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 \(A^TA\). In exact arithmetic, \(V_1\) is also the matrix of right singular vectors of A so that the columns of \(AV_1\) lie in \(\mathrm {span}(U)\), where U is the matrix of left singular vectors. However, in finite arithmetic, this is not true in general, and the deviation of \(\mathrm {span}(AV_1)\) from \(\mathrm {span}(U)\) depends on the 2-norm condition number \(\kappa (A)\). 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.