Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations (Classic Reprint)

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Excerpt from Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations Finite precision CG computations for solving an n by n symmetric positive definite linear system Ar b sometimes fail to converge after n steps, especially when n is small. In such cases, it is demonstrated that exact CG applied to the corresponding large linear system A5: b also requires more than n iterations to converge. More commonly, finite precision CG computations converge in far fewer than n steps, and the same holds for the exact CG algorithm applied to any matrix A whose eigenvalues are clustered in tiny intervals about the eigenvalues of A. Frequently, finite precision CG computations go through several steps at which there is only a modest reduction in the error and then at the next step there is a very sharp decrease in the error. This same behavior is observed in the exact CG algorithm applied to matrices A whose eigenvalues are distributed in n tight clusters about the eigenvalues of A. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.