Iterative methods for solving the non-sparse equations of quantum mechanical reactive scattering

Several iterative techniques are applied to solve the non-sparse, indefinite algebraic variational equations of the L² quantum mechanical description of three-dimensional atom-diatom scattering. We do not assume a symmetric matrix although we employ a symmetric test problem that allows comparison to the standard conjugate gradient algorithm. Convergence of general minimal residual and Lanczos algorithms is shown to be rapid, enabling large savings of computer time as compared to direct methods for large-scale calculations of selected elements or columns of the reactance matrix. As the number M of basis functions varies from 100 to 504, minimal residual calculations based on the Arnoldi basis show very smooth and stable convergence, with computer time scaling as M^1.8, and a Lanczos recursive algorithm is found to scale as M^1.5 (for off-diagonal matrix elements).