A Parallel Algorithm for the Nonsymmetric Eigenvalue Problems

This paper presents a parallel algorithm to solve the eigenvalue problem for non-symmetric matrices. The idea of homotopy is used to generate initial approximations, then the Aberth method and our modiied Aberth method are used to nd simultaneously all the eigenvalues. The advantage of this approach is that multiple eigenvalues can be handled. In addition, the multiplicities of the eigenvalues can be detected dynamically. In this paper we consider the problem of nding all the eigenvalues of a nonsymmetric matrix. Without loss of generality we assume that the matrix is irreducible upper Hessenberg matrix, A = (a ij) = 0 B B B B B B B @ a 21 a 32 0 a n?1;n?2 a n;n?1 1 C C C C C C C A : Let D be the matrix obtained from A by making a subdiagonal entry a j;j?1 zero, ! : The eigenvalues of D can be calculated by calculating the eigenvalues of A 11 and A 22 in parallel. To calculate the eigenvalues of A the following homotopy is established, H(; t) = c (1 ? t) det(D ? I) + t det(A ? I) ; where c is a random complex number. The solutions of H(; t) = 0, 1 (t); 2 (t); :::; n (t), are smooth functions of t which connect the eigenvalues of D and A. They do not meet in the t interval (0; 1) with probability one. The Aberth method and our modiied Aberth method are then applied to follow these curves simultaneously. The Aberth method 1] is an eecient iterative method of nding simultaneously all the roots of polynomials with simple roots. We modiied this method to nd all the roots of ANY polynomial without sacriicing eeciency. Both methods converge cubically. A distinguished property both of the original and of these methods is that they are globally convergent for almost all initial approximations.