The application of the generalized conjugate residual algorithm to accelerate the fast multipole method

Scattering by an arbitrarily shaped conductor can be obtained by finding the solution of an integral equation where the unknown function is the induced current distribution. The integral equation is usually discretized into a matrix equation by the method of moments (MoM) and solved by iterative techniques, for example, the conjugate gradient method (CG). In this paper, the generalized conjugate residual algorithm is employed to replace the conjugate gradient method as the iterative method and the fast multipole method (FMM) is used to expedite the matrix-vector products. Because the generalized conjugate residual method makes the search direction vectors of each iteration Z-orthogonal with respect to those obtained from earlier iterations, it converges much faster than the conjugate gradient method. Numerical examples demonstrate this advantage.