On the effectiveness of the diagonally scaled conjugate gradient algorithm on vector computers

The efficiency of the diagonally scaled conjugate gradient algorithm on vector computers for solving large sparse positive definite linear systems, which arise in the discrete approximation of partial differential equations, is demonstrated. The algorithm, which applies the conjugate gradient algorithm after scaling the matrix by its diagonals, is 100% vectorizable and utilizes the vector processing ability of the supercomputer at its best. Besides, it is simple to code and requires little memory. The effect of the diagonal scaling is analyzed, and is shown to depend on the spatial inhomogeneity of coefficients of the partial differential equation and mesh size of the discretization. Numerical experiments (on NEC SX-2) for various large and ill-conditioned problems arising from finite difference approximation of three dimensional diffusion equations show that, for isotropic problems, the present algorithm is considerably faster (2.5 to 2.9 times) than the vectorized (M)ICCG algorithm, even when the diffusion coefficient varies very rapidly in space. The present algorithm is also shown to be efficient for random sparse matrices arising in finite element analysis. [In Japanese.]