Numerical Analysis Spring 2011 1 of 5 Department of Mathematics California State University Los Angeles Master’s Degree Comprehensive Examination in NUMERICAL ANALYSIS SPRING 2011 Instructions: • Do exactly 2 problems from Part A AND 2 problems from Part B. If you attempt more than two problems in either Part A or Part B, and do not clearly indicate which two are to count, only the ﬁrst two problems will be counted toward your grade. • No calculators. • Closed books and closed notes. PART A: Do only 2 problems 1. (a) [8 points] Find the LU decomposition of the matrix 1 1 2 B = 2 4 7 , 3 11 19 that is, ﬁnd a unit lower-triangular matrix L and an upper-triangular matrix U such that B = LU . (b) Given the linear system Ax = b, where A has been factored as A = LU , and 1 0 0 2 4 4 2 1 1 0 , U= 0 1 2 , b= 0 . L= 1 0 1 0 0 1 2 1 Numerical Analysis Spring 2011 2 of 5 i. [8 points] Solve Ax = b without multiplying the matrices L and U . ii. [3 points] Give matrices L, D, and U such that A = LDU , where L is a unit lower-triangular matrix, D is a diagonal matrix, and U is a unit upper-triangular matrix. (c) [3 points each] Give one advantage of Gaussian elimination with partial pivoting over each of the following techniques for solving a nonsingular system of n linear equations in n unknowns: • Gaussian elimination without partial pivoting • Jacobi iteration 2. (a) [6 points] Let 1 1 4 1 0 (0) 1 . 2 , u = 1 4 2 , b= A= 1 3 0 2 4 Perform one Jacobi iteration for solving Au = b, with starting vector u(0) , to ﬁnd u(1) . (b) [7 points] Let B be an arbitrary 4 × 4 upper-triangular matrix with nonzero diagonal entries. Show that Gauss-Seidel iteration converges for Bx = b, for arbitrary b. (c) Let ( C= 3 1 5 7 ) . C has eigenvalues 2 and 8, and corresponding eigenvectors [−s, s]T and [s, 5s]T , respectively, where s ̸= 0. i. [6 points] Apply two iterations of the Power Method to the matrix C with initial vector x(0) = [1, 0]T to obtain x(2) , an approximation to the eigenvector of C corresponding to eigenvalue 8. ii. [6 points] Explain why the Power Method converges when applied to the matrix C using the initial value [1, 0]T . 3. (a) [7 points] Let A be an n × n matrix. Show that if ρ(A) < 1, then I − A is non-singular (invertible). 2 Numerical Analysis Spring 2011 (b) [7 points] Let 3 of 5 3 1 B= 0 0 1 3 1 0 0 1 3 1 0 0 . 1 3 Use Gershgorin’s circle theorem, together with part (a), to show that B is non-singular. (c) [3 points] Give an example to show that a diagonally-dominant matrix need not be non-singular. Be sure to show that your matrix is non-singular. (d) [8 points] Let ( C= 1 1 2 2 ) . Find the matrix resulting from performing one iteration of the QR method (for approximating eigenvalues) on C. 3 Numerical Analysis Spring 2011 4 of 5 PART B: Do only 2 problems 1. For the initial boundary value problem ∂u ∂ 2u , 0 < x < 1, t > 0 = ∂t ∂x2 ∂u = u + 1, x = 0, t > 0 ∂x ∂u = −u, x = 1, t > 0 ∂x u(x, 0) = x, 0 ≤ x ≤ 1 (a) [10 points] Show that the scheme ui,j+1 − ui,j ui−1,j − 2ui,j + ui+1,j = k h2 is consistent with the diﬀerential equation. (b) [15 points] Let h = 1/10 and k = 1/200. Using central diﬀerencing for the boundary data and the scheme above, compute u0,2 and u10,2 . 2. (a) [7 points] By ﬁnding the truncation error in approximating the ordinary derivative f ′′ (x) by Dh (f ) = f (x + 2h) − 2f (x) + f (x − 2h) h2 determine whether or not this is a consistent approximation to f ′′ (x). (b) [2 points each] Consider the parabolic PDE ut = uxx . i. Explain what it means for a diﬀerence scheme for this PDE to be stable. ii. Can a consistent explicit scheme approximating this PDE be stable but not convergent? Explain why or why not? iii. Give an example of an explicit scheme approximating this PDE which is not stable. (You need not prove that your scheme is not stable.) 4 Numerical Analysis Spring 2011 5 of 5 iv. Explain why the concept of stability does not apply to a difference scheme that is approximating an elliptic PDE. ∂2u ∂ 2u (c) Consider the PDE x − 2 = 0. ∂y∂x ∂ y i. [3 points] Determine the values of (x, y) for which this PDE is hyperbolic. ii. [4 points] Determine the characteristic curves for this PDE. iii. [3 points] Can an elliptic PDE generate characteristic curves? Why or why not? 3. (a) Consider the ﬁnite diﬀerence scheme ui,j+1 − ui,j ui+1,j − ui,j + =0 k h for the PDE ut + ux = 0. i. [6 points] Show that the scheme is unstable. ii. [3 points] Modify the scheme to make it stable. iii. [4 points] Write down an explicit consistent scheme for ut + ux = 0 that is unconditionally stable if there is any. If there is none, explain why. (b) Consider the PDE ∂u ∂u + e−y = −u2 ∂x ∂y u(x, 0) = 1, 0 < x < ∞ x2 u i. [6 points] Find the equation for the characteristic curve through the point (s, 0). ii. [6 points] Find the equation for the characteristic curve through the point (1, 1) and the value of the exact solution of the given IVP at (1, 1). 5

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