.95 0.9 0 0.2 0.4 0.6 0.8 1 x Changing BVP Integration Properties The default integration properties in the BVP solver bvp4c are selected to handle common problems. In some cases, you can improve solver performance by overriding these defaults. You do this by supplying bvp4c with an options structure that specifies one or more property values. For example, to change the value of the relative error tolerance of bvp4c from the default value of 1e-3 to 1e-4, 1 Create an options structure using the function bvpset by entering options = bvpset('RelTol', 1e-4); 2 Pass the options structure to bvp4c as follows: sol = bvp4c(odefun,bcfun,solinit,options) For a complete description of the available options, see the reference page for bvpset. 5-79 5 Differential Equations Note For other ways to improve solver efficiency, check “Using Continuation to Make a Good Initial Guess” on page 8-68 and the tutorial, “Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c,” available at http://www.mathworks.com/bvp_tutorial. 5-80 Partial Differential Equations Partial Differential Equations This section describes how to use MATLAB to solve initial-boundary value problems for partial differential equations (PDEs). It provides: • A summary of the MATLAB PDE functions and examples • An introduction to PDEs • A description of the PDE solver and its syntax • General instructions for representing a PDE in MATLAB, including an example • Instructions on evaluating the solution at specific points • A discussion about changing default integration properties • An example of solving a real-life problem PDE Function Summary MATLAB PDE Solver This is the MATLAB PDE solver. PDE Initial-Boundary Value Problem Solver pdepe Solve initial-boundary value problems for systems of parabolic and elliptic PDEs in one space variable and time. PDE Helper Function PDE Helper Function pdeval Evaluate the numerical solution of a PDE using the output of pdepe. PDE Examples These examples illustrate some problems you can solve using the MATLAB PDE solver. Click the example name to see the code in an editor. Type the example name at the command line to run it. 5-81 5 Differential Equations Note The Differential Equations Examples browser enables you to view the code for the PDE examples, and also run them. Click on the link to invoke the browser, or type odeexamples('pde')at the command line. Example Description pdex1 Simple PDE that illustrates the straightforward formulation, computation, and plotting of the solution pdex2 Problem that involves discontinuities pdex3 Problem that requires computing values of the partial derivative pdex4 System of two PDEs whose solution has boundary layers at both ends of the interval and changes rapidly for small t pdex5 System of PDEs with step functions as initial conditions Introduction to PDE Problems pdepe solves systems of parabolic and elliptic PDEs in one spatial variable x and time t , of the form ∂u ∂u ∂u ∂u –m ∂ ⎛ m ⎛ ------ x f x, t, u, -------⎞ ⎞ + s ⎛ x, t, u, -------⎞ c ⎛ x, t, u, -------⎞ ------- = x ⎝ ⎝ ⎝ ∂x⎠ ∂t ∂x⎠ ⎠ ∂x⎠ ∂x ⎝ (5-3) The PDEs hold for t 0 ≤ t ≤ t f and a ≤ x ≤ b . The interval [ a, b ] must be finite. m can be 0, 1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If m > 0 , then a ≥ 0 must also hold. In Equation 5-3, f ( x, t, u, ∂u ⁄ ∂x ) is a flux term and s ( x, t, u, ∂u ⁄ ∂x ) is a source term. The flux term must depend on ∂u ⁄ ∂x . The coupling of the partial derivatives with respect to time is restricted to multiplication by a diagonal matrix c ( x, t, u, ∂u ⁄ ∂x ) . The diagonal elements of this matrix are either identically zero or positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a parabolic equation. There must be at least one parabolic equation. An element of c that corresponds to a parabolic 5-82 Partial Differential Equations equation can vanish at isolated values of x if they are mesh points. Discontinuities in c and/or s due to material interfaces are permitted provided that a mesh point is placed at each interface. At the initial time t = t 0 , for all x the solution components satisfy initial conditions of the form u ( x, t 0 ) = u 0 ( x ) (5-4) At the boundary x = a or x = b , for all t the solution components satisfy a boundary condition of the form ∂u p ( x, t, u ) + q ( x, t )f ⎛ x, t, u, -------⎞ = 0 ⎝ ∂x ⎠ (5-5) q ( x, t ) is a diagonal matrix with elements that are either identically zero or never zero. Note that the boundary conditions are expressed in terms of the flux f rather than ∂u ⁄ ∂x . Also, of the two coefficients, only p can depend on u . MATLAB Partial Differential Equation Solver This section describes: • The PDE solver, pdepe • PDE solver basic syntax • Additional PDE solver arguments The PDE Solver The MATLAB PDE solver, pdepe, solves initial-boundary value problems for systems of parabolic and elliptic PDEs in the one space variable x and time t . There must be at least one parabolic equation in the system. The pdepe solver converts the PDEs to ODEs using a second-order accurate spatial discretization based on a fixed set of nodes specified by the user. The discretization method is described in [9]. The time integration is done with ode15s. The pdepe solver exploits the capabilities of ode15s for solving the differential-algebraic equations that arise when Equation 5-3 contains elliptic equations, and for handling Jacobians with a specified sparsity pattern. ode15s changes both the time step and the formula dynamically. 5-83 5 Differential Equations After discretization, elliptic equations give rise to algebraic equations. If the elements of the initial conditions vector that correspond to elliptic equations are not “consistent” with the discretization, pdepe tries to adjust them before beginning the time integration. For this reason, the solution returned for the initial time may have a discretization error comparable to that at any other time. If the mesh is sufficiently fine, pdepe can find consistent initial conditions close to the given ones. If pdepe displays a message that it has difficulty finding consistent initial conditions, try refining the mesh. No adjustment is necessary for elements of the initial conditions vector that correspond to parabolic equations. PDE Solver Basic Syntax The basic syntax of the solver is sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan) Note Correspondences given are to terms used in “Introduction to PDE Problems” on page 5-82. The input arguments are: m Specifies the symmetry of the problem. m can be 0 = slab, 1 = cylindrical, or 2 = spherical. It corresponds to m in Equation 5-3. pdefun Function that defines the components of the PDE. It computes the terms c , f , and s in Equation 5-3, and has the form [c,f,s] = pdefun(x,t,u,dudx) where x and t are scalars, and u and dudx are vectors that approximate the solution u and its partial derivative with respect to x . c, f, and s are column vectors. c stores the diagonal elements of the matrix c . icfun Function that evaluates the initial conditions. It has the form u = icfun(x) When called with an argument x, icfun evaluates and returns the initial values of the solution components at x in the column vector u. 5-84 Partial Differential Equations bcfun Function that evaluates the terms p and q of the boundary conditions. It has the form [pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t) where ul is the approximate solution at the left boundary xl = a and ur is the approximate solution at the right boundary xr = b . pl and ql are column vectors corresponding to p and the diagonal of q evaluated at xl. Similarly, pr and qr correspond to xr. When m > 0 and a = 0 , boundedness of the solution near x = 0 requires that the flux f vanish at a = 0 . pdepe imposes this boundary condition automatically and it ignores values returned in pl and ql. xmesh Vector [x0, x1, ..., xn] specifying the points at which a numerical solution is requested for every value in tspan. x0 and xn correspond to a and b , respectively. Second-order approximation to the solution is made on the mesh specified in xmesh. Generally, it is best to use closely spaced mesh points where the solution changes rapidly. pdepe does not select the mesh in x automatically. You must provide an appropriate fixed mesh in xmesh. The cost depends strongly on the length of xmesh. When m > 0 , it is not necessary to use a fine mesh near x = 0 to account for the coordinate singularity. The elements of xmesh must satisfy x0 < x1 < ... < xn. The length of xmesh must be ≥ 3. tspan Vector [t0, t1, ..., tf] specifying the points at which a solution is requested for every value in xmesh. t0 and tf correspond to t 0 and t f , respectively. pdepe performs the time integration with an ODE solver that selects both the time step and formula dynamically. The solutions at the points specified in tspan are obtained using the natural continuous extension of the integration formulas. The elements of tspan merely specify where you want answers and the cost depends weakly on the length of tspan. The elements of tspan must satisfy t0 < t1 < ... < tf. The length of ≥ 3. tspan must be 5-85 5 Differential Equations The output argument sol is a three-dimensional array, such that: • sol(:,:,k) approximates component k of the solution u . • sol(i,:,k) approximates component k of the solution at time tspan(i) and mesh points xmesh(:). • sol(i,j,k) approximates component k of the solution at time tspan(i) and the mesh point xmesh(j). Additional PDE Solver Arguments For more advanced applications, you can also specify as input arguments solver options and additional parameters that are passed to the PDE functions. options Structure of optional parameters that change the default integration properties. This is the seventh input argument. sol = pdepe(m,pdefun,icfun,bcfun,... xmesh,tspan,options) See “Changing PDE Integration Properties” on page 5-92 for more information. Solving PDE Problems This section describes: • The process for solving PDE problems using the MATLAB solver, pdepe • Evaluating the solution at specific points Example: A Single PDE This example illustrates the straightforward formulation, solution, and plotting of the solution of a single PDE 2 2 ∂u ∂ u π ------ = --------2 ∂t ∂x This equation holds on an interval 0 ≤ x ≤ 1 for times t ≥ 0 . At t = 0 , the solution satisfies the initial condition u ( x, 0 ) = sin πx 5-86 Partial Differential Equations At x = 0 and x = 1 , the solution satisfies the boundary conditions u ( 0, t ) = 0 πe –t ∂u + ------ ( 1, t ) = 0 ∂x Note The demo pdex1 contains the complete code for this example. The demo uses subfunctions to place all functions it requires in a single M-file. To run the demo type pdex1 at the command line. See “PDE Solver Basic Syntax” on page 5-84 for more information. 1 Rewrite the PDE. Write the PDE in the form ∂u ∂u ∂u ∂u –m ∂ ⎛ m ⎛ ------ x f x, t, u, -------⎞ ⎞ + s ⎛ x, t, u, -------⎞ c ⎛ x, t, u, -------⎞ ------- = x ⎝ ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ∂x ∂x ∂t ∂x ∂x⎠ This is the form shown in Equation 5-3 and expected by pdepe. See “Introduction to PDE Problems” on page 5-82 for more information. For this example, the resulting equation is 2 ∂u 0 ∂ 0 ∂u π ------ = x ------ ⎛ x -------⎞ + 0 ∂t ∂x ⎝ ∂x⎠ with parameter m = 0 and the terms ∂u 2 c ⎛ x, t, u, -------⎞ = π ⎝ ∂x⎠ ∂u ∂u f ⎛ x, t, u, -------⎞ = ------⎝ ⎠ ∂x ∂x ∂u s ⎛ x, t, u, -------⎞ = 0 ⎝ ∂x⎠ 5-87 5 Differential Equations 2 Code the PDE. Once you rewrite the PDE in the form shown above (Equation 5-3) and identify the terms, you can code the PDE in a function that pdepe can use. The function must be of the form [c,f,s] = pdefun(x,t,u,dudx) where c, f, and s correspond to the c , f , and s terms. The code below computes c, f, and s for the example problem. function [c,f,s] = pdex1pde(x,t,u,DuDx) c = pi^2; f = DuDx; s = 0; 3 Code the initial conditions function. You must code the initial conditions in a function of the form u = icfun(x) The code below represents the initial conditions in the function pdex1ic. function u0 = pdex1ic(x) u0 = sin(pi*x); 4 Code the boundary conditions function. You must also code the boundary conditions in a function of the form [pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t) The boundary conditions, written in the same form as Equation 5-5, are ∂u u(0, t) + 0 ⋅ ------(0, t) = 0 ∂x at x = 0 and πe –t ∂u + 1 ⋅ ------(1, t) = 0 ∂x at x = 1 The code below evaluates the components p ( x, t, u ) and q ( x, t ) of the boundary conditions in the function pdex1bc. function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) pl = ul; 5-88 Partial Differential Equations ql = 0; pr = pi * exp(-t); qr = 1; In the function pdex1bc, pl and ql correspond to the left boundary conditions ( x = 0 ), and pr and qr correspond to the right boundary condition ( x = 1 ). 