An approach to construct pre-conditioning matrices for block iteration of linear equations

An approximate block elimination (ABE) method for constructing preconditioning matrices in the block iteration of systems of linear equations is presented. Four families of preconditioning matrices based on this approach are identified for solving the 2×2 block equations with the Jacobian resulted from the 3-D one-carrier semiconductor device analysis problem. The fully coupled Newton´s method is used as a test case on a parallel computer, the Intel iPSC/2 hypercube. There are ten different preconditioning transformations including the alternate block factorization (ABF) and the modified singular perturbation (MSP) schemes. The results from computational experiments indicate that eight of these preconditioning matrices significantly extend the convergence region of the block Gauss-Seidel iteration process of Newton steps and five of them speed up the convergence with a factor of up to more than an order of magnitude