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Unformatted text preview: Homework 6 Foundations of Computational Math 1 Fall 2011 The solutions will be posted on Wenesday, 11/9/11 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 | 1 | > 1 and | 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 such that x k converges to x = A- 1 b . 6.1.b . Does your answer give a characterization of selecting x 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 ( 11 ,..., 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 F...
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