A generalized nonsymmetric Lanczos procedure. (English) Zbl 0798.65050
Summary: We present a procedure for computing interior eigenvalues and corresponding eigenvectors of certain generalized nonsymmetric eigenvalue problems which arise, for example, in the normal mode analysis of plasmas in magnetohydrodynamics. This procedure combines a generalization of our nonsymmetric Lanczos procedure with a shift and invert strategy and a Rayleigh quotient-inverse iteration procedure.
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
Keywords:
interior eigenvalues; eigenvectors; generalized nonsymmetric eigenvalue problems; magnetohydrodynamics; nonsymmetric Lanczos procedure; shift and invert strategy; Rayleigh quotient-inverse iterationReferences:
| [1] | Stewart, G. W., Introduction to Matrix Computations, ((1973), Academic Press: Academic Press New York, London), 309 · Zbl 0302.65021 |
| [2] | Cullum, J.; Willoughby, R. A., Lanczos Algorithms for Large Symmetric Eigenvalue Computations, (Programs, vol. 2 (1985)), Basel · Zbl 0427.65025 |
| [3] | Parlett, B. N., The Symmetric Eigenvalue Problem (1980), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0431.65016 |
| [4] | Ericsson, Th.; Ruhe, A., Math. Comput., 35, 1251 (1980) · Zbl 0468.65021 |
| [5] | Scott, D. S., SIAM J. Sci. Stat. Comput., 3, 68 (1982) · Zbl 0477.65027 |
| [6] | Kerner, W.; Lerbinger, K.; Gruber, R.; Tsunematsu, T., Comput. Phys. Commun., 36, 225 (1985) |
| [7] | Kerner, W.; Lerbinger, K.; Riedel, K., Phys. Fluids, 29, 2975 (1985) |
| [8] | Kerner, W.; Jakoby, A.; Lerbinger, K., J. Comput. Phys., 66, 332 (1986) · Zbl 0613.76056 |
| [9] | Kerner, W., Large Scale Eigenvalue Problems, (Cullum, J.; Willoughby, R. A., Math. Stud., 127 (1986), North-Holland: North-Holland Amsterdam), 241 · Zbl 0685.65028 |
| [10] | Smith, B. T., Matrix Eigensystem Routines - EISPACK Guide, (Lecture Notes in Comput. Sci., 6 (1976), Springer-Verlag: Springer-Verlag New York) · Zbl 0289.65017 |
| [11] | Garbow, B. S., Matrix Eigensystem Routines - EISPACK Guide Extension, (Lecture Notes in Comput. Sci., 51 (1977), Springer-Verlag: Springer-Verlag New York) · Zbl 0368.65020 |
| [12] | Kerner, W.; Lerbinger, K.; Steuerwald, J., Comput. Phys. Commun., 38, 27 (1984) |
| [13] | Pao, Y. P.; Kerner, W., Phys. Fluids, 28, 287 (1985) · Zbl 0565.76115 |
| [14] | Kaniel, S., Math. Comput., 20, 369 (1966) · Zbl 0156.16202 |
| [15] | Paige, C. C., (Ph.D. thesis (1971), University of London) |
| [16] | Paige, C. C., Linear Algebra Appl., 34, 235 (1980) · Zbl 0471.65017 |
| [17] | Wilkinson, J., The Algebraic Eigenvalue Problem (1965), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0258.65037 |
| [18] | Cullum, J.; Willoughby, R. A., Large-Scale Eigenvalue Problems, (Cullum, J.; Willoughby, R. A., Math. Stud., 127 (1986), North-Holland: North-Holland Amsterdam), 193 · Zbl 0595.00022 |
| [19] | Parlett, B. N.; Taylor, D. R.; Liu, Z. A., Math. Comput., 44, 105 (1985) · Zbl 0564.65022 |
