Related papers: Evaluation of the matrix exponential function usin…
The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when…
The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite…
The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
How to calculate the exponential of matrices in an explicit manner is one of fundamental problems in almost all subjects in Science. Especially in Mathematical Physics or Quantum Optics many problems are reduced to this calculation by…
We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We…
In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…
A well-known problem in computing some matrix functions iteratively is the lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Suppose the matrix exponential…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based…
In this work, we consider a rational approximation of the exponential function to design an algorithm for computing matrix exponential in the Hermitian case. Using partial fraction decomposition, we obtain a parallelizable method, where the…
We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss--Legendre quadrature has been conventionally…
SMT solvers use sophisticated techniques for polynomial (linear or non-linear) integer arithmetic. In contrast, non-polynomial integer arithmetic has mostly been neglected so far. However, in the context of program verification, polynomials…
A generalized exponential matrix based on the construction of kernel operators for generalized summability is defined and analyzing its main properties, generalizing the classical exponential matrix and fractional exponential matrix. This…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
In finance, economics and many other fields, observations in a matrix form are often observed over time. For example, many economic indicators are obtained in different countries over time. Various financial characteristics of many…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $n\times n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor…
We give a formula for matrix exponentials and partial fraction decompositions.
Hamiltonian matrices appear in a variety or problems in physics and engineering, mostly related to the time evolution of linear dynamical systems as for instance in ion beam optics. The time evolution is given by symplectic transfer…