Cross correlation functions Cnm(E) via Lanczos algorithms without diagonalization

It is shown how the quasiminimal residual algorithm (QMR), based on the Lanczos algorithm, can be modified to compute cross-correlation functions C-nm(E)=[Psi(n)\(E-H)(-1)\Psi(m)] without any diagonalization by recursively updating a small number of scalars. Only three Lanczos vectors need to be stored. Several left-hand side vectors [Psi(n)\ and multiple shifts E can be considered simultaneously. The new method is termed the quasiminimal recursive residue generation method (QM-RRGM) and is applied to the collinear H+H-2 problem to illustrate its convergence behavior. The properties of two different formulations of the Lanczos algorithm, the usual three-term and a coupled two-term recursion, are also discussed. The QM-RRGM exhibits smooth convergence behavior, and it is shown that the stopping criteria used in the QMR algorithm can also be used for computing correlation functions. (C) 2002 American Institute of Physics.