Geometric Approach to Symmetric Positive Definite Linear Systems

Abstract

This paper compares the performance of the conjugate gradient method and geometric approach in the case of symmetric positive definite (SPD) linear systems. This approach is based on the geometric theory of ODEs which was effectively initiated by Poncaré and Liapunov. The simplest and most obvious advantage of the geometric approach over the conjugate gradient method (the MATLAB code pcg) is that this approach can find the inverse of the underlying positive definite matrix and the solution. We present various numerical examples, which demonstrate the superiority of the geometric approach. For SPD linear systems, this approach provides much higher accuracy than the conjugate gradient method. In particular, since it is a one-stop procedure, it can avoid the growth of accumulated round-off errors to some extent.