Generate a Pseudo-Vandermonde matrix of given degree with complex array of points coordinates in Python

To generate a Pseudo-Vandermonde matrix of given degree with complex coordinates, use the polyvander2d() function from NumPy's polynomial module. This function creates a 2D Vandermonde matrix from arrays of point coordinates with specified maximum degrees for each dimension.

Syntax

numpy.polynomial.polynomial.polyvander2d(x, y, deg)

Parameters

x, y: Arrays of point coordinates with the same shape. Complex values are supported.
deg: List of maximum degrees in the form [x_deg, y_deg].

Example

Let's create a Pseudo-Vandermonde matrix using complex coordinate arrays ?

import numpy as np
from numpy.polynomial.polynomial import polyvander2d

# Create arrays of complex point coordinates
x = np.array([-2.+2.j, -1.+2.j])
y = np.array([1.+2.j, 2.+2.j])

print("X coordinates:", x)
print("Y coordinates:", y)
print("Data type:", x.dtype)
print("Shape:", x.shape)

# Generate Pseudo-Vandermonde matrix with degrees [2, 3]
x_deg, y_deg = 2, 3
result = polyvander2d(x, y, [x_deg, y_deg])

print("\nPseudo-Vandermonde matrix:")
print(result)

X coordinates: [-2.+2.j -1.+2.j]
Y coordinates: [1.+2.j 2.+2.j]
Data type: complex128
Shape: (2,)

Pseudo-Vandermonde matrix:
[[ 1. +0.j  1. +2.j -3. +4.j -11. -2.j  -2. +2.j  -6. -2.j  -2.-14.j
  26.-18.j   0. -8.j  16. -8.j  32.+24.j -16.+88.j]
 [ 1. +0.j  2. +2.j  0. +8.j -16.+16.j  -1. +2.j  -6. +2.j -16. -8.j
 -16.-48.j  -3. -4.j   2.-14.j  32.-24.j 112.+16.j]]

How It Works

The matrix contains all polynomial combinations up to the specified degrees:

• Each row corresponds to a point (x[i], y[i])
• Columns represent terms like x^i * y^j where i ? x_deg and j ? y_deg
• For degrees [2,3], we get (2+1) × (3+1) = 12 columns
• Complex arithmetic is handled automatically

Matrix Structure

For degrees [2,3], the columns represent these polynomial terms:

# Columns: 1, y, y², y³, x, xy, xy², xy³, x², x²y, x²y², x²y³

Conclusion

The polyvander2d() function efficiently generates Pseudo-Vandermonde matrices for complex coordinates. This is useful for polynomial fitting and interpolation in 2D with complex data points.