Number Of Matrix Multiplications Research Articles

On Moment Methods in Krylov Subspaces

Moment methods in Krylov subspaces for solving symmetric systems of linear algebraic equations (SLAEs) are considered. A family of iterative algorithms is proposed based on generalized Lanczos orthogonalization with an initial vector $${{v}^{0}}$$ chosen regardless of the initial residual. By applying this approach, a series of SLAEs with the same matrix, but with different right-hand sides can be solved using a single set of basis vectors. Additionally, it is possible to implement generalized moment methods that reduce to block Krylov algorithms using a set of linearly independent guess vectors $$v_{1}^{0},...,v_{m}^{0}.$$ The performance of algorithm implementations is improved by reducing the number of matrix multiplications and applying efficient parallelization of vector operations. It is shown that the applicability of moment methods can be extended using preconditioning to various classes of algebraic systems: indefinite, incompatible, asymmetric, and complex, including non-Hermitian ones.

Expansion algorithm for the density matrix

A purification algorithm for expanding the single-particle density matrix in terms of the Hamiltonian operator is proposed. The scheme works with a predefined occupation and requires less than half the number of matrix-matrix multiplications compared to existing methods at low (10%) and high (>90%) occupancies. The expansion can be used with a fixed chemical potential, in which case it is an asymmetric generalization of and a substantial improvement over grand canonical McWeeny purification. It is shown that the computational complexity, measured as the number of matrix multiplications, essentially is independent of system size even for metallic materials with a vanishing band gap.

Chebychev Response of Thin Film Optical Filters

The power reflection coefficient (reflectance) of linear, periodically homogenous, isotropic, nonabsorbing stack filters is derived using full wave analysis. The well-known ABCD characteristic matrix is used to assess the spectral response of the stack filters. Adopting this method requires a huge number of matrix multiplications as the number of layers increases. An alternative approach of evaluating the whole matrix is suggested which is shown to save computational time. Band reject response far from ideal characteristics, due to the presence of unwanted ripples surrounding the stopband, is obtained. Equating these ripples without changing the length of the structure is undertaken by varying some design parameters related to the layers constituting the filter. Namely, the index of refraction and the physical thickness of individual layers are used as optimization variables. This process requires evaluating the derivative of the reflectance with respect to each design variable while leaving other parameters unchanged. The sensitivity of the derivatives to various design parameters is also addressed, which is sometimes required in manufacturing multilayered dielectric thin films.