Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy (Classic Reprint) - Softcover

Demmel, James

Softcover

ISBN 10: 1332870732 ISBN 13: 9781332870738

Verlag: Forgotten Books, 2018

Inhaltsangabe

This book is a groundbreaking analysis of the accuracy of singular value computations for bidiagonal matrices. Singular value decomposition is a fundamental tool in many scientific and engineering applications, but the accuracy of computed singular values has been a long-standing problem. The author presents a new algorithm for computing singular values that is guaranteed to produce high relative accuracy, even for small singular values. This is a significant advance over existing algorithms, which can suffer from large relative errors in computed singular values. The book also provides a detailed error analysis of the new algorithm, showing that it is stable and accurate even in the presence of rounding errors. This makes the algorithm suitable for use in a wide variety of applications. The book's insights into the accuracy of singular value computations are significant because they provide a foundation for understanding the behavior of singular value decomposition algorithms. This knowledge can be used to improve the accuracy of these algorithms and to develop new algorithms that are more efficient and reliable.

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Reseña del editor

Excerpt from Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy

Now we present our central result of this section (a slightly weaker version originally appeared in an unpublished report [kahan])

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.

Reseña del editor

Excerpt from Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy

There are some situations where the smallest singular values are determined much more accurately by the data than a simple bound of the form f would indicate. In this paper we will show that for bidiagonal matrices the singular values are determined to the same relative precision as the individual matrix entries. In other words, if all the matrix entries are known to high relative accuracy, all the singular values are also known to high relative accuracy independent of their magnitudes. We will prove similar theorems about the eigenvalues of two kinds of symmetric tridiagonal matrices: those with zero diagonal, and diagonally dominant ones (diagonal dominance will be defined in section 5 below; it includes certain graded maniocs).

In such situations it is desirable to have an algorithm to compute the singular values or eigenvalues to the accuracy to which they are determined by the data. In this paper we present an algorithm for computing all the singular values of a bidiagonal matrix to guaranteed high relative accuracy, independent of their magnitudes. Our algorithm is a varia tion of the usual QR iteration which is used in the standard svd algorithm. Briefly, it is a hybrid algorithm of the usual QR iteration with a zero-shifted QR modified to guarantee forward stability. Numerical experience, which we report below, shows that it is generally faster than the standard algorithm, and ranges from times faster to times slower counting reduction to bidiagonal form times faster to times slower not counting reduction to bidiagonal form). This algorithm may be also be used to accurately compute all the eigenvalues of symmetric tridiagonal matrices with zero diagonal, and positive definite diagonally dominant symmetric tridiagonal matrices (this category includes graded matrices).

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.

Details zur Ausgabe

• Verlag: Forgotten Books
• Erscheinungsdatum: 2018
• ISBN 10: 1332870732
• ISBN 13: 9781332870738
• Einband: Tapa blanda
• Sprache: Englisch
• Anzahl der Seiten: 51
• Kontakt zum Hersteller: Nicht verfügbar