Diagonalizing the Adaptive SOR Iteration Method

Abstract

The SOR iteration method is a popular method for solving the large sparse systems of linear algebraic equations which approximate many partial differential equations that arise in engineering. Often the associated SOR matrix $M^{ - 1} N$ is diagonalizable except at the eigenvalue $\lambda = \omega - 1$, and the noneigenvector $p_ * $ associated with the $\lambda = \omega - 1$ (i) slows down the convergence, and (ii) in the adaptive SOR method, reduces the accuracy of the calculation of the next relaxation factor $\omega _i $. Of course, $M^{ - 1} N$ cannot be diagonalized, but the error vector can be pushed into the span of the eigenvectors of $M^{ - 1} N$, thereby eliminating the $p_ * $-coordinate of the error vector, together with its undesirable effects. This is done with the simple polynomial acceleration associated with the polynomial $P_1 ( x ) = ( x - ( \omega - 1 ) )/( 1 - ( \omega - 1 ) )$, and $P_n ( x ) = x^{n - 1} P_1 ( x ), n = 2,3, \cdots $. In the adaptive SOR method, this acceleration reduces the size of the error (i) by enabling the program to update the value of $\omega _i $, sooner, and (ii) by eliminating the contribution of $p_ * $ to the error vector. In the computer runs, using this polynomial acceleration resulted (on average) in an extra digit of accuracy over the results using the standard adaptive SOR method.