Abstract
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 programFMLSQ. 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.
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