New preconditioning for the one-sided block-Jacobi SVD algorithm

Abstract

New preconditioning for the one-sided block-Jacobi algorithm used for the computation of the singular value decomposition of a matrix A is proposed. To achieve the asymptotic quadratic convergence quickly, one can apply the Jacobi algorithm to the matrix AV1 instead of A, where V1 is the matrix of eigenvectors from the eigenvalue decomposition of the Gram matrix ATA. In exact arithmetic, V1 is also the matrix of right singular vectors of A so that the columns of AV1 lie in span(U), where U is the matrix of left singular vectors. However, in finite arithmetic, this is not true in general, and the deviation of (AV1) from span(U) depends on the 2-norm condition number κ(A). The performance of the new preconditioned one-sided block-Jacobi algorithm was compared with three other SVD procedures. In the case of well-conditioned matrix, the new algorithm is up to 25 times faster than the LAPACK Jacobi procedure DGESVJ.