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Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators

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Abstract

GivenA 1, the discrete approximation of a linear self-adjoint partial differential operator, the smallest few eigenvalues and eigenvectors ofA 1 are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operatorA 0 with known eigenvalues and eigenvectors, define the homotopyH(t)=(1−t)A 0+tA1, 0≤t≤1. If the eigenvectors ofH(t 0) are known, then they are used to determine the eigenpairs ofH(t 0+dt) via the Rayleigh quotient iteration, for some value ofdt. This is repeated untilt becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the step sizedt. A simple criterion to selectdt is given. It is shown that the iterative solver used to find the eigenvector at each step can be stabilized by applying a low-rank perturbation to the relevant matrix. By carrying out a small part of the calculation in higher precision, it is demonstrated that eigenvectors corresponding to clustered eigenvalues can be computed to high accuracy. Some numerical results for the Schrödinger eigenvalue problem are given. This algorithm will also be used to compute the bifurcation point of a parametrized partial differential equation.

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References

  1. E.L. Allgower and K. Georg,Numerical Continuation Methods (Springer, 1990).

  2. J.K. Cullum and R.A. Willoughby,Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I:Theory (Birkhäuser, 1985).

  3. E. Giladi, On the interplay between inner and outer iterations for a class of iterative methods, in:Colorado Conf. on Iterative Methods, Brechenridge, CO (1994).

  4. G. Golub and C. van Loan,Matrix Computations, 2nd ed. (John Hopkins U. P., Baltimore, 1989).

    Google Scholar 

  5. L.A. Hageman and D.M. Young,Applied Iterative Methods (Academic Press, 1981).

  6. L.J. Huang, Parallel Homotopy Algorithm for large sparse symmetric eigenproblems, PhD thesis, Michigan State University (1992).

  7. T. Kato,A Short Introduction to Perturbation Theory for Linear Operators (Springer, 1982).

  8. H.B. Keller,Lectures on Numerical Methods in Bifurcation Problems (Springer, Berlin, 1987), Tata Institute of Fundamental Research, Bombay, India.

    Google Scholar 

  9. K. Li and T.Y. Li, An algorithm for symmetric tridiagonal eigenproblems: Divide and conquer with homotopy continuation, SIAM J. Sci. Comp. 14 (1993) 735–751.

    Article  Google Scholar 

  10. T.Y. Li and N.H. Rhee, Homotopy algorithm for symmetric eigenvalue problems, Numer. Math. 55 (1989) 265–280.

    Article  Google Scholar 

  11. T.Y. Li, H. Zang and X.H. Sun, Parallel homotopy algorithm for the symmetric tridiagonal eigenvalue problem, SIAM J. Sci. Comp. 12 (1991) 469–487.

    Article  Google Scholar 

  12. T.Y. Li and Z. Zeng, Homotopy-determinant algorithm for solving non-symmetric eigenvalue problems, Math. Comp. 59 (1992) 483–502.

    Google Scholar 

  13. T.Y. Li, Z. Zeng and L. Cong, Solving eigenvalue problems of real nonsymmetric matrices with real homotopies, SIAM J. Numer. Anal. 29 (1992) 229–248.

    Article  Google Scholar 

  14. S.H. Lui, H.B. Keller and T.W.C. Kwok, Parallel homotopy method for the large sparse nonsymmetric eigenvalue problem, SIAM J. Matrix Anal. Appl. (1994), submitted.

  15. M. Oettli, A homotopy method applied to eigenvalue problems, Tech. Rep., ETH (1993).

  16. C.C. Paige and M.A. Saunders, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975) 617–629.

    Article  Google Scholar 

  17. C.C. Paige, G.P.H. Styan and P.G. Wachter, Computation of the stationary distribution of a Markov chain, J. Statist. Comp. Simul. 4 (1975) 173–186.

    Google Scholar 

  18. B.N. Parlett,The Symmetric Eigenvalue Problem (Prentice-Hall, 1980).

  19. G. Peters and J.H. Wilkinson, Inverse iteration, ill-conditioned equations and Newton's method, SIAM Rev. 21 (1979) 339–360.

    Article  Google Scholar 

  20. A. Ruhe and T. Wiberg, The method of conjugate gradients used in inverse iteration, BIT 12 (1972) 543–554.

    Article  Google Scholar 

  21. D.B. Szyld, Criteria for combining inverse and Rayleigh quotient iteration, SIAM J. Numer. Anal. 25 (1988) 1369–1375.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    S. H. Lui

  2. Computer Science Department, Stanford University, 94305-2140, Stanford, CA, USA

    G. H. Golub

Authors
  1. S. H. Lui
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  2. G. H. Golub
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Additional information

Communicated by C. Brezinski

Dedicated to Herbert Bishop Keller on the occasion of his 70th birthday

The work of this author was in part supported by RGC Grant DAG93/94.SC30.

The work of this author was in part supported by NSF Grant DMS-9403899.

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Lui, S.H., Golub, G.H. Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators. Numer Algor 10, 363–378 (1995). https://doi.org/10.1007/BF02140775

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  • DOI: https://doi.org/10.1007/BF02140775

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