hw6 - Homework 6 Foundations of Computational Math 1 Fall...

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Homework 6 Foundations of Computational Math 1 Fall 2010 The solutions will be posted on Monday, 11/8/10 Problem 6.1 Suppose you are attempting to solve Ax = b using a linear stationary iterative method deﬁned by x k = Gx k - 1 + f that is consistent with Ax = b . Suppose the eigenvalues of G are real and such that | λ 1 | > 1 and | λ i | < 1 for 2 i n . Also suppose that G has n linearly independent eigenvectors, z i , 1 i n . 6.1.a . Show that there exists an initial condition x 0 such that x k converges to x = A - 1 b . 6.1.b . Does your answer give a characterization of selecting x 0 that could be used in practice to create an algorithm that would ensure convergence? Problem 6.2 Suppose you are attempting to solve Ax = b using a linear stationary iterative method deﬁned by x k = M - 1 Nx k - 1 + M - 1 b where A = M - N . Suppose further that M = D + F where D = diag ( α 11 ,...,α nn ) and F is made up of any subset of the oﬀ-diagonal elements of A . The matrix N is therefore the remaining oﬀ-diagonal elements of A after removing those in

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This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.

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hw6 - Homework 6 Foundations of Computational Math 1 Fall...

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