5 Select mesh points for the solution. Before you use the MATLAB PDE solver, you need to specify the mesh points ( t, x ) at which you want pdepe to evaluate the solution. Specify the points as vectors t and x. The vectors t and x play different roles in the solver (see “MATLAB Partial Differential Equation Solver” on page 5-83). In particular, the cost and the accuracy of the solution depend strongly on the length of the vector x. However, the computation is much less sensitive to the values in the vector t. This example requests the solution on the mesh produced by 20 equally spaced points from the spatial interval [0,1] and five values of t from the time interval [0,2]. x = linspace(0,1,20); t = linspace(0,2,5); 6 Apply the PDE solver. The example calls pdepe with m = 0, the functions pdex1pde, pdex1ic, and pdex1bc, and the mesh defined by x and t at which pdepe is to evaluate the solution. The pdepe function returns the numerical solution in a three-dimensional array sol, where sol(i,j,k) approximates the kth component of the solution, u k , evaluated at t(i) and x(j). m = 0; sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); This example uses @ to pass pdex1pde, pdex1ic, and pdex1bc as function handles to pdepe. 5-89 5 Differential Equations Note See the function_handle (@), func2str, and str2func reference pages, and the “Function Handles” chapter of “Programming and Data Types” in the MATLAB documentation for information about function handles. 7 View the results. Complete the example by displaying the results: a Extract and display the first solution component. In this example, the solution u has only one component, but for illustrative purposes, the example “extracts” it from the three-dimensional array. The surface plot shows the behavior of the solution. u = sol(:,:,1); surf(x,t,u) title('Numerical solution computed with 20 mesh points') xlabel('Distance x') ylabel('Time t') Numerical solution computed with 20 mesh points. 1 0.8 0.6 0.4 0.2 0 2 1 1.5 0.8 1 0.6 0.4 0.5 0.2 Time t 0 0 Distance x b Display a solution profile at t f , the final value of t . In this example, t f = 5-90 Partial Differential Equations 2. See “Evaluating the Solution at Specific Points” on page 5-91 for more information. c figure plot(x,u(end,:)) title('Solution at t = 2') xlabel('Distance x') ylabel('u(x,2)') Solution at t = 2 0.14 0.12 0.1 u(x,2) 0.08 0.06 0.04 0.02 0 −0.02 0 0.1 0.2 0.3 0.4 0.5 Distance x 0.6 0.7 0.8 0.9 1 Evaluating the Solution at Specific Points After obtaining and plotting the solution above, you might be interested in a solution profile for a particular value of t, or the time changes of the solution at a particular point x. The kth column u(:,k) (of the solution extracted in step 7) contains the time history of the solution at x(k). The jth row u(j,:) contains the solution profile at t(j). Using the vectors x and u(j,:), and the helper function pdeval, you can evaluate the solution u and its derivative ∂u ⁄ ∂x at any set of points xout [uout,DuoutDx] = pdeval(m,x,u(j,:),xout) The example pdex3 uses pdeval to evaluate the derivative of the solution at xout = 0. See pdeval for details. 5-91 5 Differential Equations Changing PDE Integration Properties The default integration properties in the MATLAB PDE solver are selected to handle common problems. In some cases, you can improve solver performance by overriding these defaults. You do this by supplying pdepe with one or more property values in an options structure. sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan,options) Use odeset to create the options structure. Only those options of the underlying ODE solver shown in the following table are available for pdepe. The defaults obtained by leaving off the input argument options are generally satisfactory. “Changing ODE Integration Properties” on page 5-17 tells you how to create the structure and describes the properties. PDE Property Categories Properties Category Property Name Error control RelTol, AbsTol, NormControl Step-size InitialStep, MaxStep Example: Electrodynamics Problem This example illustrates the solution of a system of partial differential equations. The problem is taken from electrodynamics. It has boundary layers at both ends of the interval, and the solution changes rapidly for small t . The PDEs are 2 ∂ u1 ∂u ---------1 = 0.024 ------------ – F(u 1 – u 2) 2 ∂t ∂x 2 ∂ u2 ∂u 2 --------- = 0.170 ------------ + F(u 1 – u 2) 2 ∂t ∂x where F ( y ) = exp ( 5.73y ) – exp ( – 11.46y ) . The equations hold on an interval 0 ≤ x ≤ 1 for times t ≥ 0 . 5-92 Partial Differential Equations The solution u satisfies the initial conditions u 1 ( x, 0 ) ≡ 1 u 2 ( x, 0 ) ≡ 0 and boundary conditions ∂u ---------1 ( 0, t ) ≡ 0 ∂x u 2(0, t) ≡ 0 u 1(1, t) ≡ 1 ∂u 2 --------- ( 1, t ) ≡ 0 ∂x Note The demo pdex4 contains the complete code for this example. The demo uses subfunctions to place all required functions in a single M-file. To run this example type pdex4 at the command line. See “PDE Solver Basic Syntax” on page 5-84 and “Solving PDE Problems” on page 5-86 for more information. 1 Rewrite the PDE. In the form expected by pdepe, the equations are –F ( u1 – u2 ) ∂ u ∂ 0.024 ( ∂u 1 ⁄ ∂x ) + .∗ ----- 1 = -----∂t u ∂x 0.170 ( ∂u ⁄ ∂x ) F ( u1 – u2 ) 1 2 2 1 The boundary conditions on the partial derivatives of u have to be written in terms of the flux. In the form expected by pdepe, the left boundary condition is 0.024 ( ∂u 1 ⁄ ∂x ) 0 1 0 + = .∗ u2 0.170 ( ∂u 2 ⁄ ∂x ) 0 0 5-93 5 Differential Equations and the right boundary condition is u1 – 1 0 + 0 1 .∗ 0.024 ( ∂u 1 ⁄ ∂x ) 0.170 ( ∂u 2 ⁄ ∂x ) = 0 0 2 Code the PDE. After you rewrite the PDE in the form shown above, you can code it as a function that pdepe can use. The function must be of the form [c,f,s] = pdefun(x,t,u,dudx) where c, f, and s correspond to the c , f , and s terms in Equation 5-3. function [c,f,s] = pdex4pde(x,t,u,DuDx) c = [1; 1]; f = [0.024; 0.17] .* DuDx; y = u(1) - u(2); F = exp(5.73*y)-exp(-11.47*y); s = [-F; F]; 3 Code the initial conditions function. The initial conditions function must be of the form u = icfun(x) The code below represents the initial conditions in the function pdex4ic. function u0 = pdex4ic(x); u0 = [1; 0]; 4 Code the boundary conditions function. The boundary conditions functions must be of the form [pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t) The code below evaluates the components p ( x, t, u ) and q ( x, t ) (Equation 5-5) of the boundary conditions in the function pdex4bc. function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t) pl = [0; ul(2)]; ql = [1; 0]; pr = [ur(1)-1; 0]; qr = [0; 1]; 5-94 Partial Differential Equations 5 Select mesh points for the solution. The solution changes rapidly for small t . The program selects the step size in time to resolve this sharp change, but to see this behavior in the plots, output times must be selected accordingly. There are boundary layers in the solution at both ends of [0,1], so mesh points must be placed there to resolve these sharp changes. Often some experimentation is needed to select the mesh that reveals the behavior of the solution. x = [0 0.005 0.01 0.05 0.1 0.2 0.5 0.7 0.9 0.95 0.99 0.995 1]; t = [0 0.005 0.01 0.05 0.1 0.5 1 1.5 2]; 6 Apply the PDE solver. The example calls pdepe with m = 0, the functions pdex4pde, pdex4ic, and pdex4bc, and the mesh defined by x and t at which pdepe is to evaluate the solution. The pdepe function returns the numerical solution in a three-dimensional array sol, where sol(i,j,k) approximates the kth component of the solution, u k , evaluated at t(i) and x(j). m = 0; sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t); 7 View the results. The surface plots show the behavior of the solution components. u1 = sol(:,:,1); u2 = sol(:,:,2); figure surf(x,t,u1) title('u1(x,t)') xlabel('Distance x') ylabel('Time t') 5-95 5 Differential Equations u1(x,t) 1 0.8 0.6 0.4 0.2 0 2 1 1.5 0.8 1 0.6 0.4 0.5 0.2 Time t figure surf(x,t,u2) title('u2(x,t)') xlabel('Distance x') ylabel('Time t') 5-96 0 0 Distance x Partial Differential Equations u2(x,t) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 1 1.5 0.8 1 0.6 0.4 0.5 0.2 Time t 0 0 Distance x 5-97 5 Differential Equations Selected Bibliography [1] Ascher, U., R. Mattheij, and R. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, PA, 1995, p. 372. [2] Cebeci, T. and H. B. Keller, “Shooting and Parallel Shooting Methods for Solving the Falkner-Skan Boundary-layer Equation,” J. Comp. Phys., Vol. 7, 1971, pp. 289-300. [3] Hairer, E., and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991, pp. 5-8. [4] Hindmarsh, A. C., “LSODE and LSODI, Two New Initial Value Ordinary Differential Equation Solvers,” SIGNUM Newsletter, Vol. 15, 1980, pp. 10-11. [5] Hindmarsh, A. C., and G. D. Byrne, “Applications of EPISODE: An Experimental Package for the Integration of Ordinary Differential Equations,” Numerical Methods for Differential Systems, L. Lapidus and W. E. Schiesser eds., Academic Press, Orlando, FL, 1976, pp 147-166. [6] Ottesen, J. T., “Modelling of the Baroflex-Feedback Mechanism with Time-Delay,” J. Math. Biol., Vol. 36, 1997. [7] Shampine, L. F., Numerical Solution of Ordinary Differential Equations, Chapman & Hall Mathematics, 1994. [8] Shampine, L. F., and M. K. Gordon, Computer Solution of Ordinary Differential Equations, W.H. Freeman & Co., 1975. [9] Skeel, R. D. and M. Berzins, “A Method for the Spatial Discretization of Parabolic Equations in One Space Variable,” SIAM Journal on Scientific and Statistical Computing, Vol. 11, 1990, pp.1-32. 5-98 6 Sparse Matrices Function Summary (p. 6-2) A summary of the sparse matrix functions Introduction (p. 6-5) An introduction to sparse matrices in MATLAB Viewing Sparse Matrices (p. 6-12) How to obtain quantitative and graphical information about sparse matrices Adjacency Matrices and Graphs (p. 6-16) Using adjacency matrices to illustrate sparse matrices Sparse Matrix Operations (p. 6-24) A discussion of functions that perform operations specific to sparse matrices Selected Bibliography (p. 6-40) Published materials that support concepts described in this chapter 6 Sparse Matrices Function Summary The sparse matrix functions are located in the MATLAB sparfun directory. Function Summary Category Function Description Elementary sparse matrices speye Sparse identity matrix. sprand Sparse uniformly distributed random matrix. sprandn Sparse normally distributed random matrix. sprandsym Sparse random symmetric matrix. spdiags Sparse matrix formed from diagonals. sparse Create sparse matrix. full Convert sparse matrix to full matrix. find Find indices of nonzero elements. spconvert Import from sparse matrix external format. nnz Number of nonzero matrix elements. nonzeros Nonzero matrix elements. nzmax Amount of storage allocated for nonzero matrix elements. spones Replace nonzero sparse matrix elements with ones. spalloc Allocate space for sparse matrix. issparse True for sparse matrix. spfun Apply function to nonzero matrix elements. spy Visualize sparsity pattern. Full to sparse conversion Working with sparse matrices 6-2 Function Summary Function Summary (Continued) Category Function Description Graph theory gplot Plot graph, as in “graph theory.” etree Elimination tree. etreeplot Plot elimination tree. treelayout Lay out tree or forest. treeplot Plot picture of tree. colamd Column approximate minimum degree permutation. symamd Symmetric approximate minimum degree permutation. symrcm Symmetric reverse Cuthill-McKee permutation. colperm Column permutation. randperm Random permutation. dmperm Dulmage-Mendelsohn permutation. eigs A few eigenvalues. svds A few singular values. luinc Incomplete LU factorization. cholinc Incomplete Cholesky factorization. normest Estimate the matrix 2-norm. condest 1-norm condition number estimate. sprank Structural rank. bicg BiConjugate Gradients Method. bicgstab BiConjugate Gradients Stabilized Method. cgs Conjugate Gradients Squared Method. Reordering algorithms Linear algebra Linear equations (iterative methods) 6-3 6 Sparse Matrices Function Summary (Continued) Category Miscellaneous 6-4 Function Description gmres Generalized Minimum Residual Method. lsqr LSQR implementation of Conjugate Gradients on the Normal Equations. minres Minimum Residual Method. pcg Preconditioned Conjugate Gradients Method. qmr Quasi-Minimal Residual Method. symmlq Symmetric LQ method spaugment Form least squares augmented system. spparms Set parameters for sparse matrix routines. symbfact Symbolic factorization analysis. Introduction Introduction Sparse matrices are a special class of matrices that contain a significant number of zero-valued elements. This property allows MATLAB to: • Store only the nonzero elements of the matrix, together with their indices. • Reduce computation time by eliminating operations on zero elements. This section provides information about: • Sparse matrix storage • General storage information • Creating sparse matrices • Importing sparse matrices Sparse Matrix Storage For full matrices, MATLAB stores internally every matrix element. Zero-valued elements require the same amount of storage space as any other matrix element. For sparse matrices, however, MATLAB stores only the nonzero elements and their indices. For large matrices with a high percentage of zero-valued elements, this scheme significantly reduces the amount of memory required for data storage. MATLAB uses a compressed column, or Harwell-Boeing, format for storing matrices. This method uses three arrays internally to store sparse matrices with real elements. Consider an m-by-n sparse matrix with nnz nonzero entries stored in arrays of length nzmax: • The first array contains all the nonzero elements of the array in floating-point format. The length of this array is equal to nzmax. • The second array contains the corresponding integer row indices for the nonzero elements stored in the first nnz entries. This array also has length equal to nzmax. • The third array contains n integer pointers to the start of each column in the other arrays and an additional pointer that marks the end of those arrays. The length of the third array is n+1. 