Fast diagonalization of large and dense complex symmetric matrices, with applications to quantum reaction dynamics

SIAM Journal on Scientific Computing

Volumn 18, Issue 5, 1997, Pages 1412-1435

  Bar On, Ilan ^a   Ryaboy, Victor ^b  

^a UTPB   (United States)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Complex Hermitian; Complex symmetric; Diagonalization; QR; Reaction dynamics; Schroedinger equation; Skew symmetric; Supercomputers

Indexed keywords

FAST DIAGONALIZATION; HOUSEHOLDER TRANSFORMATION; SCHROEDINGER EQUATION; SKEW SYMMETRIC MATRICES; VECTOR SUPERCOMPUTERS;

ALGORITHMS; CHEMICAL REACTIONS; COMPUTER SIMULATION; EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES; PROBLEM SOLVING; QUANTUM THEORY; SUPERCOMPUTERS;

MATRIX ALGEBRA;

EID: 0031215986     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827594269056     Document Type: Article

References (35)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMML, J. DONGARRA, J. D. CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, SIAM, Philadelphia, PA, 1992.
2. E. BALSLEV AND J. COMBES, Spectral properties of many body Schroedinger operators with dilation analytic interactions, Comm. Math. Phys., 22 (1971), pp. 280-294.
3. I. BAR-ON, A New Divide and Conquer Parallel Algorithm for Computing the Eigenvalues of a Symmetric Tridiagonal Matrix, Tech. Report 832, Computer Science Department, Technion, Haifa, Israel, 1994.
4. I. BAR-ON, Interlacing properties for tridiagonal symmetric matrices with applications to parallel computing, SIAM J. Matrix Anal. Appl., 17 (1995), pp. 548-562.
5. I. BAR-UN AND B. CODENOTTI, A fast and stable parallel QR algorithm for symmetric tridiagonal matrices, Linear Algebra Appl., 220 (1995), pp. 63-96.
6. C. BISCHOF AND C. V. LOAN, The WY representation for products of Householder matrices, SIAM J. Sci. Statist. Comput., 8 (1987), pp. s1-s13.
7. B. R. JUNKER, Recent computational developments in the use of complex scaling in resonance phenomena, Adv. At. Mol. Phys., 18 (1982), pp. 207-263.
8. J. CULLUM AND R. WILLOUGHBY, Lanczos Algorithms for Large Symmetric Eigenvalues Computaions, Birkhauser Boston, Cambridge, MA, 1985.
9. D. BROWN AND J. C. LIGHT, Evaluation of thermal rate constants in the eigenbasis of a Hamiltonian with an optical potential, J. Chem. Phys., 97 (1992), pp. 5465-5471.
10. D. NEUHAUSER AND M. BAER, The time,-dependent Schroedinger equation: Application of absorbing boundary condition, J. Chem. Phys., 90 (1989), pp. 4351-4355.
11. J. J. DONGARRA AND D. SORENSEN, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sci. Statist. Comput., 8 (1987), pp. s139-s154.
12. P. EBERLEIN, On the diagonalization of complex symmetric matrices, J. Inst. Math. Appl., 7 (1971), pp. 377-383.
13. F. GANTMACHER, The Theory of Matrices, Vols. 1, 2, Chelsea, New York, 1959.
14. G. JOLICARD AND E. J. AUSTIN, Optical potential stabilization method for predicting resonance levels, Chem. Phys. Lett., 121 (1985), pp. 106-110.
15. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1989.
16. D. HELLER, A survey of parallel algorithms in numerical linear algebra, SIAM Rev., 20 (1978), pp. 740-777.
17. D. I. LAST AND M. BAER, The application of negative imaginary arrangement decoupling potential to reactive scattering: Conversion of a reactive scattering problem into a bound-type problem, J. Chem. Phys., 96 (1992), pp. 2017-2024.
18. J. SIMON, Quadratic form techniques and the Balslev-Combes theorem, Comm. Math. Phys., 27 (1972), pp. 1-9.
19. J. SIMON, Resonances in n-body quantum systems with dilation analytic potentials and the foundations of time dependent perturbation theory, Ann. Math., 97 (1973), pp. 247-274.
20. G. MORO AND J. FREED, Calculation of ESR spectra and related Fokker-Planck forms by the use of the Lanczos algorithm, J. Chem. Phys., 74 (1981), pp. 3757-3773.
21. N, LIPKIN, N. MOISEYEV, AND C. LEFORESTIER, A three-dimensional study of NelCl predissociation resonances by the complex scaled discrete variable representation method, J. Chem. Phys., 98 (1993), pp. 1888-1901.
22. N. MOISEYEV, Resonances, cross sections and partial widths by the complex coordinate method, Israel J. Chem., 31 (1991), pp. 311-322.
23. B. N. PARLETT, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
24. W. REINHARDT, Complex coordinates in the theory of atomic and molecular structure and dynamics, Ann. Rev. Phys. Chem., 33 (1982), pp. 223-255.
25. R. SCHREIBER AND C. VAN LOAN, A storage efficient WY representation for products of Householder transformations, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 53-57.
26. R. SCHREIBER AND B. PARLETT, Block reflectors: Theory and computation, SIAM J. Numer. Anal., 25 (1987), pp. 189-205.
27. M. SEATON, Complex symmetric matrices, Comput. J., 12 (1969), pp. 156-157.
28. B. SMITH, J. BOYLE, J. DONGARRA, B. GARBOW, Y. IKEBE, V. KLEMA, AND C. MOLER, Matrix Eigensystem Routines, EISPACK Guide, Lecture Notes in Comput. Sci. 6, Springer-Verlag, Berlin, 1976.
29. T. SEIDEMAN AND W. H. MILLER, Calculation of the cumulative reaction probability via a discrete variable representation with absorbing boundary condition, J. Chem. Phys., 96 (1992), pp. 4412-4422.
30. V. RYABOY AND N. MOISEYEV, Cumulative reaction probability from Siegert eigenvalues: Model studies, J. Chem. Phys., 98 (1993), pp. 9618-9623.
31. V. RYABOY AND N. MOISEYEV, Three dimensional study of predissociation resonances by the complex scaled discrete variable representation method: HCO/DCO, J. Chem. Phys., 103 (1995), pp. 4061-4068.
32. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, UK, 1965. Reprinted in Oxford Science Publications, 1988.
33. Y. K. HO, The method of complex coordinate rotation and its applications to atomic collision processes, Phys. Rep. C, 99 (1983), pp. 1-68.
34. Z. BACIC AND J. C. LIGHT, Accurate localized and delocalized vibrational states of HCN/HNC, J. Chem. Phys., 86 (1986), pp. 3065-3077.
35. Z. BACIC AND J. C. LIGHT, Highly excited vibrational levels of 'floppy' molecules: A discrete variable representation - distributed Gaussian basis approach, J. Chem. Phys., 85 (1986), pp. 4594-4604.