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Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems \(Ax=\lambda Bx\) with singular \(B\). (English) Zbl 1133.65020
Summary: In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem \( Ax=\lambda Bx\) are needed. If exact linear solvers with \( A-\sigma B\) are available, implicitly restarted Arnoldi with purification is a common approach for problems where \( B\) is positive semidefinite.
In this paper, a new approach based on implicitly restarted Arnoldi is presented that avoids most of the problems due to the singularity of \( B\). Secondly, if exact solvers are not available, Jacobi-Davidson QZ will be presented as a robust method to compute a few specific eigenvalues. The results are illustrated by numerical experiments.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F50 Computational methods for sparse matrices
Full Text: DOI
[1] K. A. Cliffe, T. J. Garratt, and A. Spence, Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations, Adv. Comput. Math. 1 (1993), no. 3-4, 337 – 356. · Zbl 0830.76048
[2] K. A. Cliffe, T. J. Garratt, and A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics, SIAM J. Matrix Anal. Appl. 15 (1994), no. 4, 1310 – 1318. · Zbl 0807.65030
[3] K. A. Cliffe, A. Spence, and S. J. Tavener, The numerical analysis of bifurcation problems with application to fluid mechanics, Acta numerica, 2000, Acta Numer., vol. 9, Cambridge Univ. Press, Cambridge, 2000, pp. 39 – 131. · Zbl 1005.65138
[4] Howard C. Elman, David J. Silvester, and Andrew J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. · Zbl 1083.76001
[5] Thomas Ericsson, A generalised eigenvalue problem and the Lánczos algorithm, Large scale eigenvalue problems (Oberlech, 1985) North-Holland Math. Stud., vol. 127, North-Holland, Amsterdam, 1986, pp. 95 – 119. · Zbl 0605.65026
[6] Thomas Ericsson and Axel Ruhe, The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems, Math. Comp. 35 (1980), no. 152, 1251 – 1268. · Zbl 0468.65021
[7] Diederik R. Fokkema, Gerard L. G. Sleijpen, and Henk A. Van der Vorst, Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils, SIAM J. Sci. Comput. 20 (1998), no. 1, 94 – 125. · Zbl 0924.65027
[8] T. J. Garratt, The numerical detection of Hopf bifurcations in large systems arising in fluid mechanics, Ph.D. thesis, University of Bath, 1991.
[9] Philip M. Gresho, David K. Gartling, J. R. Torczynski et al., Is the steady viscous incompressible two-dimensional flow over a backward-facing step at \?\?=800 stable?, Internat. J. Numer. Methods Fluids 17 (1993), no. 6, 501 – 541. · Zbl 0784.76050
[10] R. B. Lehoucq and J. A. Scott, Implicitly restarted Arnoldi methods and eigenvalues of the discretized Navier Stokes equations, Tech. Report SAND97-2712J, Sandia National Laboratory, 1997.
[11] R. B. Lehoucq and D. C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl. 17 (1996), no. 4, 789 – 821. · Zbl 0863.65016
[12] R. H. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK SOFTWARE, see: http://www.caam.rice.edu/software/ARPACK/.
[13] Karl Meerbergen and Dirk Roose, The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalues, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 1 – 20. · Zbl 0872.65038
[14] Karl Meerbergen and Alastair Spence, Implicitly restarted Arnoldi with purification for the shift-invert transformation, Math. Comp. 66 (1997), no. 218, 667 – 689. · Zbl 0864.65020
[15] K. Meerbergen, A. Spence, and D. Roose, Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices, BIT 34 (1994), no. 3, 409 – 423. · Zbl 0814.65037
[16] Bahram Nour-Omid, Beresford N. Parlett, Thomas Ericsson, and Paul S. Jensen, How to implement the spectral transformation, Math. Comp. 48 (1987), no. 178, 663 – 673. · Zbl 0638.65026
[17] Youcef Saad and Martin H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 856 – 869. · Zbl 0599.65018
[18] D. Silvester, H. Elman, and A. Ramage, Incompressible Flow and Iterative Solver Software, http://www.maths.manchester.ac.uk/ djs/ifiss/. · Zbl 1365.65326
[19] Gerard L. G. Sleijpen, Albert G. L. Booten, Diederik R. Fokkema, and Henk A. Van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36 (1996), no. 3, 595 – 633. International Linear Algebra Year (Toulouse, 1995). · Zbl 0861.65035
[20] Gerard L. G. Sleijpen and Henk A. Van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (1996), no. 2, 401 – 425. · Zbl 0860.65023
[21] Gerard L. G. Sleijpen, Henk A. van der Vorst, and Ellen Meijerink, Efficient expansion of subspaces in the Jacobi-Davidson method for standard and generalized eigenproblems, Electron. Trans. Numer. Anal. 7 (1998), 75 – 89. Large scale eigenvalue problems (Argonne, IL, 1997). · Zbl 0912.65026
[22] D. C. Sorensen, Implicit application of polynomial filters in a \?-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992), no. 1, 357 – 385. · Zbl 0763.65025
[23] -, Deflation for implicitly restarted Arnoldi methods, Tech. Report TR98-12, Rice University, 1998.
[24] Tycho van Noorden and Joost Rommes, Computing a partial generalized real Schur form using the Jacobi-Davidson method, Numer. Linear Algebra Appl. 14 (2007), no. 3, 197 – 215. · Zbl 1199.65126
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