Topic
RRQR factorization
About: RRQR factorization is a research topic. Over the lifetime, 8 publications have been published within this topic receiving 426 citations.
Papers
TL;DR: In this paper, a constructive proof of the existence of the rank-revealing QR factorization of any matrix A of size m x n with numerical rank r is given. But it is not clear how to find a rank revealing RRQR of A if A has numerical rank deficiency.
Abstract: T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rank-revealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size m x n with numerical rank r . The bounds derived in this paper that guarantee the existence of RRQR are all of order f i ,in comparison with Chan's 0(2"-') . It has been known for some time that if A is only numerically rank-one deficient, then the column permutation l7 of A that guarantees a small rnn in the QR factorization of A n can be obtained by inspecting the size of the elements of the right singular vector of A corresponding to the smallest singular value of A . To some extent, our paper generalizes this well-known result. We consider the interplay between two important matrix decompositions: the singular value decomposition and the QR factorization of a matrix A . In particular, we are interested in the case when A is singular or nearly singular. It is well known that for any A E R m X n (a real matrix with rn rows and n columns, where without loss of generality we assume rn > n) there are orthogonal matrices U and V such that where C is a diagonal matrix with nonnegative diagonal elements: We assume that a, 2 a2 2 . . 2 on 2 0 . The decomposition (0.1) is the singular value decomposition (SVD) of A , and the ai are the singular values of A . The columns of V are the right singular vectors of A , and the columns of U are the left singular vectors of A . Mathematically, in terms of the singular values, Received December 1, 1990; revised February 8, 199 1. 199 1 Mathematics Subject Classification. Primary 65F30, 15A23, 15A42, 15A15.
198 citations
TL;DR: An efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy, aided by incremental condition estimation is developed.
Abstract: We develop algorithms and implementations for computing rank-revealing QR (RRQR) factorizations of dense matrices. First, we develop an efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy, aided by incremental condition estimation. Second, we develop efficiently implementable variants of guaranteed reliable RRQR algorithms for triangular matrices originally suggested by Chandrasekaran and Ipsen and by Pan and Tang. We suggest algorithmic improvements with respect to condition estimation, termination criteria, and Givens updating. By combining the block algorithm with one of the triangular postprocessing steps, we arrive at an efficient and reliable algorithm for computing an RRQR factorization of a dense matrix. Experimental results on IBM RS/6000 SGI R8000 platforms show that this approach performs up to three times faster that the less reliable QR factorization with column pivoting as it is currently implemented in LAPACK, and comes within 15% of the performance of the LAPACK block algorithm for computing a QR factorization without any column exchanges. Thus, we expect this routine to be useful in may circumstances where numerical rank deficiency cannot be ruled out, but currently has been ignored because of the computational cost of dealing with it.
152 citations
TL;DR: The main contribution is an efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy and improved versions of theRRQR algorithms for triangular matrices orginally suggersted by Chandrasekaran and Ipsen and by Pan and Tang.
Abstract: This article describes a suite of codes as well as associated testing and timing drivers for computing rank-revealing QR (RRQR) factorizations of dense matrices. The main contribution is an efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy and improved versions of the RRQR algorithms for triangular matrices orginally suggersted by Chandrasekaran and Ipsen and by Pan and Tang, respectively, We highlight usage and features of these codes.
72 citations
TL;DR: A framework is presented for the efficient implementation of RRQR algorithms, in particular, for sparse matrices, using a restricted pivoting strategy guarded by incremental condition estimation (ICE) and the algorithm suggested by Chan and Foster to this QR-factorization.
Abstract: The rank-revealing QR-factorization (RRQR factorization) is a special QR-factorization that is guaranteed to reveal the numerical rank of the matrix under consideration. This makes the RRQR-factorization a useful tool in the numerical treatment of many rank-deficient problems in numerical linear algebra. In this paper, a framework is presented for the efficient implementation of RRQR algorithms, in particular, for sparse matrices. A sparse RRQR-algorithm should seek to preserve the structure and sparsity of the matrix as much as possible while retaining the ability to capture safely the numerical rank. To this end, the paper proposes to compute an initial QR-factorization using a restricted pivoting strategy guarded by incremental condition estimation (ICE), and then applies the algorithm suggested by Chan and Foster to this QR-factorization. The column exchange strategy used in the initial QR factorization will exploit the fact that certain column exchanges do not change the sparsity structure, and compu...
52 citations
4 Nov 1991
TL;DR: In this paper, an updating scheme for the rank revealing QR (RRQR) algorithm described earlier by T.F. Chan was investigated and applied to the direction of arrival problem, which allows tracking of moving sources by taking advantage of the simplicity of the regular QR updating scheme and the rank-revealing property of the RRQR factorization.
Abstract: The author investigates an updating scheme for the rank revealing QR (RRQR) algorithm described earlier by T.F. Chan (see Linear Algebr. Appl., vol.88, no 89, p.67-82 1987) and applies it to the direction of arrival problem. This technique allows for tracking of moving sources by taking advantage of the simplicity of the regular QR updating scheme and the rank-revealing property of the RRQR factorization. Subspace methods and the RRQR technique are reviewed. It is shown that the RRQR algorithm can be used to update signal and noise subspaces from the noise-free correlation matrix. Experimental results and comparisons with eigen-based signal and noise subspaces are presented. >
8 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2007 | 1 |
| 1998 | 2 |
| 1997 | 1 |
| 1992 | 1 |
| 1991 | 2 |