💡 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: This article addresses how to calculate the derivatives of a Hermite series—used for approximating functions with polynomials in a probabilistic setting—when its coefficients are not just scalar values but multidimensional arrays. Imagine you’ve been given a set of coefficients in an ‘n-dimensional’ array and you’re tasked to differentiate this Hermite series along … 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: Integrating a Laguerre series—a series expansion using orthogonal Laguerre polynomials—can be critical for solving differential equations or analyzing probabilistic systems in fields like physics and engineering. Python users often seek to accomplish this by setting a specific integration constant to tailor the result to boundary conditions or normalization requirements. This article explores … 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: 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: 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: 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: This article focuses on computationally evaluating a three-dimensional Hermite series at given coordinate points (x, y, z) utilizing a two-dimensional array of coefficients. The Hermite series, a polynomial analogous to Fourier series, can be applied in fields such as physics and engineering for various interpolations and approximations. Given a set of coefficients … Read more
💡 Problem Formulation: Given a two-dimensional Hermite series defined by a one-dimensional array of coefficients, the task is to evaluate the series at specific points (x, y). For example, if the input coefficients array is [a0, a1, a2, …], and the points are (x, y), the output should be the calculated value of the Hermite … Read more
💡 Problem Formulation: Evaluating a 3D Hermite series involves computing the value of a polynomial that’s approximated by Hermite functions along three dimensions at specific points (x, y, z). This task requires handling a four-dimensional (4D) array of coefficients that represent the series expansion in a 3D space. For applications in computational physics, computer graphics, … Read more
💡 Problem Formulation: This article addresses the challenge of evaluating a three-dimensional Hermite series at specific points (x, y, z) in Python. When given a set of coefficients representing the series, along with the desired evaluation points, the output is the calculated value at each point. For instance, for a series with coefficients [a0, a1, … Read more