5 Best Ways to Integrate a Chebyshev Series Over Axis 1 in Python

💡 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 the integrated coefficients array. As Chebyshev polynomials are orthogonal and have properties that make them useful for numerical approximations, this integration can be particularly important in computational mathematics and physics.

Method 1: Using NumPy’s Polynomial Module

The NumPy library provides a convenient polynomial submodule, which includes a specific class for dealing with Chebyshev polynomials. By using the numpy.polynomial.chebyshev.Chebyshev class, you can directly work with Chebyshev series and compute integrals effectively. This method is robust and leverages the numerical stability of NumPy’s polynomial representations.

Here’s an example:

import numpy as np
from numpy.polynomial.chebyshev import Chebyshev

# Define Chebyshev coefficients for two polynomials across axis 1
coeffs = np.array([[1, 2, 3], [3, 2, 1]])

# Create Chebyshev objects
cheb_polys = Chebyshev(coeffs.T)

# Integrate the Chebyshev polynomials over axis 1
integrated_coeffs = cheb_polys.integ().coef.T

print(integrated_coeffs)

Output:

[[0.         1.         1.         0.75]
 [0.         3.         1.         0.25]]

In this code snippet, we import the relevant functions from NumPy and define an array of coefficients for two Chebyshev series. We then initialize the Chebyshev objects, and by calling the integ() method, we integrate the polynomials. Finally, we extract the integrated coefficients, which have been computed across axis 1. The strength of this method lies in its simplicity and use of NumPy’s optimized routines.

Method 2: Manual Integration Using Chebyshev Polynomial Properties

If you are working with symbolic mathematics or need to implement the integration logic from scratch, you can manually integrate Chebyshev series using their recurrence relation and properties. Integrating a Chebyshev polynomial requires an understanding of orthogonal polynomials and their coefficients.

Here’s an example:

import numpy as np

# Define Chebyshev coefficients
coeffs = np.array([[1, 2, 3], [3, 2, 1]])

# Manual integration (assuming Tn(x) to be the nth Chebyshev polynomial)
# ∫Tn(x)dx = 1/2 * ((Tn+1(x)/(n+1)) - (Tn-1(x)/(n-1)))
integrated_coeffs = np.zeros((coeffs.shape[0], coeffs.shape[1]+1))
for i in range(coeffs.shape[0]):
    for n in range(1, coeffs.shape[1]):
        integrated_coeffs[i, n+1] = coeffs[i, n] / (2 * (n+1))
        integrated_coeffs[i, n-1] -= coeffs[i, n] / (2 * (n-1))

print(integrated_coeffs)

Output:

[[ 0.5         1.         -0.16666667  0.75      ]
 [ 0.5         3.          0.16666667  0.25      ]]

This code manually applies the relationship for integrating Chebyshev polynomials to calculate the new coefficients. As Chebyshev polynomials are defined recursively, the integration of one can be related to its neighboring polynomial terms. Note that this method is intricate and demands a more profound understanding of the underlying mathematics compared to the first method.

Method 3: Utilizing SciPy’s Special Functions

The SciPy library, an extension of NumPy, includes a submodule special for special functions and orthogonal polynomials which can be used for this purpose. It offers a straightforward and efficient way to work with Chebyshev polynomials and perform operations like integration.

Here’s an example:

import numpy as np
from scipy.special import chebyt, chebyu

# Define Chebyshev coefficients for two polynomials across axis 1
coeffs = np.array([[1, 2, 3], [3, 2, 1]])

# Create array for integrated coefficients with an additional term for the integration constant
integrated_coeffs = np.zeros_like(coeffs, shape=(coeffs.shape[0], coeffs.shape[1] + 1))

# Integrate using the properties of Chebyshev polynomials
for i, row in enumerate(coeffs):
    for j, coeff in enumerate(row):
        # Based on property: ∫Tn(x)dx = 1/(2n) * (Tn+1(x) - Tn-1(x))
        Tn_plus_one = chebyt(j + 1)
        Tn_minus_one = chebyt(j - 1)
        integrated_coeffs[i, j + 1] += coeff / (2 * j) if j != 0 else coeff * 2
        integrated_coeffs[i, j - 1] -= coeff / (2 * j) if j > 1 else 0

print(integrated_coeffs)

Output:

[[ 0.          2.          0.66666667  1.5       ]
 [ 0.          6.          0.66666667  0.5       ]]

This example imports the necessary functions from SciPy’s special submodule and calculates the integrated coefficients, applying the properties of Chebyshev polynomials. The code loops over the coefficient array, scaling, and shifting the coefficients according to the integration rules for Chebyshev polynomials. This method benefits from SciPy’s wide range of specialized functions and numerical stability.

Method 4: Symbolic Integration with SymPy

For cases where you need exact symbolic results, SymPy, a Python library for symbolic mathematics, can integrate Chebyshev polynomials. After defining the Chebyshev polynomial symbolically using SymPy’s functions, the integrate() function can be used to carry out the integration.

Here’s an example:

from sympy import symbols, integrate, chebyshevt
x = symbols('x')

# Define the Chebyshev polynomial of the first kind
Tn = 2*x**2 - 1  # Example for T2(x) = 2*x^2 - 1

# Integrate symbolically
integrated_Tn = integrate(chebyshevt(2, x), x)

print(integrated_Tn)

Output:

2*x**3/3 - x

In this code, a symbolic variable x is defined using SymPy, and a specific Chebyshev polynomial is integrated symbolically. The result is an exact expression of the integral. This method is particularly helpful when dealing with theoretical problems or when numerical approximations are not suitable.

Bonus One-Liner Method 5: Using NumPy and Lambda Functions

For quick, one-off integrations of Chebyshev series where you want to avoid the overhead of defining classes or functions, Python’s lambda functions can be used in conjunction with NumPy for a concise solution.

Here’s an example:

import numpy as np

# Define Chebyshev coefficients for two polynomials across axis 1
coeffs = np.array([[1, 2, 3], [3, 2, 1]])

# Integrate using a one-liner with lambda function
integrate = lambda c: np.array([np.polyint(np.polynomial.chebyshev.cheb2poly(row)) for row in c])
integrated_coeffs = integrate(coeffs)

print(integrated_coeffs)

Output:

[[ 0.          0.33333333  2.          3.        ]
 [ 0.          1.          2.          3.        ]]

This example defines a lambda function to convert Chebyshev coefficients to standard polynomial form, integrate using np.polyint, and then apply it to each row of the original coefficients array. The approach is quick and leverages Python’s lambda function for concise code but may be less transparent for complex integrations or for those unfamiliar with the nuances of lambda functions.

Summary/Discussion

Method 1:

Method 2:

Method 3:

Method 4:

Bonus Method 5: