A Jacobi eigenreduction algorithm for definite matrix pairs

Abstract

We propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ 2=I,J diagonal (neitherJ norA itself need be definite). Our Jacobi reduction works only on one matrix and usesJ-orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process. For the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization. We prove the global convergence of the method; the quadratic convergence was observed in all experiments. We apply our method in two situations: (i) eigenreducing a single real symmetric matrix and (ii) eigenreducing an overdamped quadratic matrix pencil. In both cases our method is preceded by a symmetric indefinite decomposition and performed in its "one-sided" variant on the thus obtained factors. Our method outdoes the standard methods like standard Jacobi orqr/ql in accuracy in spite of the use of hyperbolic transformations which are not orthogonal (a theoretical justification of this behaviour is made elsewhere). The accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order. In addition, in working with quadratic pencils our method is shown to either converge or to detect non-overdampedness.