Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy (Classic Reprint) - Tapa blanda
Sinopsis
High-precision singular values for bidiagonal matrices, with a faster, stable algorithm. A new approach blends the standard QR iteration with a zero-shift variant to guarantee forward stability and high relative accuracy for all singular values, regardless of their size.
This book presents the theory and the algorithm, showing how small relative changes in the data lead to small relative changes in the singular values. It also explains how the method can be used to compute eigenvalues of symmetric tridiagonal matrices and discusses convergence criteria, practical implementation details, and numerical results that compare performance with traditional methods.
- How a QR-based algorithm can achieve guaranteed high relative accuracy for every singular value of a bidiagonal matrix.
- Conditions and perturbation results that justify stability and accurate results across a range of matrix structures.
- The role of a forward-stable, zero-shift modification to the QR iteration and its practical impact on speed.
- Numerical experiments and practical considerations, including when and how the method outperforms standard approaches.
Ideal for readers of numerical linear algebra and computational mathematics seeking robust, precision-focused methods for singular value and eigenvalue problems.
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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Now we present our central result of this section (a slightly weaker version originally appeared in an unpublished report [kahan])
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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.
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Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High
Impresión bajo demandaLibrería: Forgotten Books, London, Reino Unido
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Paperback. Condición: New. Print on Demand. 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. 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. Nº de ref. del artículo: 9781332870738_0
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Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy Classic Reprint
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PAP. Condición: New. New Book. Shipped from UK. Established seller since 2000. Nº de ref. del artículo: LW-9781332870738
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Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy Classic Reprint
Librería: PBShop.store UK, Fairford, GLOS, Reino Unido
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PAP. Condición: New. New Book. Shipped from UK. Established seller since 2000. Nº de ref. del artículo: LW-9781332870738
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