On the Application of the Generalized BiConjugate Gradient Method

Abstract

For a non-Hermitian operator A , the conjugate gradient method, instead of solving for the operator equation directly, solves the normal equations A*AX = A*Y , where A* is the adjoint operator. Even though in actual computations A*A is never formed, the condition number of the original operator equation is squared in the solution of A*AX = A*Y. One possible way to reduce the condition number is through preconditioning which in some cases either require some a priori information on the distribution of the eigenvalues of the operator, or requires additional preprocessing of the operator equation. In the generalized biconjugate gradient method one solves a non-Hermitian operator equation AX = Y directly. The application of the new method results in faster convergence. The generalized biconjugate gradient method does not minimize the residual or the error in the solution at each iteration, but reduces some power norm. This method however requires an additional 2N storage locations for a nonsymmetric operator, where N is the number of degrees of freedom for X. For a symmetric non-Hermitian operator a compact form of the algorithm is possible. Numerical results are presented to illustrate the optimum property of this method.