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Tom Cwik ^{*}, Daniel S. Katz^{1}, Jean Patterson*

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Jet Propulsion Laboratory

California Institute of Technology

Pasadena, CA 91109

^{1}Cray Research, Inc.

El Segundo, CA 90245

Dense matrix solution algorithms are written to optimize both computational speed and storage. Traditional factorization algorithms are used to decompose the general complex-valued, impedance matrix resulting from the integral equation solution into factors that are used to solve for field solutions due to different excitations. The factorization algorithms can use the the in-core memory attached to the machine - therefore problem size is limited by the parameter - or out-of-core algorithms which use the associated disk to hold the impedance matrix and its factors. The out-of-core methods therefore allow the solution of much larger problems if the disk space is available. Current developments in non-factorization solution methods use preconditioned iterative algorithms, or low-rank approximations to the linear system, to extend the problem size or reduce the solution time.

In this talk, we will discuss the development of in-core and out-of-core dense matrix solvers, and their integration into existing integral equation codes on distributed memory, scalable parallel processors. Problem size exceeding 60,000 unknowns modeling the fields or currents are capable of being solved in less than 100 minutes time for the factorization using an in-core algorithm on a 1,024 processor Cray T3D. Out-of-core algorithms have been developed that run at nearly the same execution rate as the in-core algorithms, extending the problem size limit past this number. The current development in preconditioned iterative algorithms and low-rank approximations will also be discussed as applied to distributed memory, scalable parallel processing.