Bit-level systolic algorithms for real symmetric and Hermitian eigenvalue problems

Abstract

Arithmetic algorithms are presented that speed up the parallel Jacobi method for the eigen-decomposition of real symmetric and complex Hermitian matrices. The 2×2 submatrices to which the Jacobi rotations are applied form a Clifford algebra, hence they can be decomposed into a sum of even and odd part. This decomposition enables the application of the rotations from a single side instead of both, thus removing some sequentiality from the original Jacobi method. Moreover, with the help of implicit CORDIC algorithms, the rotations are evaluated and applied in a fully concurrent fashion on triangular arrays of specialized processors. The CORDIC algorithms employed in the complex case are genuine 3- and 4-dimensional generalizations of the 2-dimensional algorithm used in the real case. Because these algorithms are implicit, variants are obtained with minor modifications that perform rotations whose resolution is poor at first and slowly increases to become high in the last steps of the Jacobi method. Such variants further reduce the total computation time.