Irr: An algorithm for computing the smallest singular value of large scale matrices

For large-scale matrices, there is no practical algorithm to compute the smallest singular value with a satisfied relative accuracy. The widely used bidiagonalization Lanczos method can compute the largest singular value with good relative accuracy, but not the smallest one. In this paper we transform the smallest singular value of matrix A to the largest eigenvalue of (A^T A)^-1 and use Rayleigh-Ritz method, which is referred as Inverse-Rayleigh - Ritz (IRR) method. The technique computing quadratic form plays an important role in IRR. IRR takes no more flop cost and storage than Lanczos-like Krylov methods on A and gives more accurate results.