6-5 6 Sparse Matrices This matrix requires storage for nzmax floating-point numbers and nzmax+n+1 integers. At 8 bytes per floating-point number and 4 bytes per integer, the total number of bytes required to store a sparse matrix is 8*nzmax + 4*(nzmax+n+1) Note that the storage requirement depends upon nzmax and the number of columns, n. The memory required to store a sparse matrix containing a large number of rows but having few columns is much less that the memory required to store the transpose of this matrix: S1 = spalloc(2^20,2,1); S2 = spalloc(2,2^20,1); whos Name S1 S2 Size Bytes 1048576x2 2x1048576 24 4194320 Class double array (sparse) double array (sparse) Grand total is 2 elements using 4194344 bytes Sparse matrices with complex elements are also possible. In this case, MATLAB uses a fourth array with nnz floating-point elements to store the imaginary parts of the nonzero elements. An element is considered nonzero if either its real or imaginary part is nonzero. General Storage Information The whos command provides high-level information about matrix storage, including size and storage class. For example, this whos listing shows information about sparse and full versions of the same matrix. whos Name M_full M_sparse Size 1100x1100 1100x1100 Bytes 9680000 4404 Class double array sparse array Grand total is 1210000 elements using 9684404 bytes 6-6 Introduction Notice that the number of bytes used is much less in the sparse case, because zero-valued elements are not stored. In this case, the density of the sparse matrix is 4404/9680000, or approximately .00045%. Creating Sparse Matrices MATLAB never creates sparse matrices automatically. Instead, you must determine if a matrix contains a large enough percentage of zeros to benefit from sparse techniques. The density of a matrix is the number of non-zero elements divided by the total number of matrix elements. Matrices with very low density are often good candidates for use of the sparse format. Converting Full to Sparse You can convert a full matrix to sparse storage using the sparse function with a single argument. S = sparse(A) For example A = [ 0 0 1 0 0 2 3 0 0 0 0 4 5 0 0 0]; S = sparse(A) produces S = (3,1) (2,2) (3,2) (4,3) (1,4) 1 2 3 4 5 The printed output lists the nonzero elements of S, together with their row and column indices. The elements are sorted by columns, reflecting the internal data structure. 6-7 6 Sparse Matrices You can convert a sparse matrix to full storage using the full function, provided the matrix order is not too large. For example A = full(S) reverses the example conversion. Converting a full matrix to sparse storage is not the most frequent way of generating sparse matrices. If the order of a matrix is small enough that full storage is possible, then conversion to sparse storage rarely offers significant savings. Creating Sparse Matrices Directly You can create a sparse matrix from a list of nonzero elements using the sparse function with five arguments. S = sparse(i,j,s,m,n) i and j are vectors of row and column indices, respectively, for the nonzero elements of the matrix. s is a vector of nonzero values whose indices are specified by the corresponding (i,j) pairs. m is the row dimension for the resulting matrix, and n is the column dimension. The matrix S of the previous example can be generated directly with S = sparse([3 2 3 4 1],[1 2 2 3 4],[1 2 3 4 5],4,4) S = (3,1) (2,2) (3,2) (4,3) (1,4) 1 2 3 4 5 The sparse command has a number of alternate forms. The example above uses a form that sets the maximum number of nonzero elements in the matrix to length(s). If desired, you can append a sixth argument that specifies a larger maximum, allowing you to add nonzero elements later without reallocating the sparse matrix. Example: Generating a Second Difference Operator The matrix representation of the second difference operator is a good example of a sparse matrix. It is a tridiagonal matrix with -2s on the diagonal and 1s on 6-8 Introduction the super- and subdiagonal. There are many ways to generate it – here’s one possibility. D = sparse(1:n,1:n,-2∗ones(1,n),n,n); E = sparse(2:n,1:n-1,ones(1,n-1),n,n); S = E+D+E' For n = 5, MATLAB responds with S = (1,1) (2,1) (1,2) (2,2) (3,2) (2,3) (3,3) (4,3) (3,4) (4,4) (5,4) (4,5) (5,5) -2 1 1 -2 1 1 -2 1 1 -2 1 1 -2 Now F = full(S) displays the corresponding full matrix. F = full(S) F = -2 1 0 0 0 1 -2 1 0 0 0 1 -2 1 0 0 0 1 -2 1 0 0 0 1 -2 Creating Sparse Matrices from Their Diagonal Elements Creating sparse matrices based on their diagonal elements is a common operation, so the function spdiags handles this task. Its syntax is S = spdiags(B,d,m,n) 6-9 6 Sparse Matrices To create an output matrix S of size m-by-n with elements on p diagonals: • B is a matrix of size min(m,n)-by-p. The columns of B are the values to populate the diagonals of S. • d is a vector of length p whose integer elements specify which diagonals of S to populate. That is, the elements in column j of B fill the diagonal specified by element j of d. Note If a column of B is longer than the diagonal it’s replacing, super-diagonals are taken from the lower part of the column of B, and sub-diagonals are taken from the upper part of the column of B. As an example, consider the matrix B and the vector d. B = [ 41 52 63 74 11 22 33 44 0 0 13 24 ]; d = [-3 0 2]; Use these matrices to create a 7-by-4 sparse matrix A. A = spdiags(B,d,7,4) A = (1,1) (4,1) (2,2) (5,2) (1,3) (3,3) (6,3) (2,4) 6-10 11 41 22 52 13 33 63 24 Introduction (4,4) (7,4) 44 74 In its full form, A looks like this. full(A) ans = 11 0 0 41 0 0 0 0 22 0 0 52 0 0 13 0 33 0 0 63 0 0 24 0 44 0 0 74 spdiags can also extract diagonal elements from a sparse matrix, or replace matrix diagonal elements with new values. Type help spdiags for details. Importing Sparse Matrices from Outside MATLAB You can import sparse matrices from computations outside MATLAB. Use the spconvert function in conjunction with the load command to import text files containing lists of indices and nonzero elements. For example, consider a three-column text file T.dat whose first column is a list of row indices, second column is a list of column indices, and third column is a list of nonzero values. These statements load T.dat into MATLAB and convert it into a sparse matrix S: load T.dat S = spconvert(T) The save and load commands can also process sparse matrices stored as binary data in MAT-files. 6-11 6 Sparse Matrices Viewing Sparse Matrices MATLAB provides a number of functions that let you get quantitative or graphical information about sparse matrices. This section provides information about: • Obtaining information about nonzero elements • Viewing graphs of sparse matrices • Finding indices and values of nonzero elements Information About Nonzero Elements There are several commands that provide high-level information about the nonzero elements of a sparse matrix: • nnz returns the number of nonzero elements in a sparse matrix. • nonzeros returns a column vector containing all the nonzero elements of a sparse matrix. • nzmax returns the amount of storage space allocated for the nonzero entries of a sparse matrix. To try some of these, load the supplied sparse matrix west0479, one of the Harwell-Boeing collection. load west0479 whos Name west0479 Size Bytes Class 479x479 24576 sparse array This matrix models an eight-stage chemical distillation column. Try these commands. nnz(west0479) ans = 1887 format short e west0479 6-12 Viewing Sparse Matrices west0479 = (25,1) (31,1) (87,1) (26,2) (31,2) (88,2) (27,3) (31,3) (89,3) (28,4) . . . 1.0000e+00 -3.7648e-02 -3.4424e-01 1.0000e+00 -2.4523e-02 -3.7371e-01 1.0000e+00 -3.6613e-02 -8.3694e-01 1.3000e+02 nonzeros(west0479); ans = 1.0000e+00 -3.7648e-02 -3.4424e-01 1.0000e+00 -2.4523e-02 -3.7371e-01 1.0000e+00 -3.6613e-02 -8.3694e-01 1.3000e+02 . . . Note Use Ctrl+C to stop the nonzeros listing at any time. 6-13 6 Sparse Matrices Note that initially nnz has the same value as nzmax by default. That is, the number of nonzero elements is equivalent to the number of storage locations allocated for nonzeros. However, MATLAB does not dynamically release memory if you zero out additional array elements. Changing the value of some matrix elements to zero changes the value of nnz, but not that of nzmax. However, you can add as many nonzero elements to the matrix as desired. You are not constrained by the original value of nzmax. Viewing Sparse Matrices Graphically It is often useful to use a graphical format to view the distribution of the nonzero elements within a sparse matrix. The MATLAB spy function produces a template view of the sparsity structure, where each point on the graph represents the location of a nonzero array element. For example, spy(west0479) 0 50 100 150 200 250 300 350 400 450 0 6-14 100 200 300 nz = 1887 400 Viewing Sparse Matrices The find Function and Sparse Matrices For any matrix, full or sparse, the find function returns the indices and values of nonzero elements. Its syntax is [i,j,s] = find(S) find returns the row indices of nonzero values in vector i, the column indices in vector j, and the nonzero values themselves in the vector s. The example below uses find to locate the indices and values of the nonzeros in a sparse matrix. The sparse function uses the find output, together with the size of the matrix, to recreate the matrix. [i,j,s] = find(S) [m,n] = size(S) S = sparse(i,j,s,m,n) 6-15 6 Sparse Matrices Adjacency Matrices and Graphs This section includes: • An introduction to adjacency matrices • Instructions for graphing adjacency matrices with gplot • A Bucky ball example, including information about using spy plots to illustrate fill-in and distance • An airflow model example Introduction to Adjacency Matrices The formal mathematical definition of a graph is a set of points, or nodes, with specified connections between them. An economic model, for example, is a graph with different industries as the nodes and direct economic ties as the connections. The computer software industry is connected to the computer hardware industry, which, in turn, is connected to the semiconductor industry, and so on. This definition of a graph lends itself to matrix representation. The adjacency matrix of an undirected graph is a matrix whose (i,j)th and (j,i)th entries are 1 if node i is connected to node j, and 0 otherwise. For example, the adjacency matrix for a diamond-shaped graph looks like 1 A= 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 4 2 3 Since most graphs have relatively few connections per node, most adjacency matrices are sparse. The actual locations of the nonzero elements depend on how the nodes are numbered. A change in the numbering leads to permutation 6-16 Adjacency Matrices and Graphs of the rows and columns of the adjacency matrix, which can have a significant effect on both the time and storage requirements for sparse matrix computations. Graphing Using Adjacency Matrices The MATLAB gplot function creates a graph based on an adjacency matrix and a related array of coordinates. To try gplot, create the adjacency matrix shown above by entering A = [0 1 0 1; 1 0 1 0; 0 1 0 1; 1 0 1 0]; The columns of gplot’s coordinate array contain the Cartesian coordinates for the corresponding node. For the diamond example, create the array by entering xy = [1 3; 2 1; 3 3; 2 5]; This places the first node at location (1,3), the second at location (2,1), the third at location (3,3), and the fourth at location (2,5). To view the resulting graph, enter gplot(A,xy) The Bucky Ball One interesting construction for graph analysis is the Bucky ball. This is composed of 60 points distributed on the surface of a sphere in such a way that the distance from any point to its nearest neighbors is the same for all the points. Each point has exactly three neighbors. The Bucky ball models four different physical objects: • The geodesic dome popularized by Buckminster Fuller • The C60 molecule, a form of pure carbon with 60 atoms in a nearly spherical configuration • In geometry, the truncated icosahedron • In sports, the seams in a soccer ball The Bucky ball adjacency matrix is a 60-by-60 symmetric matrix B. B has three nonzero elements in each row and column, for a total of 180 nonzero values. This matrix has important applications related to the physical objects listed earlier. For example, the eigenvalues of B are involved in studying the chemical properties of C60. 6-17 6 Sparse Matrices To obtain the Bucky ball adjacency matrix, enter B = bucky; At order 60, and with a density of 5%, this matrix does not require sparse techniques, but it does provide an interesting example. You can also obtain the coordinates of the Bucky ball graph using [B,v] = bucky; This statement generates v, a list of xyz-coordinates of the 60 points in 3-space equidistributed on the unit sphere. The function gplot uses these points to plot the Bucky ball graph. gplot(B,v) axis equal 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −0.5 0 0.5 1 It is not obvious how to number the nodes in the Bucky ball so that the resulting adjacency matrix reflects the spherical and combinatorial symmetries of the graph. The numbering used by bucky.m is based on the pentagons inherent in the ball’s structure. 