Abstract
This work focuses on development and testing of alternative numerical methods and computational algorithms specifically designed for the solution of tridiagonal systems and for the vectorization of diffusion computations on a Control Data Corporation Cyber 205 vector computer.
Solution of tridiagonal systems of linear equations is a central part for several eficient numerical methods for multidimensional diffusion computations and is also essential for fluid flow and other physics and engineering problems. The first part of this paper deals with the numerical solution of linear symmetric positive definite tridiagonal systems. Among the method tested, a combined odd-even cyclic reduction and modified Cholesky factorization algorithm is found to be the most effective for these systems on a Cyber 205. For large tridiagonal systems, computation with this algorithm is an order of magnitude faster on a Cyber 205 than computation with the best algorithm for tridiagonal systems on a CDC-7600.
The above mentioned algorithm for solving tridiagonal systems is also utilized as a basis for a new hyper-line method for implementing the red-black cyclic Chebyshev iterative method to the solution of two-dimensional diffusion problems. The hyper-line method is found to be competitive with other alternative options developed in this work. This hyper-line methodhas an attractive feature of being compatible with so called “concurrent” iteration procedures whereby iterations n+1,…,n+k, can be started before the completion of iteration n. This feature is very effective in balancing computations and data transfer requirements for very large diffusion problems. Consequently implementation of the hyper-line method is suitable for certain iterative procedures used to solve large three-dimensional diffusion problems.
Some experience gained with Cyber 205 vector syntax statements related to diffusion comoputations is discussed in an Appendix to this paper.
Work supported by the Department of Energy under Contract DE-AC11-76PN00011.
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© 1985 Plenum Press, New York
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Abu-Shumays, I.K. (1985). Comparison of Methods and Algorithms for Tridiagonal Systems and for Vectorization of Diffusion Computations. In: Numrich, R.W. (eds) Supercomputer Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-2503-1_3
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DOI: https://doi.org/10.1007/978-1-4613-2503-1_3
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