New Fast and Accurate Jacobi SVD Algorithm. II

This paper presents new implementation of one–sided Jacobi SVD for triangular matrices and its use as the core routine in a new preconditioned Jacobi SVD algorithm, recently proposed by the authors. New pivot strategy exploits the triangular form and uses the fact that the input triangular matrix is the result of rank revealing QR factorization. If used in the preconditioned Jacobi SVD algorithm, it delivers superior performance leading to the currently fastest method for computing SVD decomposition with high relative accuracy. Furthermore, the efficiency of the new algorithm is comparable to the less accurate bidiagonalization based methods. The paper also discusses underflow issues in floating point implementation, and shows how to use perturbation theory to fix the imperfectness of machine arithmetic on some systems.