Saddle point methods, and algorithms, for non-symmetric linear equations

This paper describes methods for solving non-singular, non-symmetric linear equations whose symmetric part is positive definite. First, the solutions are characterized as saddle points of a convex-concave function. The associated primal and dual Variational principles provide quadratic, strictly convex, functions whose minima are the solutions of the original equation and which generalize the energy function for symmetric problems. Direct iterative methods for finding the saddle point are then developed and analyzed. A globally convergent algorithm for finding the saddle poitns is described. We show that requiring conjugacy of successive search directions with respect to the symmetric part of the equation is a poor strategy.