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Matrix pseudo-spectroscopy: Iterative calculation of matrix eigenvalues and eigenvectors of large matrices using a polynomial expansion of the Dirac delta function. (English) Zbl 0921.65029
Summary: The method of diagonalizing Hermitian matrices based on a polynomial expansion of the Dirac delta function \(\delta(E-H)\) is further refined so as to accelerate the convergence. Improved choices of the bases used for subspace diagonalization of the matrix, along with accuracy controls and estimates, are introduced. It is shown that the improved method can accurately deliver eigenvalues and eigenvectors in any region of the spectrum, including cases where the spacings are very small for “interior” eigenvalues. In addition, accurate values can be obtained for as many states as are desired. The method is illustrated for a model problem introduced recently in a study of another type approach.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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[1] Davidson, E.R., Comput. phys. commun., 53, 49, (1989)
[2] Morgan, R.B.; Scott, D.S.; Morgan, R.B.; Scott, D.S.; Morgan, R.B., SIAM J. sci. comput., SIAM J. sci. comput., Lin. alg. appl., 154, 289, (1991), See, e.g.
[3] Schwenke, D.W.; Truhlar, D.G., (), 303
[4] Bac̆ić, Z.; Light, J.C., Ann. rev. phys. chem., 40, 469, (1989), See, e.g.
[5] Jang, H.W.; Light, J.C., J. chem. phys., 99, 1057, (1993)
[6] Wigner, E.P.; Eisenbud, L.; Burke, P.G.; Hibbert, A.; Robb, W.D., Phys. rev., J. phys. B, 4, 153, (1971)
[7] Lanczos, C.; Res, J., Natl. bur. stand., 45, 255, (1950)
[8] Arnoldi, W.E., Quant. appl. math., 9, 17, (1951)
[9] Saad, Y., Numerical methods for large eigenvalue problems, (1992), Halsted New York · Zbl 0991.65039
[10] Kosloff, R.; Tal-Ezer, H., Chem. phys. lett., 127, 223, (1986)
[11] Neuhauser, D.; Neuhauser, D.; Neuhauser, D., J. chem. phys., J. chem. phys., J. chem. phys., 100, 5076, (1994)
[12] Erikson, T.; Ruhe, A., Math. comput., 35, 1251, (1980)
[13] R.E. Wyatt, Phys. Rev. E, in press;
[14] Sorensen, D.C., (), 228
[15] Pendergast, P.; Darakjian, Z.; Hayes, E.F.; Sorensen, D.C., J. comput. phys., 113, 201, (1994)
[16] Zhu, W.; Huang, Y.; Kouri, D.J.; Chandler, C.; Hoffman, D.K., Chem. phys. lett., 217, 73, (1994)
[17] D.J. Kouri, W. Zhu, G.A. Parker and D.K. Hoffman, in press.
[18] Löwdin, P.O., (), 185
[19] D.K. Hoffman, G.A. Parker, Y. Huang and D.J. Kouri, in preparation.
[20] G.A. Parker, Y. Huang, D.J. Kouri, D.K. Hoffman, E. Tiesinge, C. Williams and P. Jullienne, research in progress.
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