Summary
The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.
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References
Allgower, E., Georg, K.: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. SIAM Review22, 28–85 (1980)
Blattner, J.W.: Bordered matrices. J. Soc. Indust. Appl. Math.10, 528–536 (1962)
Chu, M.T.: A simple application of the homotopy method to symmetric eigenvalue problems. Linear Algebra Appl.59, 85–90 (1984)
Li, T.Y., Sauer, T., Yorke, J.: Numerical solution of a class of difficient polynomial systems. SIAM J. Numer. Anal.24, 435–451 (1987)
Longuet-Higgens, H.C., Opik, U., Pryce, M.H.L., Sack, R.A.: Studies of the John-Teller effect. Proceedings of the Royal Society244, 1–16 (1958)
Martin, R.S., Wilkinson, J.H.: Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection. Numerical Math.9, 386–393 (1967)
Ortega, J.M.: Numerical analysis: A second course. New York: Academic Press 1972
Parlett, B.N.: The symmetric eigenvalue problem, Englewood Cliffs: N.J., Prentice-Hall 1980
Rheinboldt, W.C.: Numerical analysis of parametrized nonlinear equations. New York Heidelberg Berlin: Wiley 1986
Ruhe, A.: Algorithms for the nonlinear cigenvalue problem. SIAM J. Numer. Anal.4, 674–689 (1973)
Smith, B.T.: Matrix eigensystem routines, EISPACK Guide, 2nd. New York Heidelberg Berlin: Springer 1976
Stoer, J., Bulirsch, R.: Introduction to numerical analysis. New York Heidelberg Berlin: Springer 1980
Wilkinson, J.H.: The calculation of the eigenvectors of codiagonal matrices. Computer Journal1, 90–96 (1958)
Wilkinson, J.H.: The algebraic eigenvalue problem. New York: Oxford University Press 1965
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Research was supported in part by NSF under Grant DMS-8701349