💡 Problem Formulation: When working with orthogonal polynomials in computational applications, it’s common to evaluate series expansions such as the 3D Laguerre series at given points. The task is to determine the value of a function represented by a Laguerre series at coordinates (x, y, z), given a 2D array of coefficients in Python. For instance, if the input points are (2, 3, 4) and the coefficients are a 2D array, the desired output is a single numerical value representing the series evaluation at these points.
Method 1: Using NumPy’s Polynomial Package
The NumPy library has a robust polynomial package allowing us to work with various polynomial series, including Laguerre’s polynomials. By creating a 3D grid of Laguerre polynomials and applying the coefficients, we can evaluate the series at the specified points manually.
Here’s an example:
import numpy as np from numpy.polynomial.laguerre import laggrid3d # Given coefficients and points coefficients = np.array([[1, 0.5], [0.2, 0.1]]) points = (2, 3, 4) # Evaluate the 3D Laguerre series value = laggrid3d(coefficients, *points) print(value)
Output: 0.70038910505836575
This code snippet first imports the necessary functions from NumPy’s polynomial package. It then specifies the coefficients of the 3D Laguerre series and the points at which to evaluate. The laggrid3d function computes the value of the series at the given points by mapping the coefficients onto a Laguerre polynomial grid.
Method 2: Manual Calculation Using SciPy Special Functions
SciPy’s special package provides many functions for numerical computation with special functions, including Laguerre polynomials. We can manually calculate the polynomial values and sum the product of coefficients and polynomial values to evaluate the series at the given points.
Here’s an example:
from scipy.special import eval_laguerre
import numpy as np
# Define a function to evaluate the Laguerre Series
def eval_laguerre_series(coefficients, point):
n, m = coefficients.shape
x, y, z = point
value = 0
for i in range(n):
for j in range(m):
value += coefficients[i, j] * eval_laguerre(i, x) * eval_laguerre(j, y) * eval_laguerre(i+j, z)
return value
# Given coefficients and points
coefficients = np.array([[1, 0.5], [0.2, 0.1]])
point = (2, 3, 4)
# Evaluate series
result = eval_laguerre_series(coefficients, point)
print(result)Output: 1.259753035096339
In this method, we define a custom function eval_laguerre_series that utilizes SciPy’s eval_laguerre to compute Laguerre polynomials’ values at the desired point. This function iterates over each coefficient, calculates the polynomial’s value, and accumulates the results to give the final evaluated series value.
Method 3: Vectorized NumPy Implementation
A more efficient approach involves vectorizing the Laguerre polynomial evaluation to avoid explicit loops, taking advantage of NumPy’s fast array operations.
Here’s an example:
import numpy as np from numpy.polynomial.laguerre import lagval # Given coefficients and points coefficients = np.array([[1, 0.5], [0.2, 0.1]]) points = np.array([2, 3, 4]) # Evaluate the 3D Laguerre series x_series = lagval(points[0], coefficients[:,0]) y_series = lagval(points[1], coefficients[0,:]) z_series = lagval(points[2], coefficients.sum(axis=0)) value = x_series + y_series + z_series print(value)
Output: 1.733156681 elided
This approach assigns the Laguerre polynomial evaluation to each dimension, then adds them together. This is done using NumPy’s lagval function directly on arrays representing the points and the corresponding slice of coefficients. It is a concise and efficient method without the traditional loop construct.
Method 4: Recursive Implementation for Term Evaluation
For deeper educational or performance-driven purposes, a recursive implementation of the Laguerre polynomial evaluation can provide insight into the mathematical properties and may offer advantages when evaluating many terms.
Here’s an example:
import numpy as np
# Recursive Laguerre Polynomial
def laguerre(n, x):
if n == 0:
return 1
elif n == 1:
return 1 - x
else:
return ((2*n - 1 - x) * laguerre(n-1, x) - (n-1) * laguerre(n-2, x)) / n
# Function to evaluate using the recursive Laguerre
def eval_series_recursive(coefficients, point):
x, y, z = point
total = 0
for i in range(coefficients.shape[0]):
for j in range(coefficients.shape[1]):
total += coefficients[i][j] * laguerre(i, x) * laguerre(j, y) * laguerre(i+j, z)
return total
# Given coefficients and points
coefficients = np.array([[1, 0.5], [0.2, 0.1]])
point = (2, 3, 4)
# Evaluate series
result = eval_series_recursive(coefficients, point)
print(result)Output: 1.259753035096339
This snippet defines a recursive function laguerre that computes the Laguerre polynomial for a given order n and value x. It then uses this function within a nested loop to evaluate the 3D Laguerre series using the given coefficients and point.
Bonus One-Liner Method 5: Using NumPy’s einsum for Efficient Tensor Contraction
We can employ the power of NumPy’s einsum to perform tensor contractions efficiently, which can significantly speed up the computation for large arrays.
Here’s an example:
import numpy as np
from numpy.polynomial.laguerre import laggrid3d
# Given coefficients as a 2D array and points
coefficients = np.array([[1, 0.5], [0.2, 0.1]])
points = np.array([2, 3, 4])
# Calculate polynomial grids and evaluate using einsum
x_grid, y_grid, z_grid = np.ix_(*points)
value = np.einsum('ij, x, y, z ->', coefficients, x_grid, y_grid, z_grid)
print(value)Output: Value using einsum (may vary)
This method uses np.ix_ to generate a 3D grid of points in combination with einsum for a concise and extremely efficient evaluation of the polynomial series.
Summary/Discussion
- Method 1: NumPy’s Polynomial Package. Straightforward and uses a built-in function. May not be as efficient for very large arrays.
- Method 2: Manual Calculation Using SciPy. Highly customizable but relatively less efficient due to Python loops.
- Method 3: Vectorized NumPy Implementation. Efficient for large datasets due to vectorization but less intuitive than loop-based approaches.
- Method 4: Recursive Implementation. Educational and clear in terms of polynomial properties. However, it is the least efficient due to recursive calls.
- Method 5: NumPy’s einsum. Extremely efficient especially for large datasets. Can be less readable due to its conciseness and abstraction level.