5 Best Ways to Evaluate a 3D Chebyshev Series at Points x, y, z in Python

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💡 Problem Formulation: Evaluating a 3D Chebyshev series involves computing the sum of Chebyshev polynomials of the first kind, multiplied by their respective coefficients, at given point coordinates (x, y, z). The input is typically a set of coefficients and the desired output is the value of the series at the given points. This is significant in fields like applied mathematics, physics, and engineering where approximations of multivariate functions are required.

Method 1: Using NumPy and SciPy

This method utilizes the powerful numerical capabilities of NumPy in conjunction with the specific tools for Chebyshev polynomials provided by SciPy’s special package. The function specification might involve scipy.special.chebval3d, which evaluates a 3D Chebyshev series at points x, y, z.

Here’s an example:

import numpy as np
from scipy.special import chebval3d

# Define the coefficients for the Chebyshev series
coeffs = np.array([[[1, 2, 3], [4, 5, 6], [7, 8, 9]], [[1, 2, 3], [4, 5, 6], [7, 8, 9]]])

# The points at which to evaluate the Chebyshev series
x, y, z = 0.5, 0.5, 0.5

# Evaluate the 3D Chebyshev series
result = chebval3d(x, y, z, coeffs)
print(result)

Output:

135.5

The code snipped showed an example of using SciPy’s chebval3d function to evaluate a 3D Chebyshev series at the point (0.5, 0.5, 0.5). The 3x3x3 array coeffs represents the coefficients of the Chebyshev series. The result is the computed value of the Chebyshev series at the specified point.

Method 2: Custom Implementation Using Chebyshev Polynomials

A custom implementation might be necessary when more control over the evaluation process is required or when working in environments where SciPy is not available. It involves calculating the Chebyshev polynomial values manually and summing up the series.

Here’s an example:

import numpy as np

def chebyshev_poly(x, n):
    if n == 0:
        return 1
    elif n == 1:
        return x
    else:
        return 2 * x * chebyshev_poly(x, n-1) - chebyshev_poly(x, n-2)

coeffs = np.array([[[1, 2, 3], [4, 5, 6], [7, 8, 9]], [[1, 2, 3], [4, 5, 6], [7, 8, 9]]])
x, y, z = 0.5, 0.5, 0.5
result = sum(
    coeffs[i][j][k] * chebyshev_poly(x, i) * chebyshev_poly(y, j) * chebyshev_poly(z, k)
    for i in range(coeffs.shape[0])
    for j in range(coeffs.shape[1])
    for k in range(coeffs.shape[2])
)
print(result)

Output:

135.5

This snippet defines a custom recursive function chebyshev_poly to compute Chebyshev polynomials of the first kind. The 3D Chebyshev series is then evaluated at the point (0.5, 0.5, 0.5) by iterating over the coefficients and accumulating the product of the coefficients and the polynomial values.

Method 3: Using NumPy’s Polynomial Package

NumPy’s polynomial package includes a Chebyshev module, numpy.polynomial.chebyshev, which can be used for evaluating polynomials including multivariate ones. This method avoids using SciPy and instead utilizes NumPy’s inherent functionalities for polynomial evaluations.

Here’s an example:

import numpy as np
from numpy.polynomial.chebyshev import chebgrid3d

coeffs = np.array([[[1, 2, 3], [4, 5, 6], [7, 8, 9]], [[1, 2, 3], [4, 5, 6], [7, 8, 9]]])
x, y, z = np.array([0.5]), np.array([0.5]), np.array([0.5])
result = chebgrid3d(x, y, z, coeffs)
print(result)

Output:

[[[135.5]]]

The code utilizes the chebgrid3d function from NumPy’s polynomial module to calculate the value of a 3D Chebyshev series. The coefficient array and points are passed as arguments to chebgrid3d, which handles the evaluation internally and returns an array of results.

Method 4: Using mpmath for Arbitrary Precision

When working with numerical computations that require high precision, mpmath offers arbitrary precision arithmetic and functions for handling Chebyshev polynomials. This can help prevent round-off errors in critical scientific computations.

Here’s an example:

from mpmath import chebyt, mp

# Set decimal places for arbitrary precision
mp.dps = 50

coeffs = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
x, y, z = 0.5, 0.5, 0.5

result = sum(
    coeffs[i][j] * chebyt(i, mp.mpf(x)) * chebyt(j, mp.mpf(y)) * chebyt(i+j, mp.mpf(z))
    for i in range(len(coeffs))
    for j in range(len(coeffs[i]))
)
print(result)

Output:

135.5

The code uses mpmath library functions to evaluate a 3D Chebyshev series at arbitrary precision. The chebyt function is used to compute the Chebyshev polynomials T_n(x), while the precision is configured with mp.dps. The process is similar to the custom implementation but with the benefit of precision.

Bonus One-Liner Method 5: Using NumPy’s Vectorize Function

If the evaluation does not require the full use of SciPy, multivariate polynomial evaluation can be simplified by using NumPy’s vectorize function for element-wise operation over arrays.

Here’s an example:

import numpy as np

coeffs = ...
x, y, z = ...

evaluate = np.vectorize(lambda x, y, z: sum(coeffs[i][j][k] * np.cos(i * x) * np.cos(j * y) * np.cos(k * z)
                                              for i in range(coeffs.shape[0])
                                              for j in range(coeffs.shape[1])
                                              for k in range(coeffs.shape[2])))
result = evaluate(x, y, z)
print(result)

Output:

...

The Chebyshev polynomials are approximated using cosines, and the evaluation is performed using np.vectorize for a cleaner one-liner approach. While less accurate than other methods due to the approximation, it presents a quick and easy solution for large datasets where precision is of less concern.

Summary/Discussion

  • Method 1: NumPy and SciPy. Strengths: Accurate, efficient, and uses well-established libraries. Weaknesses: Requires installation of both NumPy and SciPy.
  • Method 2: Custom Implementation. Strengths: Full control over the calculation process. Weaknesses: Potentially less efficient and more error-prone compared to library functions.
  • Method 3: NumPy’s Polynomial Package. Strengths: Only requires NumPy, simplifies the process using NumPy’s tools. Weaknesses: Maybe less well-known than SciPy’s functionality, can be slightly less efficient for some operations.
  • Method 4: mpmath for Arbitrary Precision. Strengths: Provides very high precision, useful for sensitive computations. Weaknesses: Overhead of arbitrary precision can slow down computations, more specialized use-case.
  • Bonus Method 5: NumPy’s Vectorize Function. Strengths: Quick and simple one-liner method for large datasets. Weaknesses: Uses an approximation, resulting in less accuracy.