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Algorithm 933: Reliable calculation of numerical rank, null space bases, pseudoinverse solutions, and basic solutions using suitesparseQR

Published: 03 October 2013 Publication History

Abstract

The SPQR_RANK package contains routines that calculate the numerical rank of large, sparse, numerically rank-deficient matrices. The routines can also calculate orthonormal bases for numerical null spaces, approximate pseudoinverse solutions to least squares problems involving rank-deficient matrices, and basic solutions to these problems. The algorithms are based on SPQR from SuiteSparseQR (ACM Transactions on Mathematical Software 38, Article 8, 2011). SPQR is a high-performance routine for forming QR factorizations of large, sparse matrices. It returns an estimate for the numerical rank that is usually, but not always, correct. The new routines improve the accuracy of the numerical rank calculated by SPQR and reliably determine the numerical rank in the sense that, based on extensive testing with matrices from applications, the numerical rank is almost always accurately determined when our methods report that the numerical rank should be correct. Reliable determination of numerical rank is critical to the other calculations in the package. The routines work well for matrices with either small or large null space dimensions.

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Software for Reliable calculation of numerical rank, null space bases, pseudoinverse solutions, and basic solutions using suitesparseQR

References

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  1. Algorithm 933: Reliable calculation of numerical rank, null space bases, pseudoinverse solutions, and basic solutions using suitesparseQR

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        cover image ACM Transactions on Mathematical Software
        ACM Transactions on Mathematical Software  Volume 40, Issue 1
        September 2013
        165 pages
        ISSN:0098-3500
        EISSN:1557-7295
        DOI:10.1145/2513109
        Issue’s Table of Contents
        Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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        Publication History

        Published: 03 October 2013
        Accepted: 01 March 2013
        Revised: 01 May 2012
        Received: 01 May 2011
        Published in TOMS Volume 40, Issue 1

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        Author Tags

        1. Numerical rank
        2. QR factorization
        3. null space
        4. pseudoinverse
        5. rank revealing
        6. sparse matrices

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        • (2022)Prestressed elasticity of amorphous solidsPhysical Review Research10.1103/PhysRevResearch.4.0431814:4Online publication date: 12-Dec-2022
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