Asymptotic Quadratic Convergence of the Two-Sided Serial and Parallel Block-Jacobi SVD Algorithm

Abstract

We present a proof of the global and asymptotic quadratic convergence of the serial and parallel two-sided block-Jacobi SVD algorithm with dynamic ordering. In the serial case, one pair of the off-diagonal blocks with the largest weight given as the sum of squares of Frobenius norms is annihilated. In the parallel case, using the greedy implementation of dynamic ordering and having $p$ processors, $p$ pairs of the off-diagonal blocks with largest weights, and disjoint block row and column indices are annihilated in each parallel iteration step. Additionally, the asymptotic quadratic convergence is also proved for the scaled iterated matrix, both in serial and parallel cases. Numerical examples confirm the developed theory.