9 research outputs found
Decay properties of spectral projectors with applications to electronic structure
Motivated by applications in quantum chemistry and solid state physics, we
apply general results from approximation theory and matrix analysis to the
study of the decay properties of spectral projectors associated with large and
sparse Hermitian matrices. Our theory leads to a rigorous proof of the
exponential off-diagonal decay ("nearsightedness") for the density matrix of
gapped systems at zero electronic temperature in both orthogonal and
non-orthogonal representations, thus providing a firm theoretical basis for the
possibility of linear scaling methods in electronic structure calculations for
non-metallic systems. We further discuss the case of density matrices for
metallic systems at positive electronic temperature. A few other possible
applications are also discussed.Comment: 63 pages, 13 figure
On the Iwasawa decomposition of a symplectic matrix
We consider the computation of the Iwasawa decomposition of a symplectic matrix via the QR factorization. The algorithms presented improve on the method recently described by T.-Y. Tam in [Computing Iwasawa decomposition of a symplectic matrix by Cholesky factorization, Appl. Math. Lett. (in press) doi:10.1016/j.aml.2006.03.001]. © 2006 Elsevier Ltd. All rights reserved
DECAY BOUNDS AND O(n) ALGORITHMS FOR APPROXIMATING FUNCTIONS OF SPARSE MATRICES
We establish decay bounds for the entries of f(A), where A is a sparse (in particular, banded) n × n diagonalizable matrix and f is smooth on a subset of the complex plane containing the spectrum of A. Combined with techniques from approximation theory, the bounds are used to compute sparse (or banded) approximations to f(A), resulting in algorithms that under appropriate conditions have linear complexity in the matrix dimension. Applications to various types of problems are discussed and illustrated by numerical examples