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Variational and numerical methods for symmetric matrix pencils

Published online by Cambridge University Press:  17 April 2009

Peter Lancaster  and
Qiang Ye
Peter Lancaster
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
Qiang Ye
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
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Abstract

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A review is presented of some recent advances in variational and numerical methods for symmetric matrix pencils λA – B in which A is nonsingular, A and B are hermitian, but neither is definite. The topics covered include minimax and maximin characterisations of eigenvalues, perturbation by semidefinite matrices and interlacing properties of real eigenvalues, Rayleigh quotient algorithms and their convergence properties, Rayleigh-Ritz methods employing Krylov subspaces, and a generalised Lanczos algorithm.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 1 - 17
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Beattie, C. and Fox, D.W., ‘Localization criteria and containment for Rayleigh quotient iteration’, SIAM J. Matrix Anal. Appl. 10 (1989), 80–93.CrossRefGoogle Scholar
[2]Burgoyne, N. and Cushman, R., ‘Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues’, Celestial Mech. 8 (1974), 435–443.CrossRefGoogle Scholar
[3]Courant, R., ‘Uber die Eigenwerte bei den Differentialgleichungen der mathematischen Physik’, Math Z. 7 (1920), 1–57.CrossRefGoogle Scholar
[4]Crandall, S.H., ‘Iterative procedures related to relaxation methods for eigenvalue problems’, Proc. Roy. Soc. London Ser. A 207 (1951), 416–423.Google Scholar
[5]Crawford, C.R., ‘A stable generalized eigenvalue problem’, SIAM J. Numer. Anal. 6 (1976), 854–860.CrossRefGoogle Scholar
[6]Duffin, R.J., ‘A minimax theory for overdamped networks’, Arch. Rational Mech. Anal. 4 (1955), 221–233.Google Scholar
[7]Ericsson, T. and Ruhe, A., ‘Lanczos algorithm and field of value rotations for symmetric matrix pencils’, Linear Algebra Appl. 88/89 (1987), 733–746.CrossRefGoogle Scholar
[8]Fischer, E., ‘Uber quadratische Formen mit reellen Koeffizienten’, Monatsh. Math. 16 (1905), 234–409.CrossRefGoogle Scholar
[9]Gohberg, I., Lancaster, P. and Rodman, L., Matrices and Indefinite Scalar Products (Birkhäuser, Basel, 1983).Google Scholar
[10]Inman, D.J., Vibration (Prentice Hall, Englewood Cliffs, 1989).Google Scholar
[11]Kahan, W., ‘Inclusion theorems for clusters of eigenvalues of hermitian matrices’, in Technical Report (Dept. of Comp. Sci., University of Toronto, 1967).Google Scholar
[12]Kahan, W., Parlett, B. and Jiang, E., ‘Residual bounds on approximate eigensystems of nonnormal matrices’, SIAM J. Numer. Anal. 19 (1982), 470–484.CrossRefGoogle Scholar
[13]Kaniel, S., ‘Estimates for some computational techniques in linear algebra’, Math. Comp. 20 (1966), 369–378.CrossRefGoogle Scholar
[14]Lancaster, P., ‘A generalized Rayleigh quotient iteration for lambda-matrices’, Arch. Rational Mech. Anal. 8 (1961), 309–322.CrossRefGoogle Scholar
[15]Laub, A. and Meyer, K., ‘Canonical forms for sympletic and Hamiltonian matrices’, Celestial Mechanics 9 (1974), 213–238.CrossRefGoogle Scholar
[16]Lancaster, P. and Ye, Q., ‘Inverse spectral problems for linear and quadratic matrix pencils’, Linear Alg. Appl. 107 (1988), 293–309.CrossRefGoogle Scholar
[17]Lancaster, P. and Ye, Q., ‘Variational properties and Rayleigh quotient algorithms for symmetric matrix pencils’, in Operator Theory: Advances and Applications 40, pp. 247–278 (Birkhauser, Basel, 1989).Google Scholar
[18]Lancaster, P. and Ye, Q., ‘Rayleigh-Ritz and Lanczos methods for symmetric matrix pencils’, (preprint).Google Scholar
[19]Lancaster, P. and Tismentsky, M., The Theory of Matrices (Academic Press, Orlando, 1985).Google Scholar
[20]Najman, B. and Ye, Q., ‘A minimax characterization for eigenvalues of hermitian pencils’, Linear Algebra Appl. (to appear).Google Scholar
[21]Ostrowski, A., ‘On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors’, I–VI, Arch. Rational Mech. Anal. 1–4 (1958/1959), 233–241, 423–428, 325–340, 341–347, 472–481, 153–165.Google Scholar
[22]Paige, C., The computation of eigenvalues and eigenvectors of very large sparse matrices, Ph.D. dissertation (University of London, 1971).Google Scholar
[23]Parlett, B.N., The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, N.J., 1980).Google Scholar
[24]Parlett, B.N., ‘The Rayleigh quotient iteration and some generalizations for non-normal matrices’, Math. Comp. 28 (1974), 679–693.CrossRefGoogle Scholar
[25]Parlett, B.N. and Chen, H., ‘Use of an indefinite inner product for computing damped natural modes’, (preprint).Google Scholar
[26]Parlett, B.N. and Kahan, W., ‘On the convergence of a practical QR algorithm’, Information Processing 68, I Mathematics, Software, pp. 114–118, (North-Holland, Amsterdam, 1969).Google Scholar
[27]Phillips, R.S., ‘A minimax characterization for the eigenvalues of a positive symmetric operator in a space with an indefinite metric’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), 51–59.Google Scholar
[28]Rogers, E.H., ‘A minimax theory for overdamped systems’, Arch. Rational Mech. Anal. 16 (1964), 89–96.CrossRefGoogle Scholar
[29]Saad, Y., ‘On the rate of convergence of the Lanczos and block Lanczos methods’, SIAM J. Numer. Anal. 17 (1980), 687–706.CrossRefGoogle Scholar
[30]Stewart, G.W., ‘Perturbation bounds for the definite generalized eigenvalue problem’, Linear Alg. Appl. 23 (1979), 69–85.CrossRefGoogle Scholar
[31]Temple, G., ‘The accuracy of Rayleigh's method of calculating the natural frequencies of vibrating systems’, Proc. Roy. Soc. London 211 (1952), 204–224.Google Scholar
[32]Textorius, B., ‘Minimaxprinzipe zur Bestimmung der Eigenwerte J-nichtnegativer Operatoren’, Math. Scand. 35 (1974), 105–114.CrossRefGoogle Scholar
[33]Turner, R., ‘Some variational principles for a nonlinear eigenvlaue problem’, J. Math. Anal. Appl. 17 (1967), 151–165.CrossRefGoogle Scholar
[34]Van Dooren, P. and Dewilde, P., ‘The eigenstructure of an arbitrary polynomial matrix: computational aspects’, Linear Algebra Appl. 50 (1983), 545–579.CrossRefGoogle Scholar
[35]Weierstrass, K., ‘Zur Theorie der bilinearen und quadratischen Formen’, Monatsber. Akad. Wiss. Berl. (1868), p. 310.Google Scholar
[36]Ye, Q., Variational principles and numerical algorithms for symmetric matrix pencils, Ph.D. Thesis (University of Calgary, 1989).Google Scholar