Generate a Pseudo Vandermonde matrix of the Hermite polynomial with float array of points coordinates in Python

To generate a pseudo Vandermonde matrix of the Hermite polynomial, use the hermite.hermvander2d() method in Python NumPy. This method creates a 2D Vandermonde matrix where each row corresponds to a point coordinate and columns represent polynomial basis functions of varying degrees.

Syntax

numpy.polynomial.hermite.hermvander2d(x, y, deg)

Parameters

• x, y − Arrays of point coordinates with the same shape
• deg − List of maximum degrees [x_deg, y_deg]

Example

Let's create a pseudo Vandermonde matrix using float coordinate arrays ?

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

# Create arrays of point coordinates
x = np.array([0.1, 1.4])
y = np.array([1.7, 2.8])

# Display the arrays
print("Array1...")
print(x)
print("\nArray2...")
print(y)

# Display the datatype
print("\nArray1 datatype:", x.dtype)
print("Array2 datatype:", y.dtype)

# Check the dimensions and shape
print("\nDimensions of Array1:", x.ndim)
print("Dimensions of Array2:", y.ndim)
print("\nShape of Array1:", x.shape)
print("Shape of Array2:", y.shape)

Array1...
[0.1 1.4]

Array2...
[1.7 2.8]

Array1 datatype: float64
Array2 datatype: float64

Dimensions of Array1: 1
Dimensions of Array2: 1

Shape of Array1: (2,)
Shape of Array2: (2,)

Generate Pseudo Vandermonde Matrix

Now generate the matrix with degree [2, 3] for x and y respectively ?

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

x = np.array([0.1, 1.4])
y = np.array([1.7, 2.8])

# Set degrees for x and y
x_deg, y_deg = 2, 3

# Generate pseudo Vandermonde matrix
result = H.hermvander2d(x, y, [x_deg, y_deg])
print("Pseudo Vandermonde Matrix:")
print(result)
print("\nMatrix shape:", result.shape)

Pseudo Vandermonde Matrix:
[[ 1.0000000e+00  3.4000000e+00  9.5600000e+00  1.8904000e+01
   2.0000000e-01  6.8000000e-01  1.9120000e+00  3.7808000e+00
  -1.9600000e+00 -6.6640000e+00 -1.8737600e+01 -3.7051840e+01]
 [ 1.0000000e+00  5.6000000e+00  2.9360000e+01  1.4201600e+02
   2.8000000e+00  1.5680000e+01  8.2208000e+01  3.9764480e+02
   5.8400000e+00  3.2704000e+01  1.7146240e+02  8.2937344e+02]]

Matrix shape: (2, 12)

How It Works

The matrix has dimensions (N, (x_deg+1)×(y_deg+1)) where N is the number of coordinate points. Each element represents the evaluation of Hermite polynomial basis functions at the given coordinates. The columns correspond to different polynomial terms of the form H_i(x) × H_j(y) where i ranges from 0 to x_deg and j ranges from 0 to y_deg.

Conclusion

The hermvander2d() function efficiently generates pseudo Vandermonde matrices for 2D Hermite polynomials. This is useful for polynomial fitting and interpolation in scientific computing applications.