Efficient approximation of functions of some large matrices by partial fraction expansions

Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$ is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from $A$ by a complex multiple of the identity matrix $I$ for computing $\Psi(A)\mathbf{v}$ or require inverting sequences of matrices with the same characteristics for computing $\Psi(A)$. Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that $A^{-1}$ shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests

Determinants of Block Tridiagonal Matrices

An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).Comment: 8 pages, final form. To appear on Linear Algebra and its Application

On the decay of the inverse of matrices that are sum of Kronecker products

Decay patterns of matrix inverses have recently attracted considerable interest, due to their relevance in numerical analysis, and in applications requiring matrix function approximations. In this paper we analyze the decay pattern of the inverse of banded matrices in the form $S=M \otimes I_n + I_n \otimes M$ where $M$ is tridiagonal, symmetric and positive definite, $I_n$ is the identity matrix, and $\otimes$ stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal. However, the entries in $S^{-1}$ show a non-monotonic decay, which is not caught by classical bounds. By using an alternative expression for $S^{-1}$, we derive computable upper bounds that closely capture the actual behavior of its entries. We also show that similar estimates can be obtained when $M$ has a larger bandwidth, or when the sum of Kronecker products involves two different matrices. Numerical experiments illustrating the new bounds are also reported

The Lyapunov matrix equation. Matrix analysis from a computational perspective

Decay properties of the solution $X$ to the Lyapunov matrix equation $AX + X A^T = D$ are investigated. Their exploitation in the understanding of equation matrix properties, and in the development of new numerical solution strategies when $D$ is not low rank but possibly sparse is also briefly discussed.Comment: This work is a contribution to the Seminar series "Topics in Mathematics", of the PhD Program of the Mathematics Department, Universit\`a di Bologna, Ital

Distributed NEGF Algorithms for the Simulation of Nanoelectronic Devices with Scattering

Through the Non-Equilibrium Green's Function (NEGF) formalism, quantum-scale device simulation can be performed with the inclusion of electron-phonon scattering. However, the simulation of realistically sized devices under the NEGF formalism typically requires prohibitive amounts of memory and computation time. Two of the most demanding computational problems for NEGF simulation involve mathematical operations with structured matrices called semiseparable matrices. In this work, we present parallel approaches for these computational problems which allow for efficient distribution of both memory and computation based upon the underlying device structure. This is critical when simulating realistically sized devices due to the aforementioned computational burdens. First, we consider determining a distributed compact representation for the retarded Green's function matrix $G^{R}$. This compact representation is exact and allows for any entry in the matrix to be generated through the inherent semiseparable structure. The second parallel operation allows for the computation of electron density and current characteristics for the device. Specifically, matrix products between the distributed representation for the semiseparable matrix $G^{R}$ and the self-energy scattering terms in $\Sigma^{<}$ produce the less-than Green's function $G^{<}$. As an illustration of the computational efficiency of our approach, we stably generate the mobility for nanowires with cross-sectional sizes of up to 4.5nm, assuming an atomistic model with scattering

Computationally efficient modeling and simulation of large scale systems

A method of simulating operation of a VLSI interconnect structure having capacitive and inductive coupling between nodes thereof. A matrix X and a matrix Y containing different combinations of passive circuit element values for the interconnect structure are obtained where the element values for each matrix include inductance L and inverse capacitance P. An adjacency matrix A associated with the interconnect structure is obtained. Numerical integration is used to solve first and second equations, each including as a factor the product of the inverse matrix X.sup.-1 and at least one other matrix, with first equation including X.sup.-1Y, X.sup.-1A, and X.sup.-1P, and the second equation including X.sup.-1A and X.sup.-1P

Preconditioning issues in the numerical solution of nonlinear equations and nonlinear least squares

Second order methods for optimization call for the solution of sequences of linear systems. In this survey we will discuss several issues related to the preconditioning of such sequences. Covered topics include both techniques for building updates of factorized preconditioners and quasi-Newton approaches. Sequences of unsymmetric linear systems arising in Newton-Krylov methods will be considered as well as symmetric positive definite sequences arising in the solution of nonlinear least-squares by Truncated Gauss-Newton methods

Functions of rational Krylov space matrices and their decay properties

Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, many structural properties of the reduced matrices in these subspaces are not fully understood. We advance in this analysis by deriving bounds on the entries of rational Krylov reduced matrices and of their functions, that ensure an a-priori decay of their entries as we move away from the main diagonal. As opposed to other decay pattern results in the literature, these properties hold in spite of the lack of any banded structure in the considered matrices. Numerical experiments illustrate the quality of our results