A rank-revealing method with updating, downdating, and applications

A new rank-revealing method is proposed. For a given matrix and a threshold for near-zero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the approximate...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 2005, Vol.26 (4), p.918-946
Main Authors: LI, T. Y, ZHONGGANG ZENG
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Applied mathematics
Decomposition
Exact sciences and technology
Linear algebra
Linear and multilinear algebra, matrix theory
Mathematics
Matrix
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
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Summary:A new rank-revealing method is proposed. For a given matrix and a threshold for near-zero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the approximate rank of the matrix along with an orthonormal basis for the approximate null space. When a row or column is inserted or deleted, algorithms for updating/downdating the approximate rank and null space are straightforward, stable, and efficient. Numerical results exhibiting the advantages of our code over existing packages based on two-sided orthogonal rank-revealing decompositions are presented. Also presented are applications of the new algorithm in numerical computation of the polynomial GCD as well as identification of nonisolated zeros of polynomial systems.
ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479803435282