Iterative methods for solving the non-sparse equations of quantum mechanical reactive scattering

Csilla Duneczky, Robert E. Wyatt, David Chatfield, Kenneth Haug, David W. Schwenke, Donald G. Truhlar, Yan Sun, Donald J. Kouri

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19 Scopus citations


Several iterative techniques are applied to solve the non-sparse, indefinite algebraic variational equations of the L2 quantum mechanical description of three-dimensional atom-diatom scattering. We do not assume a symmetric matrix although we employ a symmetric test problem that allows comparison to the standard conjugate gradient algorithm. Convergence of general minimal residual and Lanczos algorithms is shown to be rapid, enabling large savings of computer time as compared to direct methods for large-scale calculations of selected elements or columns of the reactance matrix. As the number M of basis functions varies from 100 to 504, minimal residual calculations based on the Arnoldi basis show very smooth and stable convergence, with computer time scaling as M1.8, and a Lanczos recursive algorithm is found to scale as M1.5 (for off-diagonal matrix elements).

Original languageEnglish (US)
Pages (from-to)357-379
Number of pages23
JournalComputer Physics Communications
Issue number1-3
StatePublished - May 1989

Bibliographical note

Funding Information:
The authors are grateful to A. Chronopoulos for helpful comments on the manuscript and to Mirjana Mladenovic, Philippe Halvick and Meis-han Zhao for assistance with the calculations. Computer time on the Cray X-MP/24 at the University of Texas System Center for High Performance Computing and on the Cray-2 at the Minnesota Supercomputer Institute is gratefully acknowledged. This work was supported in part at the University of Texas by Cray Research, Inc., the National Science Foundation (grant no. CHE 87-13555) and the Welch Foundation; at the University of Minnesota by the National Science Foundation (grant no. CHE86-62825), the Mm-nesota Supercomputer Institute and Control Data Corporation; and at the University of Houston by the donors of the Petroleum Research Fund, administered by the American Chemical Society, the National Science Foundation (grant no. CHE86-00363) and the Robert A. Welch Foundation (grant no. E-608).


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