| [20] | Saad, Y., (Lecture Notes in Mathematics, 973 (1983), Springer-Verlag: Springer-Verlag New York), 121 |
| [21] | Saad, Y., Math. Comput., 42, 567 (1984) · Zbl 0539.65013 |
| [22] | Saad, Y., Yale University Research Report U/DCS/RR-397 (1985) |
| [23] | Ho, Diem; Chatelin, F., (IBM Science Center Report (1988)), Paris, France |
| [24] | Parlett, B. N.; Saad, Y., Linear Algebra Appl., 88&89, 575 (1987) · Zbl 0623.65045 |
| [25] | Cullum, J.; Willoughby, R. A., IBM Research Report RC12835 (1987), T.J. Watson Research Center: T.J. Watson Research Center Yorktown Heights, New York |
| [26] | Bunse-Gerstner, A., (Ph.D. dissertation (1978), Universität Bielefeld: Universität Bielefeld Bielefeld, Fed. Rep. Germany) |
| [27] | Gordon, R. G.; Messenger, T., Electron Spin Relaxation in Liquids, (Muus, L. T.; Atkins, P. W., NATO Advanced Study Institute. NATO Advanced Study Institute, Spåtind, Norway, August 1971 (1972), Plenum Press: Plenum Press New York), 331 |
| [28] | Cullum, J.; Willoughby, R. A., (IBM Research Report (1989), T.J. Watson Research Center: T.J. Watson Research Center Yorktown Heights, NY), to appear |
| [29] | Simon, H. D., (Presented at the Intern. Congr. on Industrial and Applied Mathematics. Presented at the Intern. Congr. on Industrial and Applied Mathematics, Paris, France (June 1987)) |
| [30] | Saad, Y., Linear Algebra Appl., 34, 269 (1980) · Zbl 0456.65017 |
| [31] | Saad, Y., Math. Comput., 37, 105 (1981) · Zbl 0474.65019 |
| [32] | Saad, Y., SINUM, 19, 485 (1982) |
| [33] | Ruhe, A., (Lecture Notes in Mathematics, 973 (1983), Springer-Verlag: Springer-Verlag New York), 104 |
| [34] | Ericsson, Th., Large Eigenvalue Problems, (Cullum, J.; Willoughby, R. A., Math. Stud., 127 (1986), North-Holland: North-Holland Amsterdam), 95 |
| [35] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1983), The Johns Hopkins Univ. Press: The Johns Hopkins Univ. Press Baltimore, Maryland · Zbl 0559.65011 |
| [36] | Ericsson, Th., (Dold, A.; Eckmann, B., Lecture Notes in Mathematics (1983), Springer-Verlag: Springer-Verlag Berlin), 177, 973, Matrix Pencils |
| [37] | Nour-Omid, B.; Parlett, B. N.; Ericsson, T.; Jenson, P. S., Math. Comput., 48, 663 (1987) · Zbl 0638.65026 |
| [38] | Stewart, G. W., Recent Advances in Numerical Analysis, ((1987), Academic Press: Academic Press New York), 193 |
| [39] | Wilkinson, J. H., Symp. Mat., 10, 361 (1972) |
| [40] | Kahan, W.; Parlett, B. N.; Jiang, E., SINUM, 19, 470 (1982) · Zbl 0483.65024 |
| [41] | Dongarra, J. J.; Bunch, J. R.; Moler, C. B.; Stewart, G. W., LINPACK User’s Guide (1979), SIAM Press: SIAM Press Philadelphia · Zbl 0476.68025 |
| [42] | Parlett, B. N., Math. Comput., 28, 679 (1974) · Zbl 0293.65023 |
| [43] | Ostrowski, A. M., Arcv. Ratl. Mech. Anal., 4, 153 (1960) |
| [44] | Varah, J. M., Math. Comput., 24, 137 (1970) · Zbl 0195.45101 |
| [45] | Peters, G.; Wilkinson, J. H., SIAM Rev., 21, 339 (1979) · Zbl 0424.65021 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
