Efficient calculation of a normal matrix-vector product for anisotropic full-matrix least-squares refinement of macromolecular structures

A novel algorithm is described for multiplying a normal equation matrix by an arbitrary real vector using the fast Fourier transform technique during anisotropic crystallographic refinement. The matrix-vector algorithm allows one to solve normal matrix equations using the conjugate-gradients or conjugate-directions technique without explicit calculation of a normal matrix. The anisotropic version of the algorithm has been implemented in a new version of the computer program FMLSQ. The updated program has been tested on several protein structures at high resolution. In addition, rapid methods for preconditioner and normal matrix-vector product calculations are described.