Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations (Classic Reprint) - Softcover
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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 11 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 it tight clusters about the eigenvalues of A.
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Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient
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Paperback. Condition: New. Print on Demand. This book explores the intricate relationship between finite-precision and exact Lanczos and Conjugate Gradient computations applied to solve symmetric positive definite linear systems or compute eigenvalues of symmetric matrices. It offers valuable insights into the behavior of the finite-precision implementations of these algorithms through a novel analogy with their exact counterparts operating on larger matrices with specific eigenvalue clusters. By establishing this connection, the author provides a framework for understanding and predicting the convergence rates, accuracy, and other attributes of finite-precision computations in practical scenarios. The book contributes to the broader field of numerical linear algebra by deepening our understanding of the practical behavior of these important algorithms and their limitations due to finite-precision arithmetic. It offers a unique perspective that can guide researchers and practitioners working with finite-precision computations and inspire further advancements in the field. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Seller Inventory # 9781333699888_0
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Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations Classic Reprint
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Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations Classic Reprint
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