A Vectorized Conjugate-Gradient Solver for Sparse Systems of Algebraic Equations

Abstract

This article describes a preconditioned conjugate-gradient (CG) solver suitable for large sparsely populated systems of algebraic equations. The solver adopts the diagonally scaled scheme that retains the low storage requirement of the basic CG method, while improving its rate of convergence. Compared to other preconditioning schemes, such as the incomplete Cholesky factorization, the diagonally scaled scheme requires minor modifications to the basic CG method and can be implemented easily. Like the basic method, the diagonally scaled scheme is readily vectorizable and can perform better than the incomplete Cholesky factorization on vector processors. The article evaluates the performance of the solver by using the standard test case of a flow in a differentially heated square cavity at different values of the Rayleigh number. It also shows the computer-time requirements of the solver on two vector processors, in both scalar and vectorized modes, which clearly show the benefit of vectorization.