many-body problems where all sites are directly coupled by scalar operators. (If a matrix representation were con-Projection operators are used to effect ''deflation by restriction'' and it is argued that this is an optimal Lanczos algorithm for memory structed, it would be sparse.) In contrast to most electronminimization. Algorithmic optimization is constrained to dense, Herspin models, the solutions are not pure spin states, and the mitian eigensystems where a significant number of the extreme Hamiltonian is not block-diagonalizable. These examples eigenvectors must be obtained reliably and completely. The definare discussed in Section 4 and are used to illustrate the ing constraints are operator algebra without a matrix representation algorithm in considerable detail.
and semi-orthogonalization without storage of Krylov vectors. Other semi-orthogonalization strategies for Lanczos algorithms and conju-
In Section 5 we consider plane-wave density functional gate gradient techniques are evaluated within these constraints. theory of beryllium clusters. These examples are dense.
Large scale, sparse, complex numerical experiments are performed
The focus is upon the use of preconditioners and spectral on clusters of magnetic dipoles, a quantum many-body system that transformations. For an applied mathematician these is not block-diagonalizable. Plane-wave, density functional theory transformations provide a wide variety of exercises with of beryllium clusters provides examples of dense complex eigensystems. Use of preconditioners and spectral transformations is evalu-very different eigenvalue spectra. For a solid-state physiated in a preprocessor prior to a high accuracy self-consistent field cist, these evaluate methods for accelerating computation calculation.