On Relative Accuracy of the One-Sided Block-Jacobi SVD Algorithm

Abstract

We are interested in the relative accuracy of computed singular values in the serial real one-sided block-Jacobi SVD algorithm with dynamic ordering using approximate weights, and in the orthogonality of computed left and right singular vectors. Test matrices are of the form $$A=BD$$ , where B is a random matrix with a prescribed condition number $$\kappa (B)$$ and D is diagonal with given $$\kappa (D)$$ . We compare the relative accuracy of singular values as well as the orthogonality of left and right singular vectors computed by the Jacobi SVD algorithm with results computed using the SVD algorithm based on the matrix bi-diagonalization. When B is well-conditioned, the one-sided block-Jacobi algorithm inherits a high relative accuracy from its element-wise counterpart over a wide range of condition numbers $$\kappa (A)$$ .