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We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively.…

Mathematical Physics · Physics 2018-06-21 A. D. Alhaidari

We introduce a numerical method for the numerical solution of the so-called Lur'e matrix equations that arise in balancing-related model reduction and linear-quadratic infinite time horizon optimal control. Based on the fact that the set of…

Numerical Analysis · Mathematics 2011-01-07 Federico Poloni , Timo Reis

An existing solvability result for relaxed one-sided Lipschitz algebraic inclusions is substantially improved. This enhanced solvability result allows the design of a very robust numerical method for the approximation of a solution of the…

Optimization and Control · Mathematics 2013-08-19 Wolf-Jürgen Beyn , Janosch Rieger

Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an…

Mathematical Physics · Physics 2022-06-20 A. D. Alhaidari

A new Monte-Carlo method for solving linear parabolic partial differential equations is presented. Since, in this new scheme, the particles are followed backward in time, it provides great flexibility in choosing critical points in…

Numerical Analysis · Mathematics 2025-10-20 Johan Carlsson

Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale 2000). The main progress on Smale's problem is Beltr\'an-Pardo (2011) and B\"urgisser-Cucker (2010). In…

Numerical Analysis · Mathematics 2015-07-15 Diego Armentano , Carlos Beltrán , Peter Bürgisser , Felipe Cucker , Michael Shub

Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F\_1, ..., F\_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$…

Combinatorics · Mathematics 2008-05-05 Mireille Bousquet-Mélou , Arnaud Jehanne

We consider an affine process $X$ which is only observed up to an additive white noise, and we ask for its law, for some time $t > 0 $, conditional on all observations up to this time $ t $. This is a general, possibly high dimensional…

Probability · Mathematics 2018-01-25 Lukas Gonon , Josef Teichmann

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed.…

Numerical Analysis · Mathematics 2015-10-29 Petr N. Vabishchevich

This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying…

Optimization and Control · Mathematics 2018-04-25 I. L. Galabova , J. A. J. Hall

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…

Computational Physics · Physics 2009-10-31 Bogdan Mihaila , Ioana Mihaila

We present a new algorithm to solve polynomial equations, and publish its code, which is 1.6-3 times faster than the ZROOTS subroutine that is commercially available from Numerical Recipes, depending on application. The largest improvement,…

Earth and Planetary Astrophysics · Physics 2012-03-07 J. Skowron , A. Gould

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

Symbolic Computation · Computer Science 2017-04-14 Victor Y. Pan , Liang Zhao

Using Maple, we compute some analytical solutions of a modified Black-Scholes equation, recently proposed, in the case of the European put option. We show that the modified Black-Scholes equation with the European put option is exactly…

Computational Finance · Quantitative Finance 2015-08-18 Juan Ospina

We consider the problem of finding a low rank symmetric matrix satisfying a system of linear equations, as appears in phase retrieval. In particular, we solve the gauge dual formulation, but use a fast approximation of the spectral…

Optimization and Control · Mathematics 2020-06-02 Ron Estrin , Yifan Sun , Halyun Jeong , Michael Friedlander

We address asymptotic formulae for the classical Poincar\'e-Perron problem of linear differential equations with almost constant coefficients in a half line $[t_0,+\infty)$ for high order equation $n\ge 5$ and some $t_0\in\mathbb{R}$. By…

Classical Analysis and ODEs · Mathematics 2021-11-11 H. Bustos , P. Figueroa , Manuel Pinto

In the article \The State of SAT", the authors asked whether a procedure dramatically different from DPLL can be found for handling unsatisfiable instances. This study proposes a new linear programming approach to address this issue…

Computational Complexity · Computer Science 2018-08-20 Chengling Fang , Jiang Liu

We consider a method of pairwise variations for smooth optimization problems, which involve polyhedral constraints. It consists in making steps with respect to the difference of two selected extreme points of the feasible set together with…

Optimization and Control · Mathematics 2017-01-12 I. V. Konnov

In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite…

Numerical Analysis · Mathematics 2025-10-20 Gianluca Argentini

We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical solutions of the equation. Error estimates are given for Lagrange…

Numerical Analysis · Mathematics 2015-07-31 Gerard Awanou
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