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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 offdiagonal elements of A . The matrix N is therefore the remaining offdiagonal elements of A after removing those in F...
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 Fall '09
 gallivan

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