6-18 Adjacency Matrices and Graphs The vertices of one pentagon are numbered 1 through 5, the vertices of an adjacent pentagon are numbered 6 through 10, and so on. The picture on the following page shows the numbering of half of the nodes (one hemisphere); the numbering of the other hemisphere is obtained by a reflection about the equator. Use gplot to produce a graph showing half the nodes. You can add the node numbers using a for loop. k = 1:30; gplot(B(k,k),v); axis square for j = 1:30, text(v(j,1),v(j,2), int2str(j)); end 1 14 15 13 0.8 18 17 11 12 0.6 0.4 19 16 2 20 0.2 10 3 9 0 1 −0.2 22 6 4 8 5 −0.4 23 7 21 −0.6 26 24 30 25 −0.8 27 −1 −1 −0.8 −0.6 −0.4 −0.2 0 29 28 0.2 0.4 0.6 0.8 1 To view a template of the nonzero locations in the Bucky ball’s adjacency matrix, use the spy function: spy(B) 6-19 6 Sparse Matrices 0 10 20 30 40 50 60 0 10 20 30 nz = 180 40 50 60 The node numbering that this model uses generates a spy plot with 12 groups of five elements, corresponding to the 12 pentagons in the structure. Each node is connected to two other nodes within its pentagon and one node in some other pentagon. Since the nodes within each pentagon have consecutive numbers, most of the elements in the first super- and sub-diagonals of B are nonzero. In addition, the symmetry of the numbering about the equator is apparent in the symmetry of the spy plot about the antidiagonal. Graphs and Characteristics of Sparse Matrices Spy plots of the matrix powers of B illustrate two important concepts related to sparse matrix operations, fill-in and distance. spy plots help illustrate these concepts. spy(B^2) spy(B^3) spy(B^4) spy(B^8) 6-20 Adjacency Matrices and Graphs 0 0 10 10 20 20 30 30 40 40 50 50 60 60 0 20 40 nz = 420 60 0 0 10 10 20 20 30 30 40 40 50 50 60 0 20 40 nz = 780 60 0 20 40 nz = 3540 60 60 0 20 40 nz = 1380 60 Fill-in is generated by operations like matrix multiplication. The product of two or more matrices usually has more nonzero entries than the individual terms, and so requires more storage. As p increases, B^p fills in and spy(B^p) gets more dense. The distance between two nodes in a graph is the number of steps on the graph necessary to get from one node to the other. The spy plot of the p-th power of B shows the nodes that are a distance p apart. As p increases, it is possible to get to more and more nodes in p steps. For the Bucky ball, B^8 is almost completely full. Only the antidiagonal is zero, indicating that it is possible to get from any node to any other node, except the one directly opposite it on the sphere, in eight steps. 6-21 6 Sparse Matrices An Airflow Model A calculation performed at NASA’s Research Institute for Applications of Computer Science involves modeling the flow over an airplane wing with two trailing flaps. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 In a two-dimensional model, a triangular grid surrounds a cross section of the wing and flaps. The partial differential equations are nonlinear and involve several unknowns, including hydrodynamic pressure and two components of velocity. Each step of the nonlinear iteration requires the solution of a sparse linear system of equations. Since both the connectivity and the geometric location of the grid points are known, the gplot function can produce the graph shown above. In this example, there are 4253 grid points, each of which is connected to between 3 and 9 others, for a total of 28831 nonzeros in the matrix, and a density equal to 0.0016. This spy plot shows that the node numbering yields a definite band structure. 6-22 Adjacency Matrices and Graphs The Laplacian of the mesh. 0 ....................................... 500 1000 1500 2000 2500 3000 3500 4000 . . .. . .............................................. . ..... .... . . . ... . . ............................... . . . . . . . . . . . . . . . ..................................... .. .... . .... . .. .. .......... ........................................... . . ..... . .............................................................. .. ... .. . . . ............................. .. ... . .. . . ......................... .. . . .. ........... . . ....................................... . . . .. ........ .......................................... ..... .. . .. . . .. ....... ................................... .. . . . ............................... . . .. ... ................................................... . . . . .. . ...................................... . ...... .... ........................................... . . ......................... . ........................................ ................... . ................................... .. . . ................................. . ... . . . . . .. . . .. . . . .. ..................................................... ...... . . . . . . ................................................ . . . ... . ............................................ . . . . ... . .. . . .. ... .. ................................................... .. ......... . . .. .. . . ..... . ..................................................... ............ .... . . .. ......................... .. . ...... ............................................................... . . . . . .. .. .. .. .......................... . . . . . . . . .. .. .. . . ................................................................ . .. . . . ..................................................... . .. . . ....................................................... .. . .. .. . .......... .... . . . .. . . . . . ... .............................................. . ...... .. .. . .. . . . . . . . ..... . . . . .... ...................................................... . . . . . .. .................................................. . .. . ................... . . . . ... ... .............................. . . .. . . ... . .. . . ................................................... ...... .. . .. .............................................. . . ............ . . .. . .. .. .......................................................... . ... ............................ . . . ........................................................ ... . .............................. . . . ................ ... 0 1000 2000 3000 4000 nz = 28831 6-23 6 Sparse Matrices Sparse Matrix Operations Most of the MATLAB standard mathematical functions work on sparse matrices just as they do on full matrices. In addition, MATLAB provides a number of functions that perform operations specific to sparse matrices. This section discusses: • Computational considerations • Standard mathematical operations • Permutation and reordering • Factorization • Simultaneous linear equations • Eigenvalues and singular values Computational Considerations The computational complexity of sparse operations is proportional to nnz, the number of nonzero elements in the matrix. Computational complexity also depends linearly on the row size m and column size n of the matrix, but is independent of the product m*n, the total number of zero and nonzero elements. The complexity of fairly complicated operations, such as the solution of sparse linear equations, involves factors like ordering and fill-in, which are discussed in the previous section. In general, however, the computer time required for a sparse matrix operation is proportional to the number of arithmetic operations on nonzero quantities. Standard Mathematical Operations Sparse matrices propagate through computations according to these rules: • Functions that accept a matrix and return a scalar or vector always produce output in full storage format. For example, the size function always returns a full vector, whether its input is full or sparse. • Functions that accept scalars or vectors and return matrices, such as zeros, ones, rand, and eye, always return full results. This is necessary to avoid introducing sparsity unexpectedly. The sparse analog of zeros(m,n) is simply sparse(m,n). The sparse analogs of rand and eye are sprand and speye, respectively. There is no sparse analog for the function ones. 6-24 Sparse Matrix Operations • Unary functions that accept a matrix and return a matrix or vector preserve the storage class of the operand. If S is a sparse matrix, then chol(S) is also a sparse matrix, and diag(S) is a sparse vector. Columnwise functions such as max and sum also return sparse vectors, even though these vectors may be entirely nonzero. Important exceptions to this rule are the sparse and full functions. • Binary operators yield sparse results if both operands are sparse, and full results if both are full. For mixed operands, the result is full unless the operation preserves sparsity. If S is sparse and F is full, then S+F, S*F, and F\S are full, while S.*F and S&F are sparse. In some cases, the result might be sparse even though the matrix has few zero elements. • Matrix concatenation using either the cat function or square brackets produces sparse results for mixed operands. • Submatrix indexing on the right side of an assignment preserves the storage format of the operand unless the result is a scalar. T = S(i,j) produces a sparse result if S is sparse and either i or j is a vector. It produces a full scalar if both i and j are scalars. Submatrix indexing on the left, as in T(i,j) = S, does not change the storage format of the matrix on the left. Permutation and Reordering A permutation of the rows and columns of a sparse matrix S can be represented in two ways: • A permutation matrix P acts on the rows of S as P*S or on the columns as S*P'. • A permutation vector p, which is a full vector containing a permutation of 1:n, acts on the rows of S as S(p,:), or on the columns as S(:,p). For example, the statements p I P e S = = = = = [1 3 4 2 5] eye(5,5); I(p,:); ones(4,1); diag(11:11:55) + diag(e,1) + diag(e,-1) 6-25 6 Sparse Matrices produce p = 1 3 4 2 5 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 11 1 0 0 0 1 22 1 0 0 0 1 33 1 0 0 0 1 44 1 0 0 0 1 55 P = S = You can now try some permutations using the permutation vector p and the permutation matrix P. For example, the statements S(p,:) and P*S produce ans = 11 0 0 1 0 1 1 0 22 0 0 33 1 1 0 0 1 44 0 1 0 0 1 0 55 Similarly, S(:,p) and S*P' produce ans = 11 1 0 0 0 6-26 0 1 33 1 0 0 0 1 44 1 1 22 1 0 0 0 0 0 1 55 Sparse Matrix Operations If P is a sparse matrix, then both representations use storage proportional to n and you can apply either to S in time proportional to nnz(S). The vector representation is slightly more compact and efficient, so the various sparse matrix permutation routines all return full row vectors with the exception of the pivoting permutation in LU (triangular) factorization, which returns a matrix compatible with earlier versions of MATLAB. To convert between the two representations, let I = speye(n) be an identity matrix of the appropriate size. Then, P = I(p,:) P' = I(:,p) p = (1:n)*P' p = (P*(1:n)')' The inverse of P is simply R = P'. You can compute the inverse of p with r(p) = 1:n. r(p) = 1:5 r = 1 4 2 3 5 Reordering for Sparsity Reordering the columns of a matrix can often make its LU or QR factors sparser. Reordering the rows and columns can often make its Cholesky, factors sparser. The simplest such reordering is to sort the columns by nonzero count. This is sometimes a good reordering for matrices with very irregular structures, especially if there is great variation in the nonzero counts of rows or columns. The function p = colperm(S) computes this column-count permutation. The colperm M-file has only a single line. [ignore,p] = sort(full(sum(spones(S)))); This line performs these steps: 1 The inner call to spones creates a sparse matrix with ones at the location of every nonzero element in S. 6-27 6 Sparse Matrices 2 The sum function sums down the columns of the matrix, producing a vector that contains the count of nonzeros in each column. 3 full converts this vector to full storage format. 4 sort sorts the values in ascending order. The second output argument from sort is the permutation that sorts this vector. Reordering to Reduce Bandwidth The reverse Cuthill-McKee ordering is intended to reduce the profile or bandwidth of the matrix. It is not guaranteed to find the smallest possible bandwidth, but it usually does. The function symrcm(A) actually operates on the nonzero structure of the symmetric matrix A + A', but the result is also useful for asymmetric matrices. This ordering is useful for matrices that come from one-dimensional problems or problems that are in some sense “long and thin.” Approximate Minimum Degree Ordering The degree of a node in a graph is the number of connections to that node. This is the same as the number of off-diagonal nonzero elements in the corresponding row of the adjacency matrix. The approximate minimum degree algorithm generates an ordering based on how these degrees are altered during Gaussian elimination or Cholesky factorization. It is a complicated and powerful algorithm that usually leads to sparser factors than most other orderings, including column count and reverse Cuthill-McKee. Because the keeping track of the degree of each node is very time-consuming, the approximate minimum degree algorithm uses an approximation to the degree, rather than the exact degree. The following MATLAB functions implement the approximate minimum degree algorithm: • symamd — Use with symmetric matrices • colamd — Use with nonsymmetric matrices and symmetric matrices of the form A*A' or A'*A. See “Reordering and Factorization” on page 6-30 for an example using symamd. You can change various parameters associated with details of the algorithms using the spparms function. 