💡 Problem Formulation: Calculating the integral of a Hermite series with a specified lower bound is an important task in computational mathematics and physics. The Hermite series is a series expansion similar to a Fourier series that is defined by Hermite polynomials. In this article, we will explore several methods of integrating a Hermite series … Read more

💡 Problem Formulation: You’re tasked with integrating a Hermite series, a polynomial used in probability, quantum physics, and approximation theory, and then need to multiply the results by a scalar value, all before an integration constant is added. For example, given a Hermite series H_n(x) and a scalar a, you want to perform the integration … Read more

💡 Problem Formulation: In computational mathematics, evaluating polynomials on a grid is a common task. Specifically, for a 2D Laguerre series, given a 1D array of coefficients, the goal is to compute the values on the Cartesian product of vectors x and y. The input consists of a 1D array of coefficients c, and two … Read more

💡 Problem Formulation: In numerical analysis and computational mathematics, integrating a Hermite series over a specified axis is a common task. For example, we might have a two-dimensional array that represents the coefficients of a Hermite series along axis 0 and we want to perform an integration along this axis to obtain the result. The … Read more

💡 Problem Formulation: Integrating a Chebyshev series over a specific axis is a computational task often encountered in scientific computing and data analysis. The challenge is to accurately perform the integral of a series, which is represented by Chebyshev coefficients, along the desired axis in a multidimensional array. For example, given a 2D array where … Read more

💡 Problem Formulation: In scientific computing, integrating polynomial series such as Chebyshev series is a common task. Specifically, we may want to integrate a multidimensional array representing Chebyshev coefficients along a particular axis. This article provides methods for integrating a Chebyshev series over axis 0 within Python, transforming our input, an array of coefficients, into … Read more

💡 Problem Formulation: Integrating a Chebyshev series over a specific axis can be crucial in analyzing the behavior of polynomials within a certain range or domain. The task is to compute the definite integral of Chebyshev polynomials of the first kind along axis 1. Given a multidimensional array of Chebyshev coefficients, we aim to find … Read more

💡 Problem Formulation: In computational mathematics, generating a Hermite series with specific roots is a common task. This article outlines how to construct a Hermite polynomial in Python that zeros at a given set of roots. For instance, if given roots are 1, 2, 3, we desire to create a Hermite polynomial that has these … Read more

💡 Problem Formulation: We want to generate a pseudo Vandermonde matrix using Hermite polynomials for a given set of complex points. These complex points are the coordinates in the space where we wish to evaluate the Hermite polynomials. The desired output is a matrix where each column corresponds to a Hermite polynomial evaluated at all … Read more

💡 Problem Formulation: A Vandermonde matrix is pivotal in polynomial fitting and interpolation. Specifically, generating a pseudo-Vandermonde matrix with Hermite polynomials requires evaluating these polynomials at given sample points (x, y, z). The desired outcome is a matrix with each column corresponding to a Hermite polynomial evaluated at these sample points, allowing for numerical applications … Read more

💡 Problem Formulation: Subtracting one Laguerre series from another is an essential task in numerical analysis and mathematical computations involving orthogonal polynomials. Given two Laguerre series with coefficients, the goal is to perform a subtraction operation resulting in a new series representing the difference. For example, if we have Laguerre series A and B as … Read more

💡 Problem Formulation: Multiplying a Laguerre series by an independent variable is a task often encountered in numerical analysis and scientific computing. Given a series expansion of a function as a sum of Laguerre polynomials, we want to scale this series by an independent variable ‘x’. The input is a sequence of coefficients (c0, c1, … Read more