Emily Rosemary Collins

💡 Problem Formulation: When working with orthogonal polynomials, such as the Hermite polynomials in the probability (He) or physics (H) conventions, there may be the need to perform operations like raising a polynomial to a power. This article provides methods to raise a Hermite E series to a power in Python, with an emphasis on … Read more

💡 Problem Formulation: If you’re dealing with Legendre polynomials and need to evaluate them across a multidimensional array of points in Python, finding the right approach is crucial. Let’s say you have a series of Legendre polynomials and a 2D array of points ‘x’. You seek a method to efficiently compute the corresponding values at … Read more

💡 Problem Formulation: Users working in computational mathematics or physics may often need to compute derivatives of Legendre series, which represent functions as expansions of Legendre polynomials. This article provides robust solutions to differentiate a Legendre series and express first or higher-order derivatives using Python. An example input could be a Legendre series expressed by … Read more

💡 Problem Formulation: When working with Legendre polynomials in scientific and engineering computations, it is common to encounter series expansions with multidimensional coefficients. The differentiation of such series is key in various problems, such as solving differential equations, optimizing systems, and more. Consider a Legendre series expressed as a sum of Legendre polynomials P_n(x) with … Read more

💡 Problem Formulation: In computational mathematics, evaluating a two-dimensional Hermite E series at given points using a three-dimensional array of coefficients is a specific task that may arise in the context of approximation theory or spectral methods. For example, given a set of coefficients C[i][j][k] in a 3D array, and points x and y, the … Read more

💡 Problem Formulation: Given a two-dimensional Hermite E Series, a physicist or mathematician might need to evaluate the series at specific points (x, y). This could be for the purposes of statistical analysis, signal processing, or solving physics problems involving quantum harmonic oscillators. For example, if given the coefficients of a 2D Hermite E Polynomial, … Read more

💡 Problem Formulation: The challenge is to calculate the value of a 2D Laguerre series given a set of points (x, y) and a one-dimensional array of coefficients. Specifically, we want to plug the points into a polynomial which coefficients are determined by the 1D array, to assess the polynomial’s value at those points. For … Read more

💡 Problem Formulation: This article aims to address the computational task of evaluating a 3D Laguerre series at a set of points (x, y, z) using a 4D array of coefficients in Python. Given a set of Laguerre coefficients arranged in a four-dimensional array and a triplet of points, the desired output is the computed … Read more

💡 Problem Formulation: Evaluating a 2D Legendre series involves a scenario where you have two sets of variables, usually x and y , and you want to compute the series sum based on the Cartesian product of these variables using a given set of coefficients stored in a 1D array. For example, given 1D coefficients … Read more

💡 Problem Formulation: We aim to evaluate a 2D Legendre series on a grid formed by the Cartesian product of x and y coordinates using a 3D array for coefficients. Such a task is frequently encountered in numerical analysis and computational physics, where a set of coefficients a[n, m] relates to the nth and mth … Read more

💡 Problem Formulation: For scientists and mathematicians working with polynomial approximations, generating a pseudo-Vandermonde matrix using Hermite E polynomials and a set of sample points is a common problem. A Vandermonde matrix is essential for various numerical and analytical applications. With Python, the goal is to transform a given set of sample points x, y, … Read more

💡 Problem Formulation: This article seeks to provide solutions for generating a pseudo-Vandermonde matrix which incorporates Hermite E polynomials evaluated at complex array points. Given a set of complex numbers representing coordinates, the goal is to construct a matrix where each row corresponds to a complex point, and each column represents a successive Hermite E … Read more