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We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a…
A certain representation for the Heisenberg algebra in finite-difference operators is established. The Lie-algebraic procedure of discretization of differential equations with isospectral property is proposed. Using $sl_2$-algebra based…
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an…
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…
Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…
Discretizing continuous-time linear systems typically requires numerical integration. This document presents a convenient method for discretizing the dynamics, input, and process noise state-space matrices of a continuous-time linear system…
Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{G}_a$ denote the additive group of $k$. Let $n \geq 1$ and let ${\rm Mat}(n, k[T])^E$ denote the set of all exponential matrices of ${\rm Mat}(n,…
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…
A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation, by means of its differential representation. The package provides functions which enable us to solve the determining…
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…
We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
A framework for exponential time discretization of the multilayer rotating shallow water equations is developed in combination with a mimetic discretization in space. The method is based on a combination of existing exponential time…
This paper focuses on the further development of the Lie bracket approximation approach for extremum seeking systems. Classical results in this area provide extremum seeking algorithms with exponential convergence rates for quadratic-like…
Let $F^{*}$ be an approximation of a given $(a \times b)$ matrix $F$ derived by methods that are not randomized. We prove that for a given $F$ and $F^{*}$, $H$ and $T$ can be computed by randomized algorithm such that $(HT)$ is an…
An exact discretization method is being developed for solving linear systems of ordinary fractional-derivative differential equations with constant matrix coefficients (LSOFDDECMC). It is shown that the obtained linear discrete system in…
The $(u,v)$-Pad\'e approximation to a function $f$ is the (unique, up to scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$ has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin, Pazuki, and…
This article deals with the adaptive and approximative computation of the Lam\'e equations. The equations of linear elasticity are considered as boundary integral equations and solved in the setting of the boundary element method (BEM).…
The dynamic matrix method addresses the Landau-Lifshitz-Gilbert (LLG) equation in the frequency domain by transforming it into an eigenproblem. Subsequent numerical solutions are derived from the eigenvalues and eigenvectors of the dynamic…