5 Best Ways to Evaluate a 2D Hermite Series on the Cartesian Product of x and y with a 1D Array of Coefficients in Python

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💡 Problem Formulation: When dealing with Hermite polynomial series, we often want to compute the series expansion for a two-dimensional grid of points, using a one-dimensional array of coefficients. This task requires evaluating the product of the Hermite series along two separate dimensions, x and y, to achieve a two-dimensional series expansion. An example input would be x and y arrays representing the grid, and a 1D coefficients array for the Hermite series. The desired output is a 2D array with evaluated series values at all grid points.

Method 1: NumPy and Scipy

Evaluating Hermite series on a 2D grid using NumPy and Scipy libraries is efficient since these libraries are optimized for mathematical computations in Python. Consider using numpy.polynomial.hermite.hermgrid2d from Scipy, which efficiently computes the values of a 2-dimensional Hermite series at points in the Cartesian product of x and y.

Here’s an example:

import numpy as np
from numpy.polynomial.hermite import hermgrid2d

# 1D arrays for x and y
x = np.array([0, 1, 2])
y = np.array([-1, 0, 1])

# 1D array of coefficients for the Hermite series
coefficients = np.array([2, 3, 5])

# Evaluate the Hermite series on the Cartesian product of x and y
result = hermgrid2d(x, y, coefficients)

The output would be a 2D array with the evaluated Hermite series at each point on the grid.

This code first imports the necessary functions from NumPy. Then, it defines one-dimensional arrays for x and y axes and a coefficients array representing the Hermite series. The function hermgrid2d is used to evaluate the series across the cartesian product of x and y.

Method 2: Custom Implementation with NumPy

If you need more control or want to implement the series evaluation from scratch, using NumPy’s vectorization abilities is a good place to start. This method involves manually calculating the Hermite polynomials for each pair in the Cartesian product.

Here’s an example:

import numpy as np
from numpy.polynomial.hermite import hermeval

# Define the grid points for x and y
x = np.linspace(-1, 1, 3)
y = np.linspace(-1, 1, 3)

# 1D array of coefficients for the Hermite series
coeffs = np.array([1, 0, 2])

# Evaluate the series on the Cartesian grid
X, Y = np.meshgrid(x, y)
result = np.zeros_like(X)
for i in range(len(x)):
    for j in range(len(y)):
        result[i, j] = hermeval(X[i, j], coeffs) * hermeval(Y[i, j], coeffs)

The output will be the evaluated Hermite series at each point on the Cartesian product of x and y.

This snippet creates a grid for x and y using numpy.linspace and prepares an output matrix. For each point on the grid, we manually compute the Hermite series using the hermeval function and store the result. This approach provides a clear understanding of the underlying process but might not be as efficient as library functions.

Method 3: Utilizing SymPy for Symbolic Computation

SymPy, a Python library for symbolic mathematics, can compute with unlimited precision and provides functionality to handle Hermite polynomials symbolically. This can be particularly useful for understanding the series or for symbolic manipulation prior to numerical evaluation.

Here’s an example:

import numpy as np
from sympy import symbols, diff, hermite, lambdify

# Define the symbolic variables
x, y = symbols('x y')

# Symbolic Hermite polynomial coefficients
coeffs = [2, 0, -1]

# Create the symbolic Hermite polynomials
H_x = sum(c*hermite(n, x) for n, c in enumerate(coeffs))
H_y = sum(c*hermite(n, y) for n, c in enumerate(coeffs))

# Generate functions for numerical evaluation
h_x = lambdify(x, H_x, 'numpy')
h_y = lambdify(y, H_y, 'numpy')

# Evaluate on a Cartesian grid
grid_x = np.array([-1, 0, 1])
grid_y = np.array([-1, 0, 1])

result = np.array([[h_x(ix) * h_y(iy) for ix in grid_x] for iy in grid_y])

The output will be the evaluated Hermite series at each Cartesian grid point, produced using symbolic computation.

This snippet showcases using the SymPy library to handle Hermite polynomials symbolically. We define symbolic variables and coefficients, create the symbolic Hermite polynomials, and then turn them into functions that can be evaluated numerically. This method provides a deep insight into the computations but may be slower than numerical evaluations.

Method 4: Leveraging TensorFlow for Parallel Computation

TensorFlow provides tools for parallel computation, which might benefit high-performance scenarios. By using TensorFlow operations, we can evaluate Hermite series over a Cartesian grid with potential GPU acceleration.

Here’s an example:

import tensorflow as tf
import numpy as np
from tensorflow.math import add, multiply

# Define tensors for the x and y grid and coefficients
x = tf.constant([-1.0, 0.0, 1.0])
y = tf.constant([-1.0, 0.0, 1.0])
coeffs = tf.constant([1.0, 2.0, -1.0])

# Define a custom function for Hermite polynomial evaluation
def hermite_series(x, coeffs):
    val = tf.zeros_like(x)
    for i, coeff in enumerate(coeffs):
        val = add(val, multiply(coeff, tf.math.pow(x, i)))
    return val

# Evaluate the series on the Cartesian grid
result = tf.tensordot(hermite_series(x, coeffs), hermite_series(y, coeffs), axes=0)

The output will be a 2D Tensor representing the evaluated series across the Cartesian grid, leveraging TensorFlow’s parallel execution capabilities.

In this example, we use TensorFlow to handle mathematical operations. We define tensors for our grid and coefficients, and create a function to evaluate the Hermite series. We then use TensorFlow’s parallel computation abilities to apply this function across our Cartesian grid, allowing potential GPU acceleration.

Bonus One-Liner Method 5: Using NumPy’s Einstein Summation

For a condensed and potentially efficient one-liner, NumPy’s einsum function offers an elegant solution to evaluate a Hermite series on a 2D grid. This method relies on expressing the calculation as an Einstein summation, which NumPy optimizes under the hood.

Here’s an example:

import numpy as np

# Define x and y grid and coefficients
x = np.array([0, 1, 2])
y = np.array([-1, 0, 1])
coeffs = np.array([2, 3, 5])

# One-liner using einsum for Cartesian product evaluation
result = np.einsum('i,j,k->ijk', coeffs, x ** np.arange(len(coeffs))[:, None], y ** np.arange(len(coeffs))[:, None])

The output will be a 3D array that, once summed over the last axis, gives the 2D evaluated Hermite series.

Here we use numpy.einsum to efficiently perform computations through Einstein summation convention. The axes of the grids are aligned to the coefficients, and the computation of the 3D array is done succinctly. Summing over the last dimension gives the final result.

Summary/Discussion

  • Method 1: NumPy and Scipy. Using established libraries for mathematical operations. Strengths: Efficient and simple. Weaknesses: Less customizable.
  • Method 2: Custom Implementation with NumPy. Manual computation for flexibility and understanding. Strengths: Highly customizable. Weaknesses: Potentially slower than library methods.
  • Method 3: Utilizing SymPy for Symbolic Computation. Symbolic approach for precision and manipulation. Strengths: Exact results, useful for symbolic processing. Weaknesses: Slower than numerical methods.
  • Method 4: Leveraging TensorFlow for Parallel Computation. Parallelized computation using TensorFlow. Strengths: Fast with potential GPU acceleration. Weaknesses: Overhead for set-up and learning curve.
  • Bonus One-Liner Method 5: Using Einstein summation for concise code. Strengths: Elegant and possibly more efficient. Weaknesses: May be difficult to understand for those unfamiliar with Einstein summation.