Related papers: Finite difference and numerical differentiation: G…
Our approach to higher order Fourier analysis is to study the ultra product of finite (or compact) Abelian groups on which a new algebraic theory appears. This theory has consequences on finite (or compact) groups usually in the form of…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
We present the classical coordinate-free formalism for forward and backward mode ad in the real and complex setting. We show how to formally derive the forward and backward formulae for a number of matrix functions starting from basic…
In this paper, we present an implicit finite difference method for the numerical solution of the Black-Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front fixing…
This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method…
In this work, we concern with the high order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the…
In the present note we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs): from one side they could look standard, since they are, real, symmetric…
Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v-…
This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional…
This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually…
The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and…
This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the…
We propose an integral representation for the higher-order Fr\'echet derivative of analytic matrix functions $f(A)$ which unifies known results for the first-order Fr\'echet derivative of general analytic matrix functions and for…
Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of…
Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence…
We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of…
In this article, a new modified Laplace-Fourier method is developed in order to obtain the solutions of linear neutral delay differential equations. The proposed method provides a more accurate solution than the one provided by the pure…
A general explicit form for generating functions for approximating fractional derivatives is derived. To achieve this, an equivalent characterisation for consistency and order of approximations established on a general generating function…