Karlovich, Alexei Yu. "
Higher order asymptotic formulas for traces of Toeplitz matrices with symbols in Hölder-Zygmund spaces."
Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, 179. Eds. Joseph A. Ball, Yuli Eidelman, William J. Helton, Vadim Olshevsky, and James Rovnyak. Basel: Bikhäuser, 2008. 185-196.
AbstractWe prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.
Morrison, J. C., S. Boyd, L. Marsano, B. Bialecki, T. Ericsson, and J. P. Santos. "
Numerical methods for solving the Hartree-Fock equations of diatomic molecules I."
Communications in Computational Physics. 5 (2008): 959-985.
AbstractThe theory of domain decomposition is described and used to divide the variable domain of a diatomic molecule into separate regions which are solved independently. This approach makes it possible to use fast Krylov methods in the broad interior of the region while using explicit methods such as Gaussian elimination on the boundaries. As is demonstrated by solving a number of model problems, these methods enable one to obtain solutions of the relevant partial differential equations and eigenvalue equations accurate to six significant figures with a small amount of computational time. Since the numerical approach described in this article decomposes the variable space into separate regions where the equations are solved independently, our approach is very well-suited to parallel computing and offers the long term possibility of studying complex molecules by dividing them into smaller fragments that are calculated separately.