A variable-band relaxation algorithm for solving large linear systems is developed as an alternative to Gauss-Jacobi relaxation. This algorithm seeks to improve the reliability of Gauss-Jacobi relaxation by extracting a variable-sized band from the matrix and solving that band directly. This leads to a relaxation algorithm with provably better convergence properties. The algorithm can be used effectively on a massively parallel computer because band matrices can be solved in log(n) time on n/2 processors. Test results are presented which compare the convergence properties of variable-band and Gauss-Jacobi relaxation.
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