Matrix-free iterative solution strategies for large dense linear systems

A purely algebric approach to solving very large general unstructured dense linear systems, in particular, those that arise in 3D boundary integral applications is suggested. We call this technique the matrix-free approach because it allows one to avoid the necessity of storing the whole coefficient matrix in any form, which provides significant memory and arithmetic savings. We propose to approximate a non-singular coefficient matrix A by a block low-rank matrix à and to use the latter when performing matrix–vector multiplications in iterative solution algorithms. Such approximations are shown to be easily computable, and a reliable a posteriori accuracy estimate of ‖A − Ã‖₂ is derived. We prove that block low-rank approximations are sufficiently accurate for some model cases. However, even in the absence of rigorous proof of the existence of accurate approximations, one can apply the algorithm proposed to compute a block low-rank approximation and then make a decision on its practical suitability. We present numerical examples for the 3D CEM and CFD integral applications, which show that, at least for some industrial applications, the matrix-free approach is robust and cost-effective. © 1997 John Wiley & Sons, Ltd.