Spectral condition-number estimation of large sparse matrices

Haim Avron*, Alex Druinsky, Sivan Toledo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We describe a randomized Krylov-subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of the smallest singular value σ min of A. Our method estimates this value by solving a consistent linear least squares problem with a known solution using a specific Krylov-subspace method called LSQR. In this method, the forward error tends to concentrate in the direction of a right singular vector corresponding to σ min . Extensive experiments show that the method is able to estimate well the condition number of a wide array of matrices. It can sometimes estimate the condition number when running dense singular value decomposition would be impractical due to the computational cost or the memory requirements. The method uses very little memory (it inherits this property from LSQR), and it works equally well on square and rectangular matrices.

Original languageEnglish
Article numbere2235
JournalNumerical Linear Algebra with Applications
Issue number3
StatePublished - May 2019


FundersFunder number
Defense Advanced Research Projects AgencyXDATA FA8750-12-C-0323
International Business Machines Corporation1045/09
Air Force Research LaboratoryFA8750-12-C-0323
United States-Israel Binational Science Foundation2010231
Israel Academy of Sciences and Humanities
Israel Science Foundation


    • Krylov methods
    • condition-number estimation
    • randomized numerical linear algebra


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