5 Best Ways to Differentiate a Legendre Series with Multidimensional Coefficients Over Specific Axis in Python

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💡 Problem Formulation: Computing derivatives of Legendre series with multidimensional coefficients can be essential in various mathematical and engineering applications. Given a multidimensional array representing coefficients of a Legendre series, how can one perform differentiation over a specific axis to achieve a new set of coefficients that represent the derivative of the original series? For instance, given a 3D array c indexing coefficients on the last axis, one might seek dc, the derivative coefficients when differentiated with respect to the second axis.

Method 1: NumPy’s Gradient Function

NumPy provides a convenient function numpy.gradient() which computes the gradient of an N-dimensional array. By setting the axis parameter, one can differentiate along any desired axis. This method automatically handles the spacing between samples if it is uniform, or it can take a variable spacing provided by the user.

Here’s an example:

import numpy as np

coeffs = np.random.random((5, 5, 5))  # A 3D array of coefficients
dc_dx = np.gradient(coeffs, axis=1)  # Differentiate over the second axis

Output: dc_dx would be an array of the same shape as coeffs, representing the derivative with respect to the second axis.

This snippet creates a random 3-dimensional array of Legendre series coefficients and computes the derivative over the second axis using the np.gradient() function. It is straightforward, requires no additional configuration, and is suitable for uniform spacing.

Method 2: SciPy’s Special Function – legendre

The scipy.special.legendre() function generates a Legendre polynomial object which includes a method for differentiating the polynomial. When dealing with multidimensional coefficient arrays, one can apply this method iteratively over the desired axis. This method is best suited for cases where the coefficients indicate a single-variable Legendre series.

Here’s an example:

from scipy.special import legendre
import numpy as np

coeffs = np.random.random(5)  # A 1D array of coefficients for a Legendre polynomial
L = legendre(coeffs.shape[0] - 1)  # Create Legendre polynomial object
dL = L.deriv()  # Differentiate the Legendre polynomial
dc_dx = dL(coeffs)  # Apply the differentiated polynomial to the coefficients

Output: dc_dx would be an array of the derivative coefficients.

In this code, the legendre class is used to represent a Legendre polynomial. The deriv() method is then called to compute its derivative, which is subsequently applied to the coefficients array. This method ensures that the Legendre polynomial properties are used but is limited to one-dimensional series.

Method 3: Custom Differentiation Function using NumPy

For non-uniform spacing or more complex differentiation needs, writing a custom differentiation function may be necessary. This involves direct manipulation of the coefficients array using slicing and NumPy’s array operations to numerically differentiate the series along the chosen axis.

Here’s an example:

import numpy as np

def differentiate_coeffs(coeffs, axis):
    dc = np.diff(coeffs, axis=axis)
    return np.concatenate([dc, np.zeros_like(coeffs.take([-1], axis=axis))], axis=axis)

coeffs = np.random.random((5, 5, 5))  # 3D array of coefficients
dc_dx = differentiate_coeffs(coeffs, 1)  # Differentiate over the second axis

Output: dc_dx is an array where the differentiation over the specified axis has been performed, padded with zeros to match the original array shape.

This custom function employs the np.diff() method to compute the difference along the specified axis and pads the result to maintain the shape of the coefficients array. This approach allows for custom differentiation logic but comes without the built-in conveniences of a higher-level function.

Method 4: Symbolic Differentiation with SymPy

For applications requiring exact differentiation, SymPy provides symbolic computation capabilities. By constructing a symbolic representation of the Legendre series and differentiating it, one obtains expressions that can be evaluated to coefficient arrays. This is especially useful for theoretical analysis or when high precision is necessary.

Here’s an example:

from sympy import legendre, symbols, diff
import numpy as np

x = symbols('x')
coeffs = np.random.random(5)  # A 1D array of coefficients for a Legendre polynomial
poly = sum(c * legendre(i, x) for i, c in enumerate(coeffs))
dpoly_dx = diff(poly, x)
# Convert symbolic derivative to coefficient representation
dc_dx_coeffs = np.array([dpoly_dx.coeff(x, i) for i in range(len(coeffs)-1)], dtype=float)

Output: dc_dx_coeffs would be an array representing the exact derivative coefficients.

This sophisticated method leverages SymPy’s symbolic algebra system to differentiate a Legendre polynomial exactly and then extract the coefficients of the resulting derivative. While it provides high precision and symbolic differentiation, this method is more computationally intensive and is limited to one dimension.

Bonus One-Liner Method 5: NumPy’s Polynomial Derivative

NumPy’s polynomial.legendre.Legendre class handles Legendre polynomials and offers concise one-liner differentiation using the deriv() method. This is a quick and clear way to differentiate Legendre series directly.

Here’s an example:

from numpy.polynomial import Legendre

coeffs = np.random.random(5)
L = Legendre(coeffs)
dc_dx = L.deriv().coef

Output: dc_dx will give you the array of coefficient derivatives.

This one-liner code creates a Legendre instance and immediately calls the deriv() method to get the differentiated polynomial’s coefficients. It encapsulates the functionality into a simple, readable line of code, providing both convenience and clarity.

Summary/Discussion

  • Method 1: NumPy’s Gradient Function. The strength of this method is its simplicity for uniform grids. It is not suitable for non-uniform spacing or when exact derivatives are required.
  • Method 2: SciPy’s Special Function. Providing mathematically accurate differentiation for one-dimensional polynomials, it’s not directly applicable to arrays with more than one dimension without additional handling.
  • Method 3: Custom Differentiation Function using NumPy. Highly customizable and versatile for different needs. However, it may require additional work to handle edge cases and does not account for non-uniform spacing out-of-the-box.
  • Method 4: Symbolic Differentiation with SymPy. Offers exact coefficients and symbolic differentiation which is useful for theoretical purposes, yet it may be inefficient and complex for large-scale numerical computations.
  • Method 5: NumPy’s Polynomial Derivative. A concise and elegant solution for differentiating Legendre polynomials but limited to arrays that directly represent a polynomial.