Spectral Condition-Number Estimation of Large Sparse Matrices
This provides a practical, memory-efficient tool for estimating condition numbers of large sparse matrices, which is important for numerical linear algebra and scientific computing.
The paper presents a randomized Krylov-subspace method (based on LSQR) to estimate the spectral condition number of large sparse matrices, achieving accurate estimates for a wide range of matrices while using minimal memory and handling cases where dense SVD is impractical.
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 a dense SVD 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.