💡 Problem Formulation: Given an array of floating point coordinates (x, y, z), the objective is to generate a pseudo Vandermonde matrix using the Laguerre polynomials. The Vandermonde matrix is a matrix with the terms of a geometric progression in each row, which, in our case, links to the evaluation of the Laguerre polynomials at … Read more
💡 Problem Formulation: When working with polynomials in the field of mathematics and computational algebra, one may need to generate a Hermite polynomial given a set of complex roots. In Python, this task involves creating a Hermite series such that, when evaluated, the polynomial returns zero at each root. This article aims to demonstrate various … Read more
💡 Problem Formulation: This article addresses the process of constructing a Vandermonde matrix specifically for the Laguerre polynomial, given a float array representing points at which the polynomial is evaluated. For example, given a float array [0.1, 0.5, 0.9], we aim to generate a matrix that represents the Laguerre polynomials evaluated at these points, providing … Read more
💡 Problem Formulation: In computational mathematics, generating a Vandermonde matrix for polynomial bases like Laguerre polynomials is a fundamental operation, especially with complex points. Given a complex array of points, the task is to produce a Vandermonde matrix where each row corresponds to a Laguerre polynomial evaluated at these points. For example, given points [1+2j, … Read more
💡 Problem Formulation: We aim to create a Vandermonde matrix based on the values of a Laguerre polynomial, a classical orthogonal polynomial. Given an order n and a set of points x, the desired output is a matrix V whereby each column j is the Laguerre polynomial of degree j evaluated at each point in … Read more
💡 Problem Formulation: In scientific computing, evaluating polynomial series such as the Hermite series at given points is a common task. In this article, we explore how to compute the values of a 2D Hermite series at specific (x, y) coordinates using a three-dimensional array of coefficients in Python. For instance, given a set of … Read more
💡 Problem Formulation: Often in computational mathematics and physics, we need to find the roots of polynomials, which correspond to solutions of the underlying problem. A Laguerre series provides a way to approximate functions, and its roots can be important in various contexts. This article will help you compute the roots of a Laguerre series … Read more
💡 Problem Formulation: Integrating Hermite series in Python involves calculating the indefinite integral of a polynomial derived from Hermite functions and adding a constant of integration. The challenge lies in applying a symbolic computation approach to determine the antiderivative and setting a specific integration constant for completeness. For example, given a Hermite series expressed through … Read more
💡 Problem Formulation: In computational mathematics, finding the roots of a Laguerre series is essential for various applications, such as solving differential equations and modeling physical phenomena. Given a Laguerre series, the goal is to find all the complex or real zeroes of the series accurately. This article illustrates five effective methods for computing these … Read more
💡 Problem Formulation: In the realm of computational mathematics, one may need to evaluate a two-dimensional Laguerre series at a series of points defined by the Cartesian product of two vectors x and y. The challenge lies in implementing an efficient and accurate calculation to obtain a matrix of the series’ values at these points. … Read more
💡 Problem Formulation: You’re given a set of complex roots and you need to generate a Laguerre series that corresponds to these roots in Python. A Laguerre polynomial is an orthogonal polynomial used in physics and engineering which can be determined by its roots. For instance, if you’re given roots as input complex(1, 2), complex(3, … Read more
💡 Problem Formulation: When working with orthogonal polynomials in computational applications, it’s common to evaluate series expansions such as the 3D Laguerre series at given points. The task is to determine the value of a function represented by a Laguerre series at coordinates (x, y, z), given a 2D array of coefficients in Python. For … Read more