Understanding NumPy's `polynomial.chebyshev.chebvander2d()` for 2D Chebyshev Series

What it Does

This function generates a 2D pseudo-Vandermonde matrix for two-dimensional Chebyshev series. In simpler terms, it creates a matrix that captures the basis functions for fitting a polynomial function using Chebyshev polynomials in two dimensions (x and y).

Breakdown of the Function

• polynomial.chebyshev.chebvander2d(x, y, deg)
  
  • x, y: These are arrays of the same shape representing the points in the 2D space where you want to evaluate the Chebyshev series.
  • deg: This is a tuple of two integers, specifying the degree of the Chebyshev polynomials in each dimension (x and y). The first element represents the degree in x, and the second element represents the degree in y.

The Output Matrix

• (deg[0] + 1, deg[1] + 1): This part represents the number of basis functions in each dimension. There are deg[0] + 1 basis functions for the x-dimension (including constant term) and deg[1] + 1 basis functions for the y-dimension.
• x.shape: This reflects the original shape of the input arrays x and y.

The Magic Behind the Scenes

Each element of the output matrix represents a specific term in the 2D Chebyshev series expansion. The elements are constructed using Chebyshev polynomials of varying degrees in both x and y.

Relationship to chebval2d

• The product of chebvander2d(x, y, deg) and the coefficient vector c (flattened version of a multi-dimensional array) is equivalent (up to rounding errors) to the result of chebval2d(x, y, c). This relationship is helpful for least-squares fitting and evaluating multiple 2D Chebyshev series efficiently.

---

import numpy as np

# Define sample points and degree
x = np.linspace(-1, 1, 5)  # 5 points from -1 to 1
y = np.linspace(-1, 1, 4)  # 4 points from -1 to 1
deg = (2, 3)  # Degree (2 for x, 3 for y)

# Generate the 2D Chebyshev Vandermonde matrix
vander2d = np.polynomial.chebyshev.chebvander2d(x, y, deg)

# Print the shape of the matrix
print(vander2d.shape)

# Access a specific element (basis function)
# Example: element at index (2, 1, 0) represents T_0(x) * T_1(y)
element = vander2d[2, 1, 0]
print(element)

1. We import numpy as np for numerical operations.
2. We define sample points for x and y using np.linspace.
3. We define the degree tuple deg specifying the degree of the Chebyshev polynomial in each dimension.
4. We call chebvander2d with the sample points and degree to generate the 2D Vandermonde matrix and store it in vander2d.
5. We print the shape of vander2d to understand its dimensionality.
6. We access a specific element in the matrix using indexing. The first two indices represent the position in the original x and y arrays, and the last index represents the term's degree in the x-dimension (0 in this case). This element corresponds to the basis function T_0(x) * T_1(y).

---

Custom Function with Vectorization

import numpy as np

def cheb_vandermonde2d_alt1(x, y, deg):
  """
  Generates a 2D Chebyshev Vandermonde matrix using chebval2d and vectorization.

  Args:
      x (numpy.ndarray): Array of points in x-dimension.
      y (numpy.ndarray): Array of points in y-dimension.
      deg (tuple): Degree tuple (deg_x, deg_y).

  Returns:
      numpy.ndarray: The 2D Chebyshev Vandermonde matrix.
  """
  grid = np.meshgrid(x, y)  # Create a grid for x and y points
  vander2d = np.empty(grid[0].shape + (deg[0] + 1, deg[1] + 1))

  # Use chebval2d in a vectorized way to compute basis functions
  for i in range(deg[0] + 1):
    for j in range(deg[1] + 1):
      vander2d[..., i, j] = np.polynomial.chebyshev.chebval2d(grid[0], grid[1], [i, j])

  return vander2d

• Finally, it returns the constructed Vandermonde matrix.
• The core part utilizes np.polynomial.chebyshev.chebval2d in a loop to compute basis functions at each grid point in a vectorized manner. This improves efficiency compared to non-vectorized operations.
• It then creates an empty array with the desired shape for the Vandermonde matrix.
• This function creates a meshgrid using np.meshgrid to combine x and y points.

Leverage polyval2d with Manual Basis Function Construction

1. Define functions for Chebyshev polynomials of different degrees in x and y.
2. Create a loop to iterate through desired degrees in x and y.
3. Within the loop, generate the basis function (e.g., x^i * y^j) for the current degrees.
4. Use np.polynomial.chebyshev.polyval2d to evaluate the basis function at the grid points (obtained from np.meshgrid).
5. Store the evaluated results in the corresponding element of the Vandermonde matrix.

• The manual basis function approach might be useful for gaining a deeper understanding of the construction process.
• The custom function with vectorization is generally more efficient.
• Both approaches achieve the goal of creating a 2D Chebyshev Vandermonde matrix.