💡 Problem Formulation: A pseudo Vandermonde matrix for Hermite polynomials is computed from a set of points and involves evaluating Hermite polynomials at those points to create the matrix. The input is a float array of point coordinates, e.g., [1.0, 3.5, 5.2]. The desired output is a matrix where each column is the evaluation of … Read more
💡 Problem Formulation: For numerical computations involving polynomials, especially in the field of science and engineering, evaluating Legendre series at specific points is a common task. If we have a Legendre series expressed by coefficients c and an array of points x, we want to compute the series value at each point in x. For … Read more
💡 Problem Formulation: A Vandermonde matrix is a square matrix with the terms of a geometric progression in each row. When dealing with Hermite polynomials, a pseudo-Vandermonde matrix incorporates the values derived from these polynomials. This article will demonstrate how to generate such a matrix in Python, starting with a sequence of points (e.g., x … Read more
💡 Problem Formulation: Evaluating a Laguerre series involves computing the values of the polynomial at a given set of points. This article addresses the computation over multidimensional arrays in Python, which can be a common task in numerical analysis or scientific computing. For instance, given a Laguerre series with coefficients [1, -1, 0.5] and a … Read more
💡 Problem Formulation: When working with polynomial approximations in numerical methods, a common task is to evaluate a Laguerre series for a given array of points ‘x’. A Laguerre series, representing a function as an infinite sum of Laguerre polynomials, is particularly useful in physics and engineering. This article demonstrates how to compute the value … Read more
💡 Problem Formulation: When working with orthogonal polynomials and series approximations in numerical analysis, a common task might be to evaluate a Laguerre series at specified points, efficiently computing the result for each coefficient set. The input typically includes a 1D array of x points where the series will be evaluated, and a 2D array … Read more
💡 Problem Formulation: If you’re working with orthogonal polynomials in multidimensional arrays, you might encounter the need to integrate a Laguerre series—a series of Laguerre polynomials—along a specified axis. This article tackles how to perform this operation over axis 1 using Python. For an input array representing coefficients of the Laguerre series, we aim to … Read more
💡 Problem Formulation: In computational mathematics, a common problem is evaluating a 3D Laguerre series over a range of points defined by the Cartesian product of x, y, and z coordinates. The input is a set of coefficients for the series and the Cartesian grid points, while the output is the computed values of the … Read more
💡 Problem Formulation: Laguerre series, based on Laguerre polynomials, often arise in problems of physics and engineering involving exponential decay. The challenge is to integrate such series along a particular axis of a multidimensional array. In Python, efficient and accurate methods are sought for performing this task. The input would be an array defined by … 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