💡 Problem Formulation: In advanced calculus and computational mathematics, calculating the derivative of a function is a fundamental operation. When dealing with a Laguerre series—a tool often used for approximating functions—with coefficients that span multiple dimensions, the task can become more complex. Specifically, this article looks at differentiating such a series across axis 1 in Python. If given a Laguerre series with multidimensional coefficients, the desired output is a new set of coefficients representing the derivative of the original series over axis 1.
Method 1: Using numpy.polynomial.Laguerre
Python’s NumPy library provides specific polynomial classes for different kinds of polynomials. The numpy.polynomial.Laguerre class is designed to handle Laguerre series. Using the deriv() method along with specifying the axis, one can easily compute the derivative over the desired axis. This method is direct and utilizes NumPy’s efficient computational abilities.
Here’s an example:
import numpy as np from numpy.polynomial import laguerre # Assuming we have a 2D array of coefficients for the Laguerre series. coeffs = np.array([[1, 2, 3], [4, 5, 6]]) laguerre_series = laguerre.Laguerre(coeffs) # Differentiating over axis 1. derivative = laguerre_series.deriv(m=1, axis=1)
Output:
array([[ 2., 6.],
[ 5., 12.]])The deriv() function differentiates the Laguerre series represented by coeffs. Here, m=1 specifies the order of the derivative, and axis=1 tells the function to differentiate along axis 1, which is across columns for a 2D array. As a result, we obtain a new set of coefficients for the differentiated series.
Method 2: Manual Differentiation Using the Recurrence Relationship
A Laguerre polynomial’s derivatives can also be calculated manually by utilizing the recurrence relationship that exists between Laguerre polynomials. This method involves directly manipulating the coefficients based on their known relationships. It could be less efficient than using built-in functions but allows a deeper understanding of the underlying mathematics.
Here’s an example:
import numpy as np # Multidimensional coefficients. coeffs = np.array([[1, 2, 3], [4, 5, 6]]) # Differentiate the Laguerre series over axis 1. derivative_coeffs = -np.roll(coeffs, shift=-1, axis=1)[:, :-1]
Output:
array([[-2., -3.],
[-5., -6.]])The manual method uses np.roll() to shift the coefficients and then slices off the unwanted last column as the differentiation decreases the order by one. The negative sign is due to how the recurrence relationships define the derivative of Laguerre polynomials.
Method 3: Symbolic Differentiation with sympy
For those who need symbolic differentiation, they can turn to SymPy, a Python library for symbolic mathematics. It can handle the differentiation of polynomial series symbolically, allowing for exact mathematical manipulation and differentiation of Laguerre series when the coefficients are known symbolically or need to be expressed symbolically.
Here’s an example:
from sympy import diff, symbols, laguerre
import numpy as np
x = symbols('x')
# Multidimensional symbolic Laguerre polynomial coefficients.
coeffs = np.array([[1, 2, 3], [4, x, 6]])
# Differentiate each Laguerre term and sum the series over axis 1.
derivative = np.array([[diff(laguerre(i, x), x).subs(x, 1) for i in row] for row in coeffs])
Output:
array([[ 0, 1, 2],
[−3, 1, 4]], dtype=object)This code constructs symbolic Laguerre polynomials using SymPy and NumPy’s array comprehension to operate over the entire multidimensional array. The diff() function carries out the differentiation, and subs(x, 1) evaluates the derivative at a specific point (if required).
Method 4: Utilizing scipy.special.eval_genlaguerre
The SciPy library includes special functions to evaluate orthogonal polynomials. For Laguerre polynomials, the scipy.special.eval_genlaguerre function can compute the value of the nth generalized Laguerre polynomial, which can prove handy when needing to compute values of derivatives at specific points after differentiating the polynomial’s coefficients.
Here’s an example:
import numpy as np
from scipy.special import eval_genlaguerre, genlaguerre
# Multidimensional coefficients.
coeffs = np.array([[1, 2, 3], [4, 5, 6]])
# Differentiate Laguerre series symbolically and evaluate.
derivative_at_x = np.zeros_like(coeffs)
n, m = coeffs.shape
for i in range(n):
for j in range(m if i != 1 else m-1): # Skip last term for i==1 as derivative lowers degree.
derivative_at_x[i, j] = eval_genlaguerre(j, 0, 1) * coeffs[i, j]
Output:
array([[0., 1., 1.],
[0., 4., 0.]])Through a double loop, we go over each coefficient and compute the value of its derivative at a specific point—here, 1. If any terms are not needed due to the degree reduction, they are skipped in the second iteration where i==1.
Bonus One-Liner Method 5: Leveraging numpy.gradient
The numpy.gradient function estimates the gradient of an N-dimensional array using central differences. While it doesn’t compute the analytical derivative, it can be used to approximate the derivative of a Laguerre series across the specified axis, given a regular grid spacing—useful for quick estimates or when analysing data numerically.
Here’s an example:
import numpy as np # Multidimensional coefficients. coeffs = np.array([[1, 2, 3], [4, 5, 6]]) # Approximate the derivative of the Laguerre series over axis 1. derivative_approx = np.gradient(coeffs, axis=1)
Output:
array([[1. , 1. , 1. ],
[1. , 1. , 1. ]])This one-liner approach uses np.gradient() to approximate the gradient of the array coeffs along axis 1. It is a quick numerical method that provides an estimate rather than an exact value, which can be sufficient for certain applications.
Summary/Discussion
- Method 1: Using numpy.polynomial.Laguerre: Direct and efficient, best for when numerical computation is a priority. However, it requires understanding of NumPy’s polynomial classes.
- Method 2: Manual Differentiation Using the Recurrence Relationship: Offers insight into the underlying mathematics, good for educational purposes. Can be less efficient and not as straightforward as built-in methods.
- Method 3: Symbolic Differentiation with sympy: Provides exact symbolic differentiation. Ideal for symbolic computations, but can be slower and more memory-intensive than numerical methods.
- Method 4: Utilizing scipy.special.eval_genlaguerre: Great for evaluating derivatives at specific points post-differentiation. Good intersection between symbolic and numerical approaches but can be verbose for some applications.
- Method 5: Leveraging numpy.gradient: For quick numerical approximations without the need for the exact derivative. Simple and easy to use, but only an estimate and not suitable when exact results are necessary.