Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization

Given an $m \times n$ matrix $M$ with $m \geqslant n$, it is shown that there exists a permutation $\Pi $ and an integer $k$ such that the QR factorization \[ M\Pi = Q\left( {\begin{array}{*{20}c} {A_k } & {B_k } \\ {} & {C_k } \\ \end{array} } \right) \] reveals the numerical rank of $M$: t...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 17; no. 4; pp. 848 - 869
Main Authors: Gu, Ming, Eisenstat, Stanley C.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.07.1996
Subjects:
Algebra
Algorithms
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Numerical analysis
Factorization method
Linear algebra
Conditioning
Algorithm
Factorization
Numerical stability
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:Given an $m \times n$ matrix $M$ with $m \geqslant n$, it is shown that there exists a permutation $\Pi $ and an integer $k$ such that the QR factorization \[ M\Pi = Q\left( {\begin{array}{*{20}c} {A_k } & {B_k } \\ {} & {C_k } \\ \end{array} } \right) \] reveals the numerical rank of $M$: the $k \times k$ upper-triangular matrix $A_k $ is well conditioned, $\|C_k \|_2 $ is small, and $B_k $is linearly dependent on $A_k $ with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take $O(mn^2 )$ floating-point operations in the worst case.
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ISSN:1064-8275
1095-7197
DOI:10.1137/0917055