The Bidiagonal Singular Value Decomposition and Hamiltonian Mechanics

Percy Deift, James Demmel

ISBN 10: 1334196907 / ISBN 13: 9781334196904

Editore: Forgotten Books, 2018

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Print on Demand. This book delves into the complex world of singular value decomposition (SVD), a fundamental mathematical operation used to analyze matrices. It examines the process of computing the SVD of a bidiagonal matrix, a specific type of matrix that arises in various applications, including the SVD of general matrices and the eigenproblem of symmetric tridiagonal matrices. The author's exploration of this topic builds upon existing research in the field, particularly the work of Demmel and Kahan, and presents new insights regarding the accuracy of SVD computations. The book's core theme revolves around demonstrating that SVD calculations on bidiagonal matrices can achieve significantly higher accuracy than those performed on general matrices. This unexpected result stems from the nature of bidiagonal matrices, where relative errors in the entries lead to relative errors in the singular values, unlike in general matrices where absolute errors are dominant. Furthermore, the book introduces a novel Hamiltonian approach to analyze the algorithm used for SVD calculations. This approach, drawing upon the concepts of Hamiltonian mechanics, reveals a deeper understanding of how errors propagate during the computation process. The book's exploration of Hamiltonian mechanics and its application to the SVD algorithm offers a new paradigm for developing high-accuracy algorithms for solving other eigenvalue problems. 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. Codice articolo 9781334196904_0

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Titolo: The Bidiagonal Singular Value Decomposition ...

Casa editrice: Forgotten Books

Data di pubblicazione: 2018

Legatura: Paperback

Condizione: New

Tipologia articolo: print-on-demand item

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Excerpt from The Bidiagonal Singular Value Decomposition and Hamiltonian Mechanics

In summary, absolute uncertainties in the entries of a general matrix A yield absolute error bounds on its singular values, and error bounds depending on the absolute gap for its singular vectors. In contrast, relative uncertainties in the entries of a bidiagonal matrix B yield relative error bounds on its singular values, and error bounds depending on the relative gap for its singular vectors.