Approximating the inverse of a matrix for use in iterative algorithms on vector processors

Most iterative techniques for solving the symmetric positive-definite systemAx=b involve approximating the matrixA by another symmetric positive-definite matrixM and then solving a system of the formMz=d at each iteration. On a vector machine such as the CDC-STAR-100, the solution of this new system can be very time consuming. If, however, an approximationM ^?1 can be given toA ^?1, the solutionz=M ^?1 d can be computed rapidly by matrix multiplication, a fast operation on the STAR. Approximations using the Neumann expansion of the inverse ofA give reasonable forms forM ^?1 and are presented. Computational results using the conjugate gradient method for the “5-point” matrixA are given.