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The nearest definite pair for the Hermitian generalized eigenvalue problem. (English) Zbl 0947.65042
Given a pair $$(A,B)$$ of Hermitian matrices of the same order, the authors give formulae for the nearest pair of Hermitian matrices that has a Crawford number greater or equal to a given positive value. The result is expressed in terms of the inner numerical radius associated with the field of values. Then they show how to exploit the result in order to solve the generalized eigenvalue problem $$Ax=\lambda Bx$$ with nearly singular $$B$$. The paper includes the algorithm and numerical examples.
Reviewer: Z.Dostal (Ostrava)

##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices
PDFIND
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##### References:
 [1] Braconnier, T.; Higham, N.J., Computing the field of values and pseudospectra using the Lanczos method with continuation, Bit, 36, 3, 422-440, (1996) · Zbl 0861.65033 [2] Crawford, C.R., A stable generalized eigenvalue problem, SIAM J. numer. anal., 13, 6, 854-860, (1976) · Zbl 0348.15005 [3] Crawford, C.R., ALGORITHM 646 PDFIND: A routine to find a positive definite linear combination of two real symmetric matrices, ACM trans. math. software, 12, 278-282, (1986) · Zbl 0613.65041 [4] Crawford, C.R.; Moon, Y.S., Finding a positive definite linear combination of two Hermitian matrices, Linear algebra appl., 51, 37-48, (1983) · Zbl 0516.15021 [5] Elsner, L.; Sun, Ji guang, Perturbation theorems for the generalized eigenvalue problem, Linear algebra appl., 48, 341-357, (1982) · Zbl 0504.15012 [6] He, Chunyang; Watson, G.A., An algorithm for computing the numerical radius, IMA J. numer. anal., 17, 329-342, (1997) · Zbl 0880.65019 [7] Higham, N.J., Computing a nearest symmetric positive semidefinite matrix, Linear algebra appl., 103, 103-118, (1988) · Zbl 0649.65026 [8] N.J. Higham, Matrix nearness problems and applications, in: M.J.C. Gover, S. Barnett (Eds.), Applications of Matrix Theory, Oxford University Press, Oxford, 1989, pp. 1-27 [9] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. xiii+561 pp. ISBN 0-521-30586-1 · Zbl 0576.15001 [10] Li, Chi-Kwong; Mathias, R., Generalized eigenvalues of a definite Hermitian matrix pair, Linear algebra and appl., 271, 309-321, (1998) · Zbl 0890.15007 [11] Li, Ren-Cang, A perturbation bound for definite pencils, Linear algebra and appl., 179, 191-202, (1993) · Zbl 0813.15017 [12] Li, Ren-Cang, On perturbations of matrix pencils with real spectra, Math. comp., 62, 205, 231-265, (1994) · Zbl 0795.15012 [13] Maroulas, J.; Psarrakos, P., The boundary of the numerical range of matrix polynomials, Linear algebra and appl., 267, 101-111, (1997) · Zbl 0888.15012 [14] Parlett, B.N., Symmetric matrix pencils, J. comp. appl. math., 38, 373-385, (1991) · Zbl 0772.15005 [15] B.N. Parlett, The Symmetric Eigenvalue Problem, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1998, pp. xxiv+398 (unabridged, amended version of book first published by Prentice-Hall in 1980, ISBN 0-89871-402-8) · Zbl 0431.65017 [16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge, 1992, pp. xxvi+963, ISBN 0 521 43064 X · Zbl 0778.65002 [17] Stewart, G.W., Perturbation bounds for the definite generalized eigenvalue problem, Linear algebra and appl., 23, 69-85, (1979) · Zbl 0407.15012 [18] G.W. Stewart, Ji-guang Sun, Matrix Perturbation Theory, Academic Press, London, 1990, pp. xv+365, ISBN 0-12-670230-6 · Zbl 0706.65013 [19] Watson, G.A., Computing the numerical radius, Linear algebra and appl., 234, 163-172, (1996) · Zbl 0848.65030
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