zbMATH — the first resource for mathematics

Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices. (English) Zbl 0814.65037
The authors consider computation of the eigenvalue of maximum real part of a large sparse real nonsymmetric matrix, using subspace iteration preconditioned by the Cayley transform and/or shift-invert. A two step method is proposed for determining whether the set of eigenvalues so obtained includes the eigenvalue of maximum real part. The philosophy is one of robustness rather than speed, although a user has the option of performing only the first step.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F50 Computational methods for sparse matrices
eigs; IRAM
Full Text: DOI
[1] F. Chatelin,Spectral Approximation of Linear Operators Academic Press, New York, 1983. · Zbl 0517.65036
[2] F. Chatelin,Eigenvalues of Matrices, John Wiley & Sons, 1993. · Zbl 0783.65031
[3] K. N. Christodoulou and L. E. Scriven,Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem Center for Interfacial Engineering and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Ph.D. thesis, 1988. · Zbl 0677.65032
[4] K. A. Cliffe, T. J. Garratt, and A. Spence,Calculation of eigenvalues of the discretised Navier-Stokes and related equations, in The Mathematics of Finite Elements and Applications VII MAFELAP, J. R. Whiteman, ed., Academic Press, 1990, pp. 119–123.
[5] K. A. Cliffe and K. H. Winters,Convergence properties of the finite element method for Bénard convection in an infinite layer J. Comput. Phys., 60 (1985), pp. 346–351. · Zbl 0573.76082
[6] I. S. Duff and J. A. Scott,Computing selected eigenvalues of large sparse unsymmetric matrices using subspace iteration ACM Trans. Math. Software, 19 (1993), pp. 137–159. · Zbl 0888.65039
[7] T. J. Garratt,The numerical detection of Hopf bifurcations in large systems arising in fluid mechanics University of Bath, UK, Ph.D. thesis, 1991.
[8] T. J. Garratt, G. Moore, and A. Spence,A generalised Cayley transform for the numerical detection of Hopf bifurcations in large systems, Tech. Report, School of Mathematical Sciences, University of Bath, 1992. · Zbl 0834.65024
[9] T. J. Garratt, G. Moore, and A. Spence,Two methods for the numerical detection of Hopf bifurcations, in Bifurcation and Chaos: Analysis, Algorithms, Applications, R. Seydel, F. W. Schneider, and H. Troger, eds., Birkhäuser, 1991, pp. 119–123.
[10] R. F. Heinemann and A. B. Poore,Multiplicity, stability, and oscillatory dynamics of a tubular reactor Chem. Eng. Sci., 36 (1981), pp. 1411–1419.
[11] T. Manteuffel,An iterative method for solving nonsymmetric linear systems with dynamic estimation of parameters Technical Report UIUCDCS-R-75-758, Department of Computer Science, University of Illinois, Urbana-Champaign, Urbana, Illinois, 1975.
[12] T. Manteuffel,The Tchebyshev iteration for nonsymmetric linear systems Numer. Math., 28 (1977), pp. 307–327. · Zbl 0361.65024
[13] K. Meerbergen, A. Spence, and D. Roose,Shift-invert and Cayley transforms for the detection of right-most eigenvalues of nonsymmetric matrices, Report TW200 (Revised), Department of Computing Science, KU Leuven, 1993. · Zbl 0814.65037
[14] B. N. Parlett,The Symmetric Eigenvalue Problem Prentice Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, N.J., 1980.
[15] B. N. Parlett and Y. Saad,Complex shift and invert strategies for real matrices, Tech. Report YALEU/DCS/RR-424, University of California at Berkeley, Mathematics Department, 1985. · Zbl 0623.65045
[16] A. Ruhe,Rational Krylov algorithms for nonsymmetric eigenvalue problems, in IMA volumes in Mathematics and its Applications, G. Golub, M. Luskin, and A. Greenbaum, eds., Volume 60, Springer, 1993, pp. 149–164. · Zbl 0803.65045
[17] Y. Saad,Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems Math. Comp., 42 (1984), pp. 567–588. · Zbl 0539.65013
[18] Y. Saad,Numerical solution of large nonsymmetric eigenvalue problems Comput. Phys. Comm., 53 (1989), pp. 71–90. · Zbl 0798.65053
[19] Y. Saad,Numerical Methods for Large Eigenvalue Problems Manchester University Press, Manchester, 1992.
[20] R. A. Silverman,Introductory Complex Analysis Prentice-Hall, Englewood Cliffs, 1967. · Zbl 0145.29804
[21] D. C. Sorensen,Implicit application of polynomial filters in a k-step Arnoldi method SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357–385. · Zbl 0763.65025
[22] G. W. Stewart,Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices Numer. Math., 25 (1976), pp. 123–136. · Zbl 0328.65025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.