💡 Problem Formulation: When working with orthogonal polynomials in computational mathematics, one operation you might need to perform is dividing one Hermite series by another. This article explores how to achieve this in Python, with the input being two Hermite series represented by coefficient arrays and the desired output being the coefficient array of their … Read more
💡 Problem Formulation: Multiplying Hermite series involves calculating the product of two polynomial series that are expressed in terms of Hermite polynomials. In Python, a Hermite series can be represented as an array of coefficients, and we seek efficient methods to compute the product of two such series, resulting in a new set of Hermite … Read more
💡 Problem Formulation: Evaluating a 2D Chebyshev series involves computing the sum over a two-dimensional grid of values (x and y), using a Chebyshev polynomial of the first kind. Given a 3D array representing coefficients of the Chebyshev series and Cartesian products of x and y, the goal is to efficiently compute the series values. … Read more
💡 Problem Formulation: Given a Chebyshev series representation of a function, a common task is to multiply this series by an independent variable, essentially increasing the degree by one and introducing a new term. For example, if you have the Chebyshev series a_0 + a_1*T_1(x) + a_2*T_2(x) for variable x, and you want to multiply … Read more
💡 Problem Formulation: Evaluating a two-dimensional Chebyshev series involves computing the Chebyshev polynomial expansion for given coefficients over a grid of x and y values. Specifically, for a known series with coefficients C[i][j], we want to compute the value at each point (x, y) in the Cartesian product of two sets of points. For example, … Read more
💡 Problem Formulation: The task involves performing subtraction of one Chebyshev series from another in Python. Chebyshev series are used to approximate functions and are akin to Fourier series with coefficients of Chebyshev polynomials. Given two such series A and B, the aim is to compute the result C = A – B, where C … Read more
💡 Problem Formulation: This article addresses the problem of evaluating a Hermite series for a given set of coefficients at multiple points arranged in a multidimensional array in Python. For instance, if we have a set of coefficients a_n for a Hermite series, and points X in an N-dimensional grid, the task is to compute … Read more
💡 Problem Formulation: In mathematical computations and data analysis, it is often required to perform operations on series expansions such as Chebyshev series. Consider two Chebyshev series A and B, represented by their coefficients. The task is to efficiently add these series to obtain a new series C, whose coefficients are the sum of the … Read more
💡 Problem Formulation: When working with polynomial approximations, Hermite series provide a robust way to represent a broad variety of functions. In Python, evaluating these polynomials at a given array of points ‘x’ requires specific methods. In this context, the ‘x’ represents the input points where we seek the value of the Hermite series, and … Read more
💡 Problem Formulation: When working with polynomials in Python, it’s common to encounter situations where we want to eliminate small trailing coefficients that are negligible and simplify the polynomial expression. For example, given a polynomial like 5x^3 + 2x^2 + 0.001x + 0.00001, we may want to remove the terms with coefficients smaller than 0.01 … Read more
💡 Problem Formulation: We aim to evaluate a three-dimensional (3D) Chebyshev series at given points (x, y, z) utilizing a two-dimensional (2D) array of coefficients. This operation is complex as it involves polynomial computations in multiple dimensions. For an input array coeffs representing the Chebyshev coefficients and a point (x, y, z), the desired output … Read more
💡 Problem Formulation: The task is to compute the values of a two-dimensional Chebyshev series at specific points (x, y) given a one-dimensional array representing the coefficients of the series. The input is the 1D array of coefficients and the (x, y) points. The desired output is the computed series values at these points. This … Read more