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Published April 1997 | Published
Journal Article Open

Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem


A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of A(t) ≡ (1 − t)A0 + tA1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: • bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented.

Additional Information

© 1997 Society for Industrial and Applied Mathematics. Received by the editors September 7, 1994; accepted for publication (in revised form) by R. Freund April 29, 1996. The work of this author [S.H.L.] was supported in part by RGC grant DAG92/93.SC16. We thank the referees for suggesting numerous improvements to the original draft. This paper is dedicated to Gene H. Golub on the occasion of his 65th birthday.

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