6-28 Sparse Matrix Operations For details on the algorithms used by colamd and symamd, see [5]. The approximate degree the algorithms use is based on [1]. Factorization This section discusses four important factorization techniques for sparse matrices: • LU, or triangular, factorization • Cholesky factorization • QR, or orthogonal, factorization • Incomplete factorizations LU Factorization If S is a sparse matrix, the following command returns three sparse matrices L, U, and P such that P*S = L*U. [L,U,P] = lu(S) lu obtains the factors by Gaussian elimination with partial pivoting. The permutation matrix P has only n nonzero elements. As with dense matrices, the statement [L,U] = lu(S) returns a permuted unit lower triangular matrix and an upper triangular matrix whose product is S. By itself, lu(S) returns L and U in a single matrix without the pivot information. The three-output syntax [L,U,P] = lu(S) selects P via numerical partial pivoting, but does not pivot to improve sparsity in the LU factors. On the other hand, the four-output syntax [L,U,P,Q]=lu(S) selects P via threshold partial pivoting, and selects P and Q to improve sparsity in the LU factors. You can control pivoting in sparse matrices using lu(S,thresh) where thresh is a pivot threshold in [0,1]. Pivoting occurs when the diagonal entry in a column has magnitude less than thresh times the magnitude of any 6-29 6 Sparse Matrices sub-diagonal entry in that column. thresh = 0 forces diagonal pivoting. thresh = 1 is the default. MATLAB automatically allocates the memory necessary to hold the sparse L and U factors during the factorization. MATLAB does not use any symbolic LU prefactorization to determine the memory requirements and set up the data structures in advance. Reordering and Factorization. If you obtain a good column permutation p that reduces fill-in, perhaps from symrcm or colamd, then computing lu(S(:,p)) takes less time and storage than computing lu(S). Two permutations are the symmetric reverse Cuthill-McKee ordering and the symmetric minimum degree ordering. r = symrcm(B); m = symamd(B); The three spy plots produced by the lines below show the three adjacency matrices of the Bucky Ball graph with these three different numberings. The local, pentagon-based structure of the original numbering is not evident in the other three. spy(B) spy(B(r,r)) spy(B(m,m)) Original 0 0 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 0 6-30 Minimum Degree Reverse Cuthill−McKee 0 10 20 30 nz = 180 40 50 60 60 0 10 20 30 nz = 180 40 50 60 60 0 10 20 30 nz = 180 40 50 60 Sparse Matrix Operations The reverse Cuthill-McKee ordering, r, reduces the bandwidth and concentrates all the nonzero elements near the diagonal. The approximate minimum degree ordering, m, produces a fractal-like structure with large blocks of zeros. To see the fill-in generated in the LU factorization of the Bucky ball, use speye(n,n), the sparse identity matrix, to insert -3s on the diagonal of B. B = B - 3*speye(n,n); Since each row sum is now zero, this new B is actually singular, but it is still instructive to compute its LU factorization. When called with only one output argument, lu returns the two triangular factors, L and U, in a single sparse matrix. The number of nonzeros in that matrix is a measure of the time and storage required to solve linear systems involving B. Here are the nonzero counts for the three permutations being considered. Original lu(B) 1022 Reverse Cuthill-McKee lu(B(r,r)) 968 Approximate minimum degree lu(B(m,m)) 636 Even though this is a small example, the results are typical. The original numbering scheme leads to the most fill-in. The fill-in for the reverse Cuthill-McKee ordering is concentrated within the band, but it is almost as extensive as the first two orderings. For the minimum degree ordering, the relatively large blocks of zeros are preserved during the elimination and the amount of fill-in is significantly less than that generated by the other orderings. The spy plots below reflect the characteristics of each reordering. 6-31 6 Sparse Matrices Minimum Degree Reverse Cuthill−McKee Original 0 0 0 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 0 10 20 30 nz = 1022 40 50 60 60 0 60 10 20 30 nz = 968 40 50 60 0 10 20 30 nz = 636 40 50 60 Cholesky Factorization If S is a symmetric (or Hermitian), positive definite, sparse matrix, the statement below returns a sparse, upper triangular matrix R so that R'*R = S. R = chol(S) chol does not automatically pivot for sparsity, but you can compute minimum degree and profile limiting permutations for use with chol(S(p,p)). Since the Cholesky algorithm does not use pivoting for sparsity and does not require pivoting for numerical stability, chol does a quick calculation of the amount of memory required and allocates all the memory at the start of the factorization. You can use symbfact, which uses the same algorithm as chol, to calculate how much memory is allocated. QR Factorization MATLAB computes the complete QR factorization of a sparse matrix S with [Q,R] = qr(S) but this is usually impractical. The orthogonal matrix Q often fails to have a high proportion of zero elements. A more practical alternative, sometimes known as “the Q-less QR factorization,” is available. 6-32 Sparse Matrix Operations With one sparse input argument and one output argument R = qr(S) returns just the upper triangular portion of the QR factorization. The matrix R provides a Cholesky factorization for the matrix associated with the normal equations, R'*R = S'*S However, the loss of numerical information inherent in the computation of S'*S is avoided. With two input arguments having the same number of rows, and two output arguments, the statement [C,R] = qr(S,B) applies the orthogonal transformations to B, producing C = Q'*B without computing Q. The Q-less QR factorization allows the solution of sparse least squares problems minimize Ax – b with two steps [c,R] = qr(A,b) x = R\c If A is sparse, but not square, MATLAB uses these steps for the linear equation solving backslash operator x = A\b Or, you can do the factorization yourself and examine R for rank deficiency. It is also possible to solve a sequence of least squares linear systems with different right-hand sides, b, that are not necessarily known when R = qr(A) is computed. The approach solves the “semi-normal equations” R'*R*x = A'*b with x = R\(R'\(A'*b)) 6-33 6 Sparse Matrices and then employs one step of iterative refinement to reduce roundoff error r = b - A*x e = R\(R'\(A'*r)) x = x + e Incomplete Factorizations The luinc and cholinc functions provide approximate, incomplete factorizations, which are useful as preconditioners for sparse iterative methods. The luinc function produces two different kinds of incomplete LU factorizations, one involving a drop tolerance and one involving fill-in level. If A is a sparse matrix, and tol is a small tolerance, then [L,U] = luinc(A,tol) computes an approximate LU factorization where all elements less than tol times the norm of the relevant column are set to zero. Alternatively, [L,U] = luinc(A,'0') computes an approximate LU factorization where the sparsity pattern of L+U is a permutation of the sparsity pattern of A. For example, load west0479 A = west0479; nnz(A) nnz(lu(A)) nnz(luinc(A,1e-6)) nnz(luinc(A,'0')) shows that A has 1887 nonzeros, its complete LU factorization has 16777 nonzeros, its incomplete LU factorization with a drop tolerance of 1e-6 has 10311 nonzeros, and its lu('0') factorization has 1886 nonzeros. The luinc function has a few other options. See the luinc reference page for details. The cholinc function provides drop tolerance and level 0 fill-in Cholesky factorizations of symmetric, positive definite sparse matrices. See the cholinc reference page for more information. 6-34 Sparse Matrix Operations Simultaneous Linear Equations There are two different classes of methods for solving systems of simultaneous linear equations: • Direct methods are usually variants of Gaussian elimination. These methods involve the individual matrix elements directly, through matrix factorizations such as LU or Cholesky factorization. MATLAB implements direct methods through the matrix division operators / and \, which you can use to solve linear systems. • Iterative methods produce only an approximate solution after a finite number of steps. These methods involve the coefficient matrix only indirectly, through a matrix-vector product or an abstract linear operator. Iterative methods are usually applied only to sparse matrices. Direct Methods Direct methods are usually faster and more generally applicable than indirect methods, if there is enough storage available to carry them out. Iterative methods are usually applicable to restricted cases of equations and depend upon properties like diagonal dominance or the existence of an underlying differential operator. Direct methods are implemented in the core of MATLAB and are made as efficient as possible for general classes of matrices. Iterative methods are usually implemented in MATLAB M-files and may make use of the direct solution of subproblems or preconditioners. Using a Different Preordering. If A is not diagonal, banded, triangular, or a permutation of a triangular matrix, backslash (\) reorders the indices of A to reduce the amount of fill-in — that is, the number of nonzero entries that are added to the sparse factorization matrices. The new ordering, called a preordering, is performed before the factorization of A. In some cases, you might be able to provide a better preordering than the one used by the backslash algorithm. To use a different preordering, first turn off the automatic preordering that backslash performs by default, using the function spparms as follows: spparms('autoamd',0); spparms('autommd',0); 6-35 6 Sparse Matrices Now, assuming you have created a permutation vector p that specifies a preordering of the indices of A, apply backslash to the matrix A(:,p), whose columns are the columns of A, permuted according to the vector p. x = A (:,p) \ b; x(p) = x; spparms('autoamd',1); spparms('autommd',1); The commands spparms('autoamd',1) and spparms('autommd',1) turns the automatic preordering back on, in case you use A\b later without specifying an appropriate preordering. Iterative Methods Nine functions are available that implement iterative methods for sparse systems of simultaneous linear systems. Functions for Iterative Methods for Sparse Systems Function Method bicg Biconjugate gradient bicgstab Biconjugate gradient stabilized cgs Conjugate gradient squared gmres Generalized minimum residual lsqr LSQR implementation of Conjugate Gradients on the Normal Equations minres Minimum residual pcg Preconditioned conjugate gradient qmr Quasiminimal residual symmlq Symmetric LQ These methods are designed to solve Ax = b or min b – Ax . For the Preconditioned Conjugate Gradient method, pcg, A must be a symmetric, positive definite matrix. minres and symmlq can be used on symmetric 6-36 Sparse Matrix Operations indefinite matrices. For lsqr, the matrix need not be square. The other five can handle nonsymmetric, square matrices. All nine methods can make use of preconditioners. The linear system Ax = b is replaced by the equivalent system –1 –1 M Ax = M b The preconditioner M is chosen to accelerate convergence of the iterative method. In many cases, the preconditioners occur naturally in the mathematical model. A partial differential equation with variable coefficients may be approximated by one with constant coefficients, for example. Incomplete matrix factorizations may be used in the absence of natural preconditioners. The five-point finite difference approximation to Laplace's equation on a square, two-dimensional domain provides an example. The following statements use the preconditioned conjugate gradient method preconditioner M = R'*R, where R is the incomplete Cholesky factor of A. A = delsq(numgrid('S',50)); b = ones(size(A,1),1); tol = 1.e-3; maxit = 10; R = cholinc(A,tol); [x,flag,err,iter,res] = pcg(A,b,tol,maxit,R',R); Only four iterations are required to achieve the prescribed accuracy. Background information on these iterative methods and incomplete factorizations is available in [2] and [7]. 