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Solving a system of linear equations with a block tridiagonal symmetric positive definite coefficient matrix
5
Goal
Solve a system of linear equations with a Cholesky-factored symmetric positive definite block tridiagonal coefficient matrix.
Solution
Given a coefficient symmetric positive definite block tridiagonal matrix (with square blocks each of the same
NB-by-NB size) is LLT factored, the solving stage consists of:
1.
Solve the system of linear equations with a lower bidiagonal coefficient matrix which is composed of N by N blocks of size NB by NB and with diagonal blocks which are lower triangular matrices:
a.
Solve the N local systems of equations with lower triangular diagonal blocks of size NB by NB which are used as coefficient matrices and respective parts of the right hand side vectors.
b.
Update the local right hand sides.
2.
Solve the system of linear equations with an upper bidiagonal coefficient matrix which is composted of block size N by N blocks of size NB by NB and with diagonal blocks which are upper triangular matrices.
a.
Solve the local systems of equations.
b.
Update the local right hand sides.
Source code: see the
BlockTDS_SPD/source/dpbltrs.f
file in the samples archive available at http:// software.intel.com/en-us/mkl_cookbook_samples.
Cholesky factorization of a symmetric positive definite block tridiagonal matrix
…
…
CALL DTRSM('L', 'L', 'N', 'N', NB, NRHS, 1D0, D, LDD, F, LDF)
DO K = 2, N
CALL DGEMM('N', 'N', NB, NRHS, NB, -1D0, B(1,(K-2)*NB+1), LDB, F((K-2)*NB+1,1), LDF, 1D0,
F((K-1)*NB+1,1), LDF)
CALL DTRSM('L','L', 'N', 'N', NB, NRHS, 1D0, D(1,(K-1)*NB+1), LDD, F((K-1)*NB+1,1), LDF)
END DO
CALL DTRSM('L', 'L', 'T', 'N', NB, NRHS, 1D0, D(1,(N-1)*NB+1), LDD, F((N-1)*NB+1,1), LDF)
DO K = N-1, 1, -1
CALL DGEMM('T', 'N', NB, NRHS, NB, -1D0, B(1,(K-1)*NB+1), LDB, F(K*NB+1,1), LDF, 1D0,
F((K-1)*NB+1,1), LDF)
CALL DTRSM('L','L', 'T', 'N', NB, NRHS, 1D0, D(1,(K-1)*NB+1), LDD, F((K-1)*NB+1,1), LDF)
END DO
…
43
5
Intel
®
Math Kernel Library Cookbook
Routines Used
Task Routine
Solve a local system of linear equations
Update the local right hand sides
DTRSM
DGEMM
Discussion
Consider a system of linear equations described by:
Description
Solves a triangular matrix equation.
Computes a matrix-matrix product with general matrices.
Assume that matrix A is a symmetric positive definite block tridiagonal coefficient matrix with all blocks of
to give:
Then the algorithm to solve the system of equations is:
1.
Solve the system of linear equations with a lower bidiagonal coefficient matrix in which the diagonal blocks are lower triangular matrices:
Y
1
=L
1
-1
F do i=2,N
1
//trsm()
G i
Y i end do
=F
=L i i
- C
-1
G i i - 1
Y i - 1
//trsm()
//gemm()
2.
Solve the system of linear equations with an upper bidiagonal coefficient matrix in which the diagonal blocks are upper triangular matrices:
X
N
=L
N
-T
Y
N
//trsm() do i=N-1,1,-1
Z i
X i end do
=F
=L i i
-C
-T i
T
Z
X i i + 1
//gemm()
//trsm()
44
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Table of contents
- 5 Legal Information
- 3 Contents
- 11 Getting Help and Support
- 13 Notational Conventions
- 15 Related Information
- 17 Intel® Math Kernel Library Recipes
- 19 Finding an approximate solution to a stationary nonlinear heat equation
- 23 Factoring general block tridiagonal matrices
- 33 Solving a system of linear equations with an LU-factored block tridiagonal coefficient matrix
- 39 Factoring block tridiagonal symmetric positive definite matrices
- 43 Solving a system of linear equations with a block tridiagonal symmetric positive definite coefficient matrix
- 45 Computing principal angles between two subspaces
- 49 Computing principal angles between invariant subspaces of block triangular matrices
- 53 Evaluating a Fourier integral
- 55 Using Fast Fourier Transforms for computer tomography image reconstruction
- 59 Noise filtering in financial market data streams
- 65 Using the Monte Carlo method for simulating European options pricing
- 71 Using the Black-Scholes formula for European options pricing
- 77 Multiple simple random sampling without replacement
- 81 Bibliography