Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems

Abstract

The eigenvalues of a matrix A are the zeros of its characteristic polynomial \[ f ( λ ) = det [ A − λ I ] . f(\lambda ) = \det [A - \lambda I]. \] With Hyman’s method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigenvalues without computing their corresponding eigenvectors. Like all homotopy methods, our method solves the eigenvalue problem by following eigen-value paths of a real homotopy whose regularity is established to the extent necessary. The inevitable bifurcation and possible path jumping are handled by effective processes. The numerical results of our algorithm, and a comparison with its counterpart, subroutine HQR in EISPACK, are presented for upper Hessenberg matrices of numerous dimensions, with randomly generated entries. Although the main advantage of our method lies in its natural parallelism, the numerical results show our algorithm to be strongly competitive also in serial mode.