6-37 6 Sparse Matrices Eigenvalues and Singular Values Two functions are available which compute a few specified eigenvalues or singular values. svds is based on eigs which uses ARPACK [6]. Functions to Compute a Few Eigenvalues or Singular Values Function Description eigs Few eigenvalues svds Few singular values These functions are most frequently used with sparse matrices, but they can be used with full matrices or even with linear operators defined by M-files. The statement [V,lambda] = eigs(A,k,sigma) finds the k eigenvalues and corresponding eigenvectors of the matrix A which are nearest the “shift” sigma. If sigma is omitted, the eigenvalues largest in magnitude are found. If sigma is zero, the eigenvalues smallest in magnitude are found. A second matrix, B, may be included for the generalized eigenvalue problem Av = λBv The statement [U,S,V] = svds(A,k) finds the k largest singular values of A and [U,S,V] = svds(A,k,0) finds the k smallest singular values. For example, the statements L = numgrid('L',65); A = delsq(L); 6-38 Sparse Matrix Operations set up the five-point Laplacian difference operator on a 65-by-65 grid in an L-shaped, two-dimensional domain. The statements size(A) nnz(A) show that A is a matrix of order 2945 with 14,473 nonzero elements. The statement [v,d] = eigs(A,1,0); computes the smallest eigenvalue and eigenvector. Finally, L(L>0) = full(v(L(L>0))); x = -1:1/32:1; contour(x,x,L,15) axis square distributes the components of the eigenvector over the appropriate grid points and produces a contour plot of the result. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 The numerical techniques used in eigs and svds are described in [6]. 6-39 6 Sparse Matrices Selected Bibliography [1] Amestoy, P. R., T. A. Davis, and I. S. Duff, “An Approximate Minimum Degree Ordering Algorithm,” SIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 4, Oct. 1996, pp. 886-905. [2] Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994. [3] Davis, T.A., Gilbert, J. R., Larimore, S.I., Ng, E., Peyton, B., “A Column Approximate Minimum Degree Ordering Algorithm,” Proc. SIAM Conference on Applied Linear Algebra, Oct. 1997, p. 29. [4] Gilbert, John R., Cleve Moler, and Robert Schreiber, “Sparse Matrices in MATLAB: Design and Implementation,” SIAM J. Matrix Anal. Appl., Vol. 13, No. 1, January 1992, pp. 333-356. [5] Larimore, S. I., An Approximate Minimum Degree Column Ordering Algorithm, MS Thesis, Dept. of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 1998, available at http://www.cise.ufl.edu/tech_reports/ [6] Lehoucq, R. B., D. C. Sorensen, C. Yang, ARPACK Users’ Guide, SIAM, Philadelphia, 1998. [7] Saad, Yousef, Iterative Methods for Sparse Linear Equations. PWS Publishing Company, 1996. 6-40 7 Nondouble Data Types Introduction (p. 7-2) Introduces the nondouble data types in MATLAB. Integer Mathematics (p. 7-4) Describes how MATLAB performs operations on integer data types. Single-Precision Mathematics (p. 7-17) Describes how MATLAB performs operations on data type single. 7 Nondouble Data Types Introduction MATLAB provides several data types that you can assign to numbers or numerical variables. By default, MATLAB assigns numbers the data type double, which means that they are double-precision floating-point numbers that are 64 bits in length. MATLAB performs operations on numbers of type double using double-precision arithmetic. For most numerical purposes, double is the recommended data type. Besides double, MATLAB also provides several other data types that require less memory than double. These include • single — Single-precision floating-point numbers that are 32 bits in length • int8 and uint8 — Signed and unsigned integers that are 8 bits in length • int16 and uint16 — Signed and unsigned integers that are 16 bits in length • int32 and uint32 — Signed and unsigned integers that are 32 bits in length These data types are useful if you need to conserve memory, for example, if you are working with very large data sets such as image files. The following sections explain the issues you need to keep in mind when performing operations on nondouble data types: • “Integer Mathematics” on page 7-4 explains how to perform operations on numbers of integer data type. • “Single-Precision Mathematics” on page 7-17 explains how to perform operations on numbers of type single. “Data Types” in the MATLAB Programming documentation provides more information about these data types. The following MATLAB functions return output of type single or an integer data type when you call them with the optional input argument datatype, which is a string containing one of data types listed above: • eye • ones • zeros 7-2 Introduction For example, ones(2,2,'int8') returns a 2-by-2 matrix of ones of type int8. ans = 1 1 1 1 In addition, the following functions return output of type single when you call them with the input argument 'single': • eps • Inf • NaN See the reference pages for these functions for more information. 7-3 7 Nondouble Data Types Integer Mathematics This section describes operations on integer data types in MATLAB. The section covers the following topics: • “Integer Data Types” on page 7-4 • “Largest and Smallest Values for Integer Data Types” on page 7-5 • “Integer Arithmetic” on page 7-6 • “Example — Digitized Signals” on page 7-8 • “Warnings for Integer Data Types” on page 7-15 Integer Data Types Integer data types in MATLAB take on integer values in a specified range. For example, an integer of type int8 can take any of the 28 possible values of signed 8-bit integers in the range -128 to 127. Integer data types are useful for storing data that can be described using only integers, such as image files. The following table lists the integer data types that support arithmetic operations and their ranges. Data Type Description Range of Values int8 Signed 8-bit integer -128 to 127 uint8 Unsigned 8-bit integer 0 to 255 int16 Signed 16-bit integer -215 to 215 - 1 uint16 Unsigned 16-bit integer 0 to 216 - 1 int32 Signed 32-bit integer -231 to 231 - 1 uint32 Unsigned 32-bit integer 0 to 232 - 1 To assign an integer data type to a number or variable, use one of the functions listed in the first column of the table. For example, x = int8(5) 7-4 Integer Mathematics sets the value of x to be 5 with data type int8. You can verify the data type of the result using the class command. class(x) ans = int8 When you convert a number to one of the integer data types, MATLAB rounds the result to the nearest integer. For example, int8(2.7) ans = 3 For numbers that are halfway between two integers, MATLAB rounds up if the number is positive and rounds down if the number is negative. For example, int8(2.5) ans = 3 int8(-2.5) ans = -3 Largest and Smallest Values for Integer Data Types For each integer data type, there is a largest and smallest integer that you can represent with that data type. The table in “Integer Data Types” on page 7-4 lists the largest and smallest values for each integer data type in the “Range of Values” column. You can also return these values with the intmax and intmin commands. For example, 7-5 7 Nondouble Data Types intmax('int8') ans = 127 intmin('int8') ans = -128 If you convert a number that is larger than the maximum value of an integer data type to that data type, MATLAB returns the maximum value. This is called saturating on overflow. int8(300) ans = 127 Similarly, if you convert a number that is smaller than the minimum value of the integer data type, MATLAB returns the minimum value. You can make MATLAB return a warning when your input is outside the range an integer data type. “Warnings for Integer Data Types” on page 7-15 explains how to turn these warnings on. Integer Arithmetic MATLAB can perform arithmetic operations on arrays of the same integer data type, and the result has the same type. For example, x = int16(5) + int16(9) x = 14 class(x) 7-6 Integer Mathematics returns ans = int16 For a list of the operations that support integer data types, see “Nondouble Data Type Support” in the arithmetic operators reference page. When the result of an operation exceeds the maximum value of the data type, MATLAB returns the maximum value. For example, int8(100)+int8(100) ans = 127 MATLAB returns 127, which is the maximum value for numbers of type int8. Similarly, if the result is less than the minimum value of the data type, MATLAB returns the minimum value. You can make MATLAB return a warning when the result of an operation is outside the range an integer data type. “Warnings for Integer Data Types” on page 7-15 explains how to turn these warnings on. MATLAB computes elementwise division, A./B and A.\B, where A and B are arrays of integer data type, using elementwise double-precision arithmetic and then converting the result back to the original integer data type. For example, int8(4)./int8(3) ans = 1 MATLAB computes 4/3 in double precision and then converts the result to int8, rounding 4/3 to 1. 7-7 7 Nondouble Data Types Combining Integer Data Types with Scalars of Type double You can combine scalars or arrays of an integer data type with scalars (but not arrays) of type double and the result has the same integer data type. For example, class(5*int8(3)) ans = int8 However, you cannot combine scalars or arrays of an integer data type with scalars or arrays of a different integer data type or data type single. For all binary operations in which one operand is an array of integer data type and the other is a scalar double, MATLAB computes the operation using elementwise double-precision arithmetic and then converts the result back to the original integer data type. For example, int8([1 2 3 4 5])*0.8 ans = 1 2 2 3 4 MATLAB computes [1 2 3 4 5]*0.8 in double precision and then converts the result to int8. Note that the second and third entries of [1 2 3 4 5]*0.8, which are 1.6 and 2.4, are both rounded to the nearest integer, which is 2. Example — Digitized Signals This section describes how you can use integer data types when modeling a digital communication system, such as a telephone network. A digital telephone converts an analog signal—your voice—to a digital signal before transmission. While the analog signal takes on real number values, the digital signal takes on only a finite set of integer values. If you are modeling a digital communication system using MATLAB, you can model these practical implementation effects and save memory by storing the digital signal as an integer data type rather than as type double. 7-8 Integer Mathematics Source Coding To convert an analog, or source, signal to a digital signal, a digital telephone samples the signal at discrete time intervals and encodes, or quantizes, the sampled values, which are real numbers, as integers. The encoding process is called source coding. One simple way to quantize a sampled signal is to 1 Partition the range of the signal into a finite number of intervals. 2 Assign each sampled value an integer based on the interval of the partition the value lies in. For example, if the signal is a sine wave, whose range is [-1 1], you could partition the range into four equal intervals, labeled 0, 1, 2, and 3, as shown in the following figure. 1 3 0.5 2 0 1 −0.5 0 −1 −10 −5 0 5 10 The quantized value at this sample time is 2. The vertical lines correspond to the sample times. For example, if you sample the signal at time -6, its value lies in the interval [0 0.5], so the quantized value is 2. Typically, the sample times are closer together and the number of intervals in the partition is larger, to make the encoding more accurate. The following table 7-9 7 Nondouble Data Types defines a partition of [-1 1] into 256 intervals, which are assigned integer values from -128 to 127, the range of data type int8. Interval Quantized Value [-Inf, -255/256] -128 (-255/256, -253/256] -127 (-253/256, -251/256] -126 ... ... (-3/256, -1/256] -1 (-1/256, 1/256) 0 [1/256 3/256) 1 ... ... [249/256, 251/256) 125 [251/256, 253/256) 126 [253/256, Inf] 127 You can use the function int8 to compute the quantized value of a sample whose value is x by the formula quantize = int8(128*x) For example, int8(128*.37) ans = 47 Note that any samples greater than 1 have the quantized value 127, the maximum value for data type int8, due to saturation, so they cannot be distinguished by this quantization scheme. To distinguish such samples, you 7-10 Integer Mathematics would need to enlarge the range of values that are partitioned. Similarly, any samples less than -1 have the quantized value -128, the minimum value for int8. As an illustration, suppose you sample a sine wave signal at time intervals of .01. The following code converts the sampled values to integers of data type int8 and plots the result: sample_times = [-2*pi:.01: 2*pi]; source = sin(sample_times); signal = int8(128*source); plot(sample_times, signal, '.') 150 100 50 0 −50 −100 −150 −8 −6 −4 −2 0 2 4 6 8 While the curve appears to be smooth, you can magnify a portion of it by clicking the magnify icon on the toolbar and then clicking the plot three times at one of the peaks, as shown in the following figure. 7-11 7 Nondouble Data Types Click the magnify icon once Click three times The result shows that the plot is actually made up of discrete points with integer y-values. 134 132 130 128 126 124 122 120 118 −5.2 −5.1 −5 −4.9 −4.8 −4.7 −4.6 −4.5 −4.4 −4.3 The Communcations Toolbox provides several functions that implement more sophisticated source coding schemes. 