Generate a Pseudo-Vandermonde matrix of given degree and x, y, z sample points in Python

To generate a pseudo Vandermonde matrix of given degree and x, y, z sample points, use the polynomial.polyvander3d() function in NumPy. This method returns a pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). The parameters x, y, z are arrays of point coordinates with the same shape, and deg is a list of maximum degrees of the form [x_deg, y_deg, z_deg].

Syntax

numpy.polynomial.polynomial.polyvander3d(x, y, z, deg)

Parameters

x, y, z: Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any elements are complex. Scalars are converted to 1-D arrays.

deg: List of maximum degrees of the form [x_deg, y_deg, z_deg].

Example

Let's create arrays of point coordinates and generate a pseudo Vandermonde matrix ?

import numpy as np
from numpy.polynomial.polynomial import polyvander3d

# Create arrays of point coordinates with the same shape
x = np.array([1, 2])
y = np.array([3, 4])
z = np.array([5, 6])

print("Array x:", x)
print("Array y:", y)
print("Array z:", z)
print("Shape:", x.shape)

# Generate pseudo Vandermonde matrix with degrees [2, 3, 4]
x_deg, y_deg, z_deg = 2, 3, 4
result = polyvander3d(x, y, z, [x_deg, y_deg, z_deg])

print("\nPseudo Vandermonde Matrix shape:", result.shape)
print("\nFirst few columns of the matrix:")
print(result[:, :10])

Array x: [1 2]
Array y: [3 4]
Array z: [5 6]
Shape: (2,)

Pseudo Vandermonde Matrix shape: (2, 60)

First few columns of the matrix:
[[1.00000e+00 5.00000e+00 2.50000e+01 1.25000e+02 6.25000e+02 3.00000e+00
  1.50000e+01 7.50000e+01 3.75000e+02 1.87500e+03]
 [1.00000e+00 6.00000e+00 3.60000e+01 2.16000e+02 1.29600e+03 4.00000e+00
  2.40000e+01 1.44000e+02 8.64000e+02 5.18400e+03]]

Understanding the Output

The pseudo Vandermonde matrix has shape (2, 60) because:

• 2 sample points (length of input arrays)
• 60 columns = (x_deg + 1) × (y_deg + 1) × (z_deg + 1) = 3 × 4 × 5 = 60

Each row corresponds to a sample point, and each column represents a term in the polynomial expansion x^i × y^j × z^k where i ? x_deg, j ? y_deg, k ? z_deg.

Example with Different Degrees

Let's try with smaller degrees to see a more manageable output ?

import numpy as np
from numpy.polynomial.polynomial import polyvander3d

# Create sample points
x = np.array([1, 2])
y = np.array([3, 4]) 
z = np.array([5, 6])

# Generate matrix with degrees [1, 1, 1]
result = polyvander3d(x, y, z, [1, 1, 1])

print("Matrix with degrees [1, 1, 1]:")
print("Shape:", result.shape)
print("Matrix:")
print(result)

Matrix with degrees [1, 1, 1]:
Shape: (2, 8)
Matrix:
[[ 1.  5.  3. 15.  1.  5.  3. 15.]
 [ 1.  6.  4. 24.  2. 12.  8. 48.]]

Conclusion

The polyvander3d() function generates a pseudo Vandermonde matrix for 3D polynomial fitting. The matrix dimensions depend on the degree parameters, with each row representing a sample point and columns representing polynomial terms up to the specified degrees.