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

To generate a Vandermonde matrix of given degree and sample points (x, y, z), use the polynomial.polyvander3d() in Python NumPy. The method returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). The parameter, x, y, z are the arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. The parameter, deg is the 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

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

Creating Sample Arrays

First, let's create arrays of point coordinates with the same shape ?

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

# Create arrays of point coordinates
x = np.array([1.5, 2.3])
y = np.array([3.7, 4.4])
z = np.array([5.3, 6.6])

print("x coordinates:", x)
print("y coordinates:", y)
print("z coordinates:", z)
print("\nArray shape:", x.shape)
print("Data type:", x.dtype)

x coordinates: [1.5 2.3]
y coordinates: [3.7 4.4]
z coordinates: [5.3 6.6]

Array shape: (2,)
Data type: float64

Generating the Vandermonde Matrix

Now generate the pseudo-Vandermonde matrix with specified degrees ?

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

x = np.array([1.5, 2.3])
y = np.array([3.7, 4.4])
z = np.array([5.3, 6.6])

# Set degrees for x, y, and z
x_deg, y_deg, z_deg = 2, 2, 2

# Generate the Vandermonde matrix
result = polyvander3d(x, y, z, [x_deg, y_deg, z_deg])
print("Vandermonde matrix shape:", result.shape)
print("\nVandermonde matrix:")
print(result)

Vandermonde matrix shape: (2, 27)

Vandermonde matrix:
[[1.0000000e+00 5.3000000e+00 2.8090000e+01 3.7000000e+00 1.9610000e+01
  1.0393300e+02 1.3690000e+01 7.2557000e+01 3.8455210e+02 1.5000000e+00
  7.9500000e+00 4.2135000e+01 5.5500000e+00 2.9415000e+01 1.5589950e+02
  2.0535000e+01 1.0883550e+02 5.7682815e+02 2.2500000e+00 1.1925000e+01
  6.3202500e+01 8.3250000e+00 4.4122500e+01 2.3384925e+02 3.0802500e+01
  1.6325325e+02 8.6524222e+02]
 [1.0000000e+00 6.6000000e+00 4.3560000e+01 4.4000000e+00 2.9040000e+01
  1.9166400e+02 1.9360000e+01 1.2777600e+02 8.4332160e+02 2.3000000e+00
  1.5180000e+01 1.0018800e+02 1.0120000e+01 6.6792000e+01 4.4082720e+02
  4.4528000e+01 2.9388480e+02 1.9396397e+03 5.2900000e+00 3.4914000e+01
  2.3043240e+02 2.3276000e+01 1.5362160e+02 1.0139026e+03 1.0241440e+02
  6.7593504e+02 4.4611713e+03]]

Understanding the Matrix Structure

The Vandermonde matrix has shape (n_points, n_terms) where n_terms = (x_deg+1) × (y_deg+1) × (z_deg+1) ?

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

x = np.array([1.0, 2.0])
y = np.array([1.0, 2.0]) 
z = np.array([1.0, 2.0])

# Simple case with degree 1 for each dimension
result = polyvander3d(x, y, z, [1, 1, 1])
print("For degrees [1,1,1]:")
print("Matrix shape:", result.shape)
print("Number of terms:", (1+1)*(1+1)*(1+1))
print("\nMatrix:")
print(result)

For degrees [1,1,1]:
Matrix shape: (2, 8)
Number of terms: 8

Matrix:
[[1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 2. 4. 2. 4. 8. 4. 8.]]

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 data point and columns representing polynomial terms of increasing complexity.