7-12 Integer Mathematics Combining Two Signals Suppose you want to model two signals transmitted over the same channel, such as two people speaking at the same time into separate telephones on the same line. When these signals are combined in a channel, their values are added together. To illustrate this, the following code creates two signals and plots them separately: source1=1/3*sin(2*sample_times)+2/3*cos(sample_times); source2=3/4*sin(3*sample_times)+1/4*cos(sample_times); signal1 = int8(128*source1); signal2 = int8(128*source2); plot(sample_times,signal1) hold on plot(sample_times,signal2,'color','red') legend('Signal1', 'Signal2') hold off ' 150 Signal1 Signal2 100 50 0 −50 −100 −150 −8 −6 −4 −2 0 2 4 6 8 7-13 7 Nondouble Data Types The following code adds the signals and plots the result. plot(sample_times, signal1 + signal2, 'color', 'black') legend('Signal1 + Signal2') 150 Signal 1 + Signal 2 100 50 0 −50 −100 −150 −8 −6 −4 −2 0 2 4 6 8 Notice that the tops of the peaks are truncated at 127, the maximum value for int8, while the bottoms of the valleys are truncated at -128, the minimum value for int8. This occurs because the sum of the signals in the truncated regions lies outside the original range [-1 1], so it saturates to 127 or -128. One way to deal with this is to first average the source signals before quantizing them, so that their average lies in the range [-1 1]. The following code quantizes the average and plots the result along with the previous plot. hold on avg_signal=int8(128*(mean([source1; source2]))); plot(sample_times, avg_signal, 'color', 'magenta', ... 'linestyle', '--') legend('Signal1 + Signal2', 'Average of signals') 7-14 Integer Mathematics 150 Signal1 + Signal2 Average of signals 100 50 0 −50 −100 −150 −8 −6 −4 −2 0 2 4 6 8 Warnings for Integer Data Types You can use the intwarning('on') command to make MATLAB return a warning message when it converts a number outside the range of an integer data type to that data type or when the result of an arithmetic operation overflows. For example, intwarning('on') int16(50000) Warning: Out of range value converted to intmin('int16') or intmax('int16'). ans = 32767 There are four possible warning messages that you can turn on using intwarning. The following example illustrates all four warning messages. intwarning('on') int8([NaN Inf pi])+1000 7-15 7 Nondouble Data Types Warning: NaN converted to int8(0). Warning: Out of range value converted to intmin('int8') or intmax('int8'). Warning: Conversion rounded non-integer floating point value to nearest int8 value. Warning: Out of range value or NaN computed in integer arithmetic. ans = 127 127 127 To turn these warnings off (their default state when you start MATLAB), enter intwarning('off') Turning Warnings On or Off Temporarily When writing M-files that contain integer data types, it is sometimes convenient to temporarily turn integer warnings on and then return the states of the warnings ('on' or 'off') to their previous settings. The following commands illustrate how to do this: oldState = intwarning('on'); int8(200) Warning: Out of range value converted to intmin('int8') or intmax('int8'). ans = 127 intwarning(oldState) To temporarily turn the warnings off, change the first line of the preceding code to oldState = intwarning('off'); 7-16 Single-Precision Mathematics Single-Precision Mathematics This section describes operations on single-precision numbers — that is, numbers of type single. Because MATLAB stores numbers of type single using 32 bits, they require less memory than numbers of type double, which use 64 bits. However, because they are stored with fewer bits, numbers of type single are represented to less precision than numbers of type double. This section covers the following topics: • “Data Type single” on page 7-17 • “Single-Precision Arithmetic” on page 7-18 • “The Function eps” on page 7-19 • “Example — Writing M-Files for Different Data Types” on page 7-21 • “Largest and Smallest Numbers of Type double and single” on page 7-23 Data Type single To assign the data type single to a numbers or variable, use the command single. For example, a = single(5) sets the value of a to be 5 with data type single. Storing a number as type single require only half as much memory as storing it as type double. You can compare how many bytes of memory are used to store 5 as type single versus type double using the whos command. b = 5; whos Name a b Size 1x1 1x1 Bytes 4 8 Class single array double array When you convert a number of type double to type single, MATLAB rounds the number to the nearest single-precision number. This can change the stored value slightly. 7-17 7 Nondouble Data Types For example, format long single(3.14) ans = 3.1400001 You can return an upper bound for how much the stored value of a number changes when you convert it to single using the eps command, as described in “The Function eps” on page 7-19. Single-Precision Arithmetic You can combine two numbers of type single and the result is of type single. For example, x = single(2)*single(3) x = 6 You can verify that the result has data type single with the class command. class(x) ans = single You can combine scalars or arrays of type single with scalars or arrays of type double, and the result has type single. For example, x = single(8) + 3 x = 11 7-18 Single-Precision Mathematics class(x) ans = single However, you cannot combine scalars or arrays of type single with scalars or arrays of an integer data type. The Function eps Because there are only finitely many double-precision numbers, you cannot represent all numbers in double-precision storage. On any computer, there is a small gap between each double-precision number and the next larger double-precision number. You can determine the size of this gap, which limits the precision of your results, using the eps function. For example, to find the distance between 5 and the next larger double-precision number, enter format long eps(5) ans = 8.881784197001252e-016 This tells you that there are no double-precision numbers between 5 and 5 + eps(5). If a double-precision computation returns the answer 5, the result is only accurate to within eps(5). The value of eps(x) depends on x: as x gets larger, so does eps(x). For example, eps(50) ans = 7.105427357601002e-015 so that eps(50) is larger than eps(5). If you enter eps with no input argument, MATLAB returns the value of eps(1), the distance from 1 to the next larger double-precision number. 7-19 7 Nondouble Data Types Similarly, there are gaps between any two single-precision numbers. If x has type single, eps(x) returns the distance between x and the next larger single-precision number. For example, x = single(5); eps(x) returns ans = 4.7684e-007 Note that this result is larger than eps(5). Because there are fewer single-precision numbers than double-precision numbers, the gaps between the single-precision numbers are larger than the gaps between double-precision numbers. This means that results in single-precision arithmetic are less precise than in double-precision. For a number x of type double, eps(single(x)) gives you an upper bound for the amount that x is rounded when you convert it from double to single. For example, when you convert the double-precision number 3.14 to single, it is rounded by double(single(3.14) - 3.14 ans = 1.0490e-007 The amount that 3.14 is rounded is less than eps(single(3.14)) ans = 2.3842e-007 7-20 Single-Precision Mathematics Example — Writing M-Files for Different Data Types If you write an M-file that works with data of type single or double, the M-file might need to return different answers depending on the data type. The following example illustrates this. Computing the Ratios of Fibonacci Numbers The Fibonacci numbers are the numbers fn defined recursively by f1 = 1 f2 = 1 fn + 2 = fn + 1 + fn The first seven numbers in the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13. As n gets larger, the ratio of the n+1st Fibonacci number divided by the nth Fibonacci number tends to the golden mean, ( 1 + 5 ) ⁄ 2 . That is, fn + 1 1 + 5 lim -----------= ----------------2 n → ∞ fn Suppose you want to compute how large n must be so that the ratio fn + 1 1 + 5 ------------ – ----------------2 fn is within eps of the golden mean. The answer depends on whether you are computing in single or double-precision arithmetic, because the value of eps((1+sqrt(5))/2) depends on the data type of the golden mean. eps((1+sqrt(5))/2) ans = 2.2204e-016 7-21 7 Nondouble Data Types while eps(single((1+sqrt(5))/2)) ans = 1.1921e-007 You can write an M-file to compute the answer in either case, by passing in the data type as an input argument. The following code shows how to do this. function count = fib(data_type) f_current = ones(1,1,data_type); f_next = f_current; golden_mean = (1+sqrt(5))/2*ones(1,1,data_type); count = 0; while abs(f_next/f_current - golden_mean) >= eps(golden_mean) count = count + 1; temp = f_next; f_next = f_next + f_current; f_current = temp; end The output count is the smallest integer for which fn + 1 1 + 5 ------------ – ----------------2 fn is smaller than eps(golden_mean). For double-precision arithmetic, the answer is fib('double') ans = 39 7-22 Single-Precision Mathematics For single-precision arithmetic, the answer is fib('single') ans = 17 Largest and Smallest Numbers of Type double and single This section explains the largest and smallest numbers of data types double and single. This section covers the following topics: • “Largest Double- and Single-Precision Numbers” on page 7-23 • “Smallest Positive Double- and Single-Precision Numbers” on page 7-24 Largest Double- and Single-Precision Numbers The MATLAB command realmax returns the largest value that you can represent as a double-precision floating-point number. realmax ans = 1.7977e+308 When the result of an operation on numbers of type double exceeds realmax, MATLAB returns Inf. Similarly, the MATLAB command realmax('single') returns the largest value that you can represent as a single-precision number. realmax('single') ans = 3.4028e+038 Note that realmax for type double is much larger than realmax('single'), because the range of numbers that you can represent in single-precision is more limited than in double-precision. 7-23 7 Nondouble Data Types When the result of an operation on numbers of type single exceeds realmax('single'), MATLAB returns Inf of class single. For example, (realmax('single')/2)^2 ans = Inf Because realmax is larger than realmax('single'), performing the same computation in double precision returns a finite answer. (double(realmax('single'))/2)^2 ans = 2.8948e+076 Smallest Positive Double- and Single-Precision Numbers The MATLAB command realmin returns the smallest positive normalized floating-point number that you can represent in double precision. realmin ans = 2.2251e-308 When the result of a computation on numbers of type double is a positive number that is less than realmin, MATLAB returns either 0 or a subnormal floating-point number, that is, one that is not in standard form. Similarly, there is a smallest positive normalized floating-point number that you can represent in single precision, whose value is returned by realmin('single'). realmin('single') ans = 1.1755e-038 7-24 Single-Precision Mathematics Because realmin is less than realmin('single'), operations that return a nonzero double-precision result in standard form might return 0 or a subnormal answer when you do the same operations in single precision. References The following references provide more information about floating-point arithmetic. [1] Moler, Cleve, “Floating Points,” MATLAB News and Notes, Fall, 1996. A PDF version is available on the MathWorks Web site at http://www.mathworks.com/company/newsletters/news_notes/pdf/Fall96 Cleve.pdf [2] Moler, Cleve, Numerical Computing with MATLAB, S.I.A.M. A PDF version is available on the MathWorks Web site at http://www.mathworks.com/moler/. 