A fully parallel method for tridiagonal eigenvalue problem

Kuiyuan Li

doi:10.1155/s0161171294001043

Abstract

In this paper, a fully parallel method for finding all eigenvalues of a real matrix pencil (A, B) is given, where A and B are real symmetric tridiagonal and B is positive definite. The method is based on the homotopy continuation coupled with the strategy ?Divide‐Conquer? and Laguerre iterations. The numerical results obtained from implementation of this method on both single and multiprocessor computers are presented. It appears that our method is strongly competitive with other methods. The natural parallelism of our algorithm makes it an excellent candidate for a variety of advanced architectures.

Highlights

Nhen B is a well-conditioned generalized eigenvalue problem positive definite matrix, real symmetric can be reduced to the form
It is sufficient to assume that all the eigenvalues of matrix pencils
BI Bll and (AI, are available and we want to compute all eigenvalues of matrix pencil (A,B)

Summary

Introduction

Nhen B is a well-conditioned generalized eigenvalue problem positive definite matrix, real symmetric can be reduced to the form (2)L- 1AL- T(LTz) A(LTz)LLT. where A and B are real n xn symmetric matrices and B=There are many very efficient algorithms for (2), for instant, the QR algorithm [8], the D&C algorithm [3], the bisection algorithm [5] and the homotopy algorithm [6]. Nhen B is a well-conditioned generalized eigenvalue problem positive definite matrix, real symmetric can be reduced to the form We shall present a parallel homotopy method for finding all the eigenvalues or all eigenpairs of a matrix pencil (A,B), where A and B are both real symmetric tridiagonal and B is positive definite.

Results

Conclusion