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We describe a randomized Krylov-subspace method for estimating the spectralcondition number of a real matrix A or indicating that it is numerically rankdeficient. The main difficulty in estimating the condition number is theestimation of the smallest singular value \sigma_{\min} of A. Our methodestimates this value by solving a consistent linear least-squares problem witha known solution using a specific Krylov-subspace method called LSQR. In thismethod, the forward error tends to concentrate in the direction of a rightsingular vector corresponding to \sigma_{\min}. Extensive experiments show thatthe method is able to estimate well the condition number of a wide array ofmatrices. It can sometimes estimate the condition number when running a denseSVD would be impractical due to the computational cost or the memoryrequirements. The method uses very little memory (it inherits this propertyfrom LSQR) and it works equally well on square and rectangular matrices.

Spectral Condition-Number Estimation of Large Sparse Matrices