7-25 7 Nondouble Data Types 7-26 Index A additional parameters BVP example 5-68, 5-71 adjacency matrix and graphing 6-16 Bucky ball 6-17 defined 6-16 distance between nodes 6-21 node 6-16 numbering nodes 6-18 airflow modeling 6-22 amp1dae demo 5-38 anonymous functions representing mathematical functions 4-3 arguments, additional 4-30 B ballode demo 5-28 bandwidth of sparse matrix, reducing 6-28 Basic Fitting interface 3-28 batonode demo 5-38 bicubic interpolation 2-12 bilinear interpolation 2-12 boundary conditions BVP 5-59 BVP example 5-65 PDE 5-83 PDE example 5-88 Boundary Value Problems. See BVP Brusselator system (ODE example) 5-25 brussode demo 5-25 Buckminster Fuller dome 6-17 Bucky ball 6-17 burgersode demo 5-38 BVP 5-57 defined 5-59 rewriting as first-order system 5-64 BVP solver 5-60 basic syntax 5-61 evaluate solution at specific points 5-68 examples boundary condition at infinity (shockbvp) 5-71 Mathieu’s Equation (mat4bvp) 5-64 multipoint terms 5-79 rapid solution changes (shockbvp) 5-68 singular terms 5-75 initial guess 5-68 multipoint terms 5-79 performance 5-79 representing problems 5-63 singular terms 5-75 unknown parameters 5-67 BVP solver properties querying property structure 5-81 C cat sparse operands 6-25 characteristic polynomial of matrix 2-4 characteristic roots of matrix 2-4 chol sparse matrices 6-25 Cholesky factorization 1-28 sparse matrices 6-32 closest point searches Delaunay triangulation 2-24 colamd minimum degree ordering 6-28 colmmd column permutation 6-30 Index-1 Index colperm 6-27 comparing sparse and full matrix storage 6-6 complex values in sparse matrix 6-6 computational functions applying to sparse matrices 6-24 computational geometry multidimensional 2-26 two-dimensional 2-18 contents of sparse matrix 6-12 convex hulls multidimensional 2-27 two-dimensional 2-20 convolution 2-5 correlation coefficients 3-10 covariance 3-10 creating sparse matrix 6-8 cubic interpolation multidimensional 2-17 one-dimensional 2-11 spline 2-11 curve fitting 3-21 Basic Fitting interface 3-28 error bounds 3-27 exponential 3-25 polynomial 2-6, 3-21 curves computing length 4-27 Cuthill-McKee reverse ordering 6-28 D DAE solution of 5-2 data analysis Index-2 column-oriented 3-7 data filtering. See filtering data fitting. See curve fitting data gridding multidimensional 2-17 data. See also multivariate data statistical data univariate data DDE 5-45 rewriting as first-order system 5-50 DDE solver 5-47 basic syntax 5-48 discontinuities 5-53 evaluating solution at specific points 5-52 examples cardiovascular model (ddex2) 5-54 straightforward example (ddex1) 5-49 performance 5-56 representing problems 5-49 ddex1 demo 5-49 ddex2 demo 5-54 decomposition eigenvalue 1-39 Schur 1-42 singular value 1-43 deconvolution 2-5 Delaunay tessellations 2-29 Delaunay triangulation 2-20 closest point searches 2-24 Delay Differential Equations. See DDE density sparse matrix 6-7 derivatives polynomial 2-5 determinant of matrix 1-23 diag 6-25 Index diagonal creating sparse matrix from 6-9 difference equations 3-39 differential equations 5-1 boundary value problems for ODEs 5-57 initial value problems for DAEs 5-2 initial value problems for DDEs 5-45 initial value problems for ODEs 5-2 partial differential equations 5-81 differential-algebraic equations. See DAE direct methods systems of sparse equations 6-35 discontinuities DDE solver 5-53 discrete Fourier transform. See Fourier transforms displaying sparse matrices 6-14 distance between nodes 6-21 dot product 1-8 E eigenvalues 1-39 of sparse matrix 6-38 eigenvectors 1-39 electrical circuits DAE example 5-38 Emden’s equation example 5-76 error bounds curve fitting 3-27 error tolerance effects of too large (ODE) 5-43 machine precision 5-41 event location (ODE) advanced example 5-31 simple example 5-28 exponential curve fitting 3-25 eye derivation of the name 1-10 sparse matrices 6-24 F factorization 6-29 Cholesky 1-28 Hermitian positive definite 1-29 incomplete 6-34 LU 1-30 partial pivoting 1-30 positive definite 1-28 QR 1-31 sparse matrices 6-29 Cholesky 6-32 LU 6-29 triangular 6-29 fast Fourier transform. See Fourier transforms fem1ode demo 5-22 fem2ode demo 5-38 fill-in of sparse matrix 6-21 filtering difference equations 3-39 find function sparse matrices 6-15 finite differences 3-11 finite element discretization (ODE example) 5-22 first-order differential equations representation for BVP solver 5-64 representation for DDE solver 5-50 Fourier analysis 3-42 concepts 3-43 Fourier transforms 3-42 calculating sunspot periodicity 3-44 Index-3 Index FFT-based interpolation 2-12 length vs. speed 3-49 phase and magnitude of transformed data 3-47 fsbvp demo 5-71 full 6-25, 6-28 function functions 4-1 functions mathematical. See mathematical functions optimizing 4-8 G Gaussian elimination 1-30 geodesic dome 6-17 geometric analysis multidimensional 2-26 two-dimensional 2-18 global minimum 4-26 global variables 4-30 gplot 6-17 graph characteristics 6-20 defined 6-16 H hb1dae demo 5-35 hb1ode demo 5-38 Hermitian positive definite matrix 1-29 higher-order ODEs rewriting as first-order ODEs 5-5 I iburgersode demo 5-38 identity matrix 1-10 Index-4 ihb1dae demo 5-38 importing sparse matrix 6-11 incomplete factorization 6-34 infeasible optimization problems 4-26 initial conditions ODE 5-4 ODE example 5-10 PDE 5-83 PDE example 5-88 initial guess (BVP) example 5-65 quality of 5-68 initial value problems DDE 5-45 defined 5-4 ODE and DAE 5-2 initial-boundary value PDE problems 5-81 inner product 1-7 integer mathematics 7-4 integration double 4-28 numerical 4-27 triple 4-27 See also differential equations integration interval DDE 5-48 PDE (MATLAB) 5-85 interpolation 2-9 comparing methods graphically 2-13 FFT-based 2-12 multidimensional 2-16 scattered data 2-34 one-dimensional 2-10 speed, memory, smoothness 2-11 three-dimensional 2-16 two-dimensional 2-12 Index inverse of matrix 1-23 iterative methods sparse matrices 6-36 sparse systems of equations 6-35 K Kronecker tensor matrix product 1-11 L least squares 6-33 length of curve, computing 4-27 linear algebra 1-4 linear equations minimal norm solution 1-26 overdetermined systems 1-18 rectangular systems 1-24 underdetermined systems 1-20 linear interpolation multidimensional 2-17 one-dimensional 2-10 linear systems of equations direct methods (sparse) 6-35 full 1-13 iterative methods (sparse) 6-35 sparse 6-35 linear transformation 1-4 load sparse matrices 6-11 Lobatto IIIa BVP solver 5-61 LU factorization 1-30 sparse matrices and reordering 6-29 M mat4bvp demo 5-59 mat4bvp demo 5-64 mathematical functions as function input arguments 4-1 finding zeros 4-21 minimizing 4-8 numerical integration 4-27 plotting 4-5 representing in MATLAB 4-3 mathematical operations sparse matrices 6-24 Mathieu’s equation (BVP example) 5-64 matrices 1-4 as linear transformation 1-4 characteristic polynomial 2-4 characteristic roots 2-4 creation 1-4 determinant 1-23 full to sparse conversion 6-7 identity 1-10 inverse 1-23 iterative methods (sparse) 6-36 orthogonal 1-31 pseudoinverse 1-24 rank deficiency 1-20 symmetric 1-7 triangular 1-28 matrix operations addition and subtraction 1-6 division 1-13 exponentials 1-36 multiplication 1-8 powers 1-35 transpose 1-7 matrix products Kronecker tensor 1-11 max 6-25 M-files Index-5 Index representing mathematical functions 4-3 minimizing mathematical functions of one variable 4-8 of several variables 4-9 options 4-13 minimum degree ordering 6-28 Moore-Penrose pseudoinverse 1-24 multidimensional data gridding 2-17 interpolation 2-16 multidimensional interpolation 2-16 scattered data 2-26 multistep solver (ODE) 5-6 multivariate data matrix representation 3-3 vehicle traffic sample data 3-3 N NaNs propagation 3-13 removing from data 3-14 nearest neighbor interpolation multidimensional 2-17 one-dimensional 2-10 three-dimensional 2-16 two-dimensional 2-12 nnz 6-12 nodes 6-16 distance between 6-21 numbering 6-18 nonstiff ODE examples rigid body (rigidode) 5-19 nonzero elements maximum number in sparse matrix 6-8 number in sparse matrix 6-12 sparse matrix 6-12 Index-6 storage for sparse matrices 6-5 values for sparse matrices 6-12 visualizing for sparse matrices 6-20 nonzeros 6-12 normalizing data 3-22 norms vector and matrix 1-12 numerical integration 4-27 computing length of curve 4-27 double 4-28 triple 4-27 nzmax 6-12, 6-14 O objective function 4-1 return values 4-26 ODE coding in MATLAB 5-10 defined 5-4 overspecified systems 5-39 solution of 5-2 ODE solver evaluate solution at specific points 5-15 ODE solver properties fixed step sizes 5-41 ODE solvers 5-5 algorithms Adams-Bashworth-Moulton PECE 5-6 Bogacki-Shampine 5-6 Dormand-Prince 5-6 modified Rosenbrock formula 5-7 numerical differentiation formulas 5-7 backwards in time 5-43 basic example stiff problem 5-12 basic syntax 5-7 Index calling 5-10 examples 5-18 minimizing output storage 5-40 minimizing startup cost 5-40 multistep solver 5-6 nonstiff problem example 5-9 nonstiff problems 5-6 one-step solver 5-6 overview 5-5 performance 5-17 problem size 5-40 representing problems 5-9 sampled data 5-43 stiff problems 5-6, 5-12 troubleshooting 5-39 one-dimensional interpolation 2-10 ones sparse matrices 6-24 one-step solver (ODE) 5-6 optimization 4-8 helpful hints 4-25 options parameters 4-13 troubleshooting 4-25 See also minimizing mathematical functions orbitode demo 5-31 Ordinary Differential Equations. See ODE orthogonal matrix 1-31 outer product 1-7 outliers removing from statistical data 3-15 output functions 4-14 overdetermined rectangular matrices 1-18 overspecified ODE systems 5-39 P Partial Differential Equations. See PDE partial fraction expansion 2-7 PDE 5-81 defined 5-82 discretized 5-42 PDE examples (MATLAB) 5-81 PDE solver (MATLAB) 5-83 basic syntax 5-84 evaluate solution at specific points 5-91 examples electrodynamics problem 5-92 simple PDE 5-86 performance 5-92 representing problems 5-86 PDE solver (MATLAB) properties 5-92 pdex1 demo 5-86 pdex2 demo 5-82 pdex3 demo 5-82 pdex4 demo 5-92 pdex5 demo 5-82 performance de-emphasizing an ODE solution component 5-42 improving for BVP solver 5-79 improving for DDE solver 5-56 improving for ODE solvers 5-17 improving for PDE solver 5-92 permutations 6-25 plotting mathematical functions 4-5 polynomial curve fitting 3-21 regression 3-17 polynomial interpolation 2-10 polynomials basic operations 2-2 Index-7 Index calculating coefficients from roots 2-3 calculating roots 2-3 curve fitting 2-6 derivatives 2-5 evaluating 2-4 multiplying and dividing 2-5 partial fraction expansion 2-7 representing as vectors 2-3 preconditioner sparse matrices 6-34 property structure (BVP) querying 5-81 pseudoinverse of matrix 1-24 Q QR factorization 1-31, 6-32 quad, quadl functions differ from ODE solvers 5-39 quadrature. See numerical integration underdetermined systems 1-20 regression linear-in-the-parameters 3-18 multiple 3-20 polynomial 3-17 reorderings 6-25 for sparser factorizations 6-27 LU factorization 6-29 minimum degree ordering 6-28 reducing bandwidth 6-28 representing mathematical functions 4-3 residuals analyzing 3-23 exponential data fit 3-27 rigid body (ODE example) 5-19 rigidode demo 5-19 Robertson problem DAE example 5-35 ODE example 5-38 roots polynomial 2-3 R rand sparse matrices 6-24 rank deficiency detecting 1-33 rectangular matrices 1-20 sparse matrices 6-33 rectangular matrices identity 1-10 overdetermined systems 1-18 pseudoinverse 1-24 QR factorization 1-31 rank deficient 1-20 singular value decomposition 1-43 Index-8 S sampled data with ODE solvers 5-43 save 6-11 scalar as a matrix 1-5 scalar product 1-8 scattered data multidimensional interpolation 2-34 multidimensional tessellation 2-26 triangulation and interpolation 2-18 Schur decomposition 1-42 seamount data set 2-19 Index second difference operator example 6-8 shockbvp demo 5-68 single-precision mathematics 7-17 singular value matrix decomposition 1-43 size sparse matrices 6-24 solution changes, rapid making initial guess 5-68 verifying consistent behavior 5-71 solving linear systems of equations full 1-13 sparse 6-35 sort 6-28 sparse function converting full to sparse 6-7 sparse matrix advantages 6-5 and complex values 6-6 Cholesky factorization 6-32 computational considerations 6-24 contents 6-12 conversion from full 6-7 creating 6-7 directly 6-8 from diagonal elements 6-9 density 6-7 distance between nodes 6-21 eigenvalues 6-38 fill-in 6-21 importing 6-11 linear systems of equations 6-35 LU factorization 6-29 and reordering 6-29 mathematical operations 6-24 nonzero elements 6-12 maximum number 6-8 specifying when creating matrix 6-8 storage 6-5, 6-12 values 6-12 nonzero elements of sparse matrix number of 6-12 operations 6-24 permutation 6-25 preconditioner 6-34 propagation through computations 6-24 QR factorization 6-32 reordering 6-25 storage 6-5 for various permutations 6-27 viewing 6-12 triangular factorization 6-29 viewing contents graphically 6-14 viewing storage 6-12 visualizing 6-20 sparse ODE examples Brusselator system (brussode) 5-25 spconvert 6-11 spdiags 6-9 speye 6-24 spones 6-27 spparms 6-35 sprand 6-24 spy 6-14 spy plot 6-20 startup cost minimizing for ODE solvers 5-40 statistical data missing values 3-13 normalizing 3-22 outliers 3-15 preprocessing 3-13 removing NaNs 3-14 See also multivariate data Index-9 Index See also univariate data statistics descriptive 3-7 stiff ODE examples Brusselator system (brussode) 5-25 differential-algebraic problem (hb1dae) 5-35 finite element discretization (fem1ode) 5-22 van der Pol (vdpode) 5-20 stiffness (ODE), defined 5-12 storage minimizing for ODE problems 5-40 permutations of sparse matrices 6-27 sparse and full, comparison 6-6 sparse matrix 6-5 viewing for sparse matrix 6-12 sum counting nonzeros in sparse matrix 6-28 sparse matrices 6-25 sunspot periodicity calculating using Fourier transforms 3-44 symamd minimum degree ordering 6-28 symmetric matrix transpose 1-7 node 6-16 threebvp demo 5-59 three-dimensional interpolation 2-16 transfer functions using partial fraction expansion 2-7 transpose complex conjugate 1-8 unconjugated complex 1-8 triangular factorization sparse matrices 6-29 triangular matrix 1-28 triangulation closest point searches 2-24 Delaunay 2-20 scattered data 2-18 Voronoi diagrams 2-25 See also tessellation tricubic interpolation 2-16 trilinear interpolation 2-16 troubleshooting (ODE) 5-39 twobvp demo 5-59 two-dimensional interpolation 2-12 comparing methods graphically 2-13 symrcm column permutation 6-30 reducing sparse matrix bandwidth 6-28 systems of equations. See linear systems of equations T tessellations, multidimensional Delaunay 2-29 Voronoi diagrams 2-31 theoretical graph 6-16 example 6-17 Index-10 U underdetermined rectangular matrices 1-20 unitary matrices QR factorization 1-31 univariate data 3-3 unknown parameters (BVP) 5-67 example 5-64 V van der Pol example 5-20 Index simple, nonstiff 5-9 simple, stiff 5-12 vdpode demo 5-20 vector products dot or scalar 1-8 outer and inner 1-7 vectors column and row 1-5 multiplication 1-7 vehicle traffic sample data 3-3 visualizing sparse matrix 6-20 visualizing solver results BVP 5-66 DDE 5-51 ODE 5-11 PDE 5-90 Voronoi diagrams multidimensional 2-31 two-dimensional 2-25 Z zeros of mathematical functions 4-21 zeros sparse matrices 6-24 Index-11

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