SOLUTION OF LARGE LINEAR SYSTEMS ON PIPELINED SIMD MACHINES

Abstract

We developed a direct out-of-core solver for dense non-symmetric linear systems of ‘arbitrary’ size N×N. The algorithm fully employs the Basic Linear Algebra Subprograms (BLAS), and can therefore easily be adapted to different computer architectures by using the corresponding optimized routines. We used blocked versions of left-looking and right-looking variants of LU decomposition to perform most of the operations in Level 3 BLAS, to reduce the number of I/O operations and to minimize the CPU time usage. The storage requirements of the algorithm are only 2N×NB data elements where NB≪N. Depending on the sustained floating point performance and the sustained I/O rate of the given hardware, we derived formulas that allow for choosing optimal values of NB to balance between CPU time and I/O time. We tested the algorithm by means of linear systems derived from 3D-BEM for strongly and weakly singular integral equations and from interpolation problems for scattered data on closed surfaces in ℝ3. It took only about 2⋅5 CPU minutes on a 5 GFLOPS vector computer SNI S600/20 to solve a linear system of size 10000, which corresponds to a performance of 4⋅3 GFLOPS; a value of NB=650 gives a reasonable I/O time and the necessary main storage size is about 13 Mwords. In addition, we compared the algorithm with (1) an out-of-core version of GMRES and (2) a wavelet transform followed by in-core GMRES after thresholding. At least for boundary integral equations of classical boundary value problems of potential theory, the out-of-core version of GMRES is superior to the direct out-of-core solver and the wavelet transform since the algorithm converged after at most 5 iteration steps. It took about 17 s to solve a system with 8192 unknowns compared with 146 s for direct out-of-core and 402 s for wavelet transform followed by in-core GMRES. © 1997 by John Wiley & Sons, Ltd.