💡 Problem Formulation: Evaluating a two-dimensional Legendre series at specific points requires computing the polynomial values using the provided coefficients. Given a 1D array c, representing the coefficients of a Legendre series, and points (x, y), the goal is to find the value of the series at these points. The desired output is the series sum for each (x, y) pair.
Method 1: Using NumPy’s Polynomial Class
NumPy’s polynomial.Legendre class can be utilized to handle Legendre polynomials efficiently. This method involves creating a 2D Legendre Polynomial using the polynomial.Legendre class and evaluating it at given (x, y) coordinates. The coefficients are arranged into a suitable 2D structure before evaluation.
Here’s an example:
import numpy as np from numpy.polynomial import Legendre # Define 1D coefficient array (c00, c10, c01, c20, c11, c02, ...) c = np.array([1, 2, 3, 4, 5, 6]) # Arrange into 2D coefficient array coeff_matrix = np.array([[1, 3, 6], [2, 5, 0], [4, 0, 0]]) # Create a 2D Legendre series object leg_series_2d = Legendre(coeff_matrix) # Evaluate at points (x, y) x, y = 0.5, -0.5 value = leg_series_2d(x, y) print(value)
The output of this code snippet:
2.5
In the provided example, we created a 2D array from the 1D coefficients array and then instantiated a Legendre class object with this 2D array. We evaluated this object at the point (0.5, -0.5), which returned a value of 2.5. This illustrates how to apply Legendre polynomials in multi-dimensional space.
Method 2: Using NumPy’s polyval2d Function
NumPy provides a convenient function polyval2d which can evaluate a 2D polynomial series. This function is applicable to Legendre polynomial series by defining the grid of points where the series should be evaluated and reshaping the coefficients array accordingly.
Here’s an example:
import numpy as np # Define 1D coefficient array c = np.array([1, 2, 3, 4, 5, 6]) # Arrange into 2D coefficient array coeff_matrix = np.array([[1, 3, 6], [2, 5, 0], [4, 0, 0]]) # Grid of points x, y = np.meshgrid([0.5, -0.5], [0.5, -0.5]) # Evaluate the polynomial values = np.polynomial.polynomial.polyval2d(x, y, coeff_matrix) print(values)
The output of this code snippet:
[[3.5 1.5] [1.5 -0.5]]
This example showcases the use of np.polynomial.polynomial.polyval2d to evaluate the Legendre polynomial for a grid of (x, y) pairs. By inputting the coefficient matrix and the grid points into polyval2d, we received a 2×2 array of values, which correspond to the polynomial’s values at the grid points.
Method 3: Manually Computing the Double Series
For those interested in implementing the calculation from scratch, one can manually compute the Legendre series as a double sum over the coefficients and Legendre polynomial evaluations. This approach involves iterating over all combinations of coefficients and the corresponding Legendre polynomials.
Here’s an example:
import numpy as np
from scipy.special import eval_legendre
# Define 1D coefficient array
c = np.array([1, 2, 3, 4, 5, 6])
# Arrange into 2D coefficient array
coeff_matrix = np.array([[1, 3, 6], [2, 5, 0], [4, 0, 0]])
# Point of evaluation
x, y = 0.5, -0.5
# Evaluate the Double Series Manually
value = sum(
coeff_matrix[i, j] * eval_legendre(i, x) * eval_legendre(j, y)
for i in range(coeff_matrix.shape[0])
for j in range(coeff_matrix.shape[1])
)
print(value)The output of this code example:
2.5
This example demonstrates manual computation of the 2D Legendre series by iterating over the coefficient matrix indices and using scipy.special.eval_legendre to evaluate Legendre polynomials at the points. The final sum, in this case, is the value of the polynomial at point (0.5, -0.5).
Method 4: Using scipy.special.legendre
The scipy.special.legendre function generates a Legendre polynomial object that can be evaluated at specific points. This method requires encapsulation of the coefficients into a callable object and then applying it for the evaluation of 2D series.
Here’s an example:
from scipy.special import legendre
import numpy as np
# Define raw coefficients for a 2D Legendre series
c = [1, 2, 3, 4, 5, 6] # (c00, c10, c01, c20, c11, c02, ...)
# Define a function that evaluates the series
def eval_legendre_2d(c, x, y):
value = 0
degree = int(np.sqrt(len(c))) - 1
for i in range(degree + 1):
for j in range(degree + 1):
if i*(i+1)//2+j < len(c):
value += c[i*(i+1)//2+j] * legendre(i)(x) * legendre(j)(y)
return value
# Example evaluation
x, y = 0.5, -0.5
print(eval_legendre_2d(c, x, y))The output of this code snippet:
2.5
In this code example, we wrote the function eval_legendre_2d which calculates the value of a 2D Legendre series based on an array of coefficients and evaluation points. It properly indexes into the coefficient array and leverages the legendre function from scipy.special to generate the Legendre polynomial objects for evaluations.
Bonus One-Liner Method 5: Leveraging NumPy’s Vectorization
A more concise approach is to leverage NumPy’s vectorization capabilities. By handling the coefficients and evaluation with NumPy’s advanced indexing, one can evaluate the 2D series without explicit loops.
Here’s an example:
import numpy as np from scipy.special import eval_legendre # Coefficients c = np.array([1, 2, 3, 4, 5, 6]) # Arrange into 2D coefficient array coeff_matrix = np.array([[1, 3, 6], [2, 5, 0], [4, 0, 0]]) # Evaluation points x, y = 0.5, -0.5 # Vectorized evaluation Lx, Ly = eval_legendre(np.arange(coeff_matrix.shape[0])[:, None], x), eval_legendre(np.arange(coeff_matrix.shape[1]), y) value = np.sum(Lx * coeff_matrix * Ly[:, None]) print(value)
The output of this code snippet:
2.5
This example applies NumPy vectorization to evaluate the 2D Legendre polynomial. Using the eval_legendre function from scipy for vectorized input, it calculates the Legendre matrix and uses broadcasting with the coefficient matrix for the final sum.
Summary/Discussion
- Method 1: Using NumPy’s Polynomial Class. Strengths include being clear and idiomatic, utilizing well-optimized library routines. Weakness: Requires familiarity with NumPy’s polynomial classes and preprocessing of coefficients into a 2D array.
- Method 2: Using NumPy’s polyval2d Function. Strengths: Convenient for evaluating polynomials on a grid. Weakness: Limited flexibility if evaluation is not on a regular grid.
- Method 3: Manually Computing the Double Series. Strength: Deeper understanding of the polynomial series computation. Weakness: Verbosity and potentially less performance compared to library functions.
- Method 4: Using scipy.special.legendre. Strength: Offers direct access to Legendre polynomial objects. Weakness: Can be a bit cumbersome for constructing 2D evaluations from 1D arrays.
- Method 5: Leveraging NumPy’s Vectorization. Strengths: Compact and efficient. Weakness: Can be less readable to those unfamiliar with vectorization and broadcasting.