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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please send solutions of (d) with full explanation then I will give up rate.
(a) Consider a system of equations QX = b (E1) where Q is an m xn real matrix, X is an n x 1 vector and b is an m x1 vector where b2 0. Let X = (#1, 2,...,)" satisfies (El) where , > 0, if i < k and æ, = 0 for i > k. Given a vector Y = (y1, V2, ... Yn)" such that y, = 0, for i > k and y; #0 for i < k, and QY = 0, find the maximum value of A such that X- AY 2 0, and X + AY 2 0. (b) Consider minimizing of f(X) = 1/2x"QX – 6" X where Q is a 10 x 10 matrix, and X, and b are 10 x 1 matrices. For which of the following the convergence will be faster? Justify. Case 1. Eigenvalues of Q are 5.0, 5.1, 5.3, 5.4, 5.7, 6.0, 6.1, 6,4, 6.8, and 7.0 Case 2. Eigenvalues of Q are 48, 50, 53, 55, 56, 60, 61, 62, 64, and 65. (c) State the pros and cons of conjugate gradient method in comparison to steepest descent method. (d) Given a vector u = (1, 1, 1) find another two vectors v and w such that u, u and w are orthogonal to each other.
Transcribed Image Text:(a) Consider a system of equations QX = b (E1) where Q is an m xn real matrix, X is an n x 1 vector and b is an m x1 vector where b2 0. Let X = (#1, 2,...,)" satisfies (El) where , > 0, if i < k and æ, = 0 for i > k. Given a vector Y = (y1, V2, ... Yn)" such that y, = 0, for i > k and y; #0 for i < k, and QY = 0, find the maximum value of A such that X- AY 2 0, and X + AY 2 0. (b) Consider minimizing of f(X) = 1/2x"QX – 6" X where Q is a 10 x 10 matrix, and X, and b are 10 x 1 matrices. For which of the following the convergence will be faster? Justify. Case 1. Eigenvalues of Q are 5.0, 5.1, 5.3, 5.4, 5.7, 6.0, 6.1, 6,4, 6.8, and 7.0 Case 2. Eigenvalues of Q are 48, 50, 53, 55, 56, 60, 61, 62, 64, and 65. (c) State the pros and cons of conjugate gradient method in comparison to steepest descent method. (d) Given a vector u = (1, 1, 1) find another two vectors v and w such that u, u and w are orthogonal to each other.
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