Convergence of a Two-Stage Richardson Iterative Procedure for Solving Systems of Linear Equations (Classic Reprint)

Excerpt from Convergence of a Two-Stage Richardson Iterative Procedure for Solving Systems of Linear Equations

One motivation for our work is that the two-stage method can be effectively used to solve nonsymmetric systems where M is symmetric positive definite and N is~skewfsymmetric. In this case the symmetric/ skew-symmetric splitting A -n:is used to precondition the outer itera tion, and a symmetric splitting-m ml-m2 can be used to precondition each inner iteration, using a direct method to solve a system of the form Mly c at each step of-each inner iteration. (see Manteuffel (1977) for alternative approaches for nonsymmetric systems.) Numerical results from applying the Richardson method to such nonsymmetric systems are given in Section 3. We also present-numerical results using the conjugate gradient method for the outer iteration. Analysis of the latter procedure would be very interesting, but this seems more diffi cult.

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