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This paper describes a novel algorithm for multiplying a matrix of normal equations by an arbitrary real vector using the fast Fourier transform technique. The algorithm allows full-matrix least-squares refinement of macromolecular structures without explicit calculation of the normal matrix. The resulting equations have been implemented in a new computer program, FMLSQ. A preliminary version of the program has been tested on several protein structures. The consequences for crystallographic refinement of macromolecules are discussed in detail.

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Full matrix–vector derivation

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