Block jacobi preconditioning of the conjugate gradient method on a vector processor

Abstract

The preconditioned conjugate gradient method is well established for solving linear systems of equations that arise from the discretization of partial differential equations. Point and block Jacobi preconditioning are both common preconditioning techniques. Although it is reasonable to expect that block Jacobi preconditioning is more effective, block preconditioning requires the solution of triangular systems of equations that are difficult to vectorize. We present an implementation of block Jacobi for vector computers, especially for the Cray Y-MP/264, and discuss several techniques to improve vectorization. We present these in a progression to show the effect on performance. For the model problem, resulting from a self-adjoint operator, the final implementation of one block Jacobi step uses almost the same amount of time as one point Jacobi step on the Cray Y-MP/264 despite the solution of triangular systems.