(15 points) Assume that A and M are symmetric and positive definite, with A M -...

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(15 points) Assume that A and M are symmetric and positive definite, with A M - N. = (a). Show that if the rank of the n x n matrix R is k < n, then the matrix I + R has n k eigenvalues equal to 1. = 1 (b). Suppose the rank of N is k < n. How many iterations will CG take to solve the linear system Ax b using M as a preconditioner? Justify your answer. (c). Use the Gerschgorin theorem (see HW2) to show that eigenvalues of M-A are close to those of I if ||N|| < and & is small enough. (You may use any matrix norm that you find convenient.) Under this assumption, what can you say about the convergence of CG using M as a preconditioner for A? How small is "small enough"? (15 points) Assume that A and M are symmetric and positive definite, with A M - N. = (a). Show that if the rank of the n x n matrix R is k < n, then the matrix I + R has n k eigenvalues equal to 1. = 1 (b). Suppose the rank of N is k < n. How many iterations will CG take to solve the linear system Ax b using M as a preconditioner? Justify your answer. (c). Use the Gerschgorin theorem (see HW2) to show that eigenvalues of M-A are close to those of I if ||N|| < and & is small enough. (You may use any matrix norm that you find convenient.) Under this assumption, what can you say about the convergence of CG using M as a preconditioner for A? How small is "small enough"?