Complex generalized minimal residual algorithm for iterative solution of quantum-mechanical reactive scattering equations

A complex GMRes (generalized minimum residual) algorithm is presented and used to solve dense systems of linear equations arising in variational basis-set approaches to quantum-mechanical reactive scattering. The examples presented correspond to physical solutions of the Schrödinger equation for the reactions O+HD→OH+D, D+H2→HD+H, and H+H2→H2+H. It is shown that the computational effort for solution with GMRes depends upon both the dimension of the linear system and the total energy of the reaction. In several cases with dimensions in the range 1110–5632, GMRes outperforms the LAPACK direct solver, with speedups for the linear equation solution as large as a factor of 23. In other cases, the iterative algorithm does not converge within a reasonable time. These convergence differences can be correlated with ‘‘indices of diagonal dominance,’’ which we define in detail and which are relatively easy to compute. Furthermore, we find that for a given energy, the computational effort for GMRes can vary with dimension as favorably as M1.7, where M is the dimension of the linear system, whereas the computer time for a direct method is approximately proportional to the cube of the dimension of the linear system.