New Preconditioning for the One-Sided Block-Jacobi SVD Algorithm

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 AV(1) instead of A, where V-1 is the matrix of eigenvectors from the eigenvalue decomposition of the Gram matrix A(T) A. In exact arithmetic, V-1 is also the matrix of right singular vectors of A so that the columns of AV(1) 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 span(AV(1)) from span(U) depends on the 2-norm condition number.(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.