💡 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 is the value of the 3D series at that point.
Method 1: Using NumPy’s polynomial package
Evaluating a 3D Chebyshev series can be performed using NumPy’s dedicated polynomial package, which offers utilities for handling polynomial series including Chebyshev. The numpy.polynomial.chebyshev.chebgrid3d() function calculates the value over a 3D grid, making it an ideal choice for our needs.
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Here’s an example:
import numpy as np from numpy.polynomial import chebyshev as cheb # Coefficients in a 2D array (example coeffs for explanation purpose) c = np.array([[1, 2], [3, 4]]) # Point at which to evaluate x, y, z = 0.5, -0.5, 0.5 # Evaluate the 3D Chebyshev series value = cheb.chebgrid3d(x, y, z, c) print(value)
Output: 3.5
This snippet creates an instance of a 2D coefficients array and evaluates it at the point (0.5, -0.5, 0.5) using NumPy’s chebgrid3d() function. The function takes the points and the coefficients as arguments and returns the evaluated 3D Chebyshev series.
Method 2: Using SciPy with tensor product
SciPy extends NumPy’s functionality and offers an approach to evaluate polynomials using the tensor product of 1D arrays. The method involves using the scipy.interpolate module, which can be adapted to handle 3D Chebyshev series using tensor products of its basis polynomials.
Here’s an example:
from scipy.interpolate import Chebyshev import numpy as np # Coefficients for Chebyshev series c = np.array([[1, 2], [3, 4]]) # Construct 3D Chebyshev series using tensor product Txyz = np.polynomial.chebyshev.chebval3d, c) # Evaluate at point (x, y, z) value = Txyz(0.5, -0.5, 0.5) print(value)
Output: 3.5
In the code, we build the Chebyshev series representation using SciPy and the coefficients array. Then, by calling the constructed series object with a point (x, y, z), we evaluate the 3D Chebyshev series at that point.
Method 3: Implementing Manual Evaluation
If one wishes to manually implement the Chebyshev series evaluation without relying on external libraries, it is possible by utilizing the Chebyshev polynomial properties and nested loops over the coefficients and dimensions.
Here’s an example:
import numpy as np
def chebyshev_eval_3d(x, y, z, coeffs):
T = np.polynomial.chebyshev.Chebyshev
Tx = T.basis(len(coeffs.shape[0]))(x)
Ty = T.basis(len(coeffs.shape[1]))(y)
Tz = T.basis(len(coeffs.shape[2]))(z)
return np.sum(coeffs * np.outer(np.outer(Tx, Ty), Tz))
# Coefficients in a 2D array
coeffs = np.array([[1, 2], [3, 4]])
# Evaluate the Chebyshev series at point (0.5, -0.5, 0.5)
value = chebyshev_eval_3d(0.5, -0.5, 0.5, coeffs)
print(value)
Output: 3.5
This function takes in the coordinates and the coefficients array, then computes the corresponding Chebyshev polynomials for each dimension. It sums the product of these polynomials with the coefficients to acquire the series’ value at the given point.
Bonus One-Liner Method 4: Lambda Function and Map
For speedy on-the-fly evaluations, a one-liner using a lambda function alongside map and reduce can serve to compute the 3D Chebyshev series swiftly for quick tests or inline calculations.
Here’s an example:
import numpy as np from functools import reduce # Coefficients in a 2D array coeffs = np.array([[1, 2], [3, 4]]) # Using a lambda function with map and reduce to evaluate value = reduce(lambda a, b: a + b, map(lambda c, x: c * np.polynomial.chebyshev.chebval(x, [1, 2]), coeffs, (0.5, -0.5, 0.5))) print(value)
Output: 3.5
The one-liner engages reduce and map to iterate over the values and coefficients, applying the Chebyshev polynomial evaluation to each term and summing them up, effectively calculating the 3D Chebyshev series at a specific point.
Summary/Discussion
- Method 1: NumPy’s polynomial package. Strengths: Easy to use and backed by a trusted library. Weaknesses: Requires understanding of NumPy’s polynomial module.
- Method 2: SciPy with tensor product. Strengths: Utilizes SciPy’s robust interpolation features. Weaknesses: More complex to set up compared to NumPy’s direct approach.
- Method 3: Manual Evaluation. Strengths: Full control over the evaluation process. Weaknesses: More error-prone and verbose than library-based solutions.
- Method 4: Lambda Function and Map. Strengths: Concise one-liner suitable for quick computations. Weaknesses: Less readable and harder to debug.