An Economical Algorithm for the Solution of Elliptic Difference Equations Independent of User-Supplied Parameters.

摘要

An adaptive algorithm to be used with factorization methods for solving sparse systems of linear equations that arise in the numerical solution of partial differential equations is developed. Factorization methods compute the solution of the system Ax=q by truncating the sequence of iterates (x sub n) defined using (A+B)(x sub(n+1)) = (A+B)(x sub n) - (tau sub n)(A(x sub n)-q),where B is given by the factorization and the parameters tau sub n are determined by the algorithm. Previous algorithms required user-supplied estimates of the extreme eigenvalues of the iteration matrix for the definition of the parameters. The adaptive algorithm,however,uses information obtained during the iterations to calculate a sequence of parameters that approach the best set. Necessary and sufficient conditions for the convergence of the iteration are derived for arbitrary A and B and constant parameter tau,for positive definite A and Hermitian B and for both A and A+B positive definite. (Author)