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

To generate a pseudo Vandermonde matrix of the Hermite_e polynomial, use the hermite_e.hermevander2d() function in NumPy. This method returns a pseudo-Vandermonde matrix where each row corresponds to a point coordinate and each column represents polynomial basis functions up to specified degrees.

Syntax

numpy.polynomial.hermite_e.hermevander2d(x, y, deg)

Parameters

The function accepts the following parameters:

• x, y: Arrays of point coordinates with the same shape. Data types are converted to float64 or complex128 automatically.
• deg: List specifying maximum degrees as [x_deg, y_deg].

Complete Example

Let's create a complete example demonstrating the generation of a pseudo Vandermonde matrix ?

import numpy as np
from numpy.polynomial import hermite_e 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("Array x:", x)
print("Array y:", y)

# Display array properties
print("\nArray x datatype:", x.dtype)
print("Array y datatype:", y.dtype)
print("Array x dimensions:", x.ndim)
print("Array y dimensions:", y.ndim)
print("Array x shape:", x.shape)
print("Array y shape:", y.shape)

# Generate pseudo Vandermonde matrix
x_deg, y_deg = 2, 3
result = H.hermevander2d(x, y, [x_deg, y_deg])
print("\nPseudo Vandermonde matrix:")
print(result)

Array x: [0.1 1.4]
Array y: [1.7 2.8]

Array x datatype: float64
Array y datatype: float64
Array x dimensions: 1
Array y dimensions: 1
Array x shape: (2,)
Array y shape: (2,)

Pseudo Vandermonde matrix:
[[ 1.000000e+00  1.700000e+00  1.890000e+00 -1.870000e-01  1.000000e-01
   1.700000e-01  1.890000e-01 -1.870000e-02 -9.900000e-01 -1.683000e+00
  -1.871100e+00  1.851300e-01]
 [ 1.000000e+00  2.800000e+00  6.840000e+00  1.355200e+01  1.400000e+00
   3.920000e+00  9.576000e+00  1.897280e+01  9.600000e-01  2.688000e+00
   6.566400e+00  1.300992e+01]]

Understanding the Output

The resulting matrix has dimensions (2, 12). Each row corresponds to a coordinate pair (x[i], y[i]), and the 12 columns represent different polynomial basis functions formed by combinations of Hermite_e polynomials up to degrees (2, 3).

Matrix Dimensions

The number of columns in the pseudo Vandermonde matrix is calculated as (x_deg + 1) × (y_deg + 1) = (2 + 1) × (3 + 1) = 12 columns.

Conclusion

The hermite_e.hermevander2d() function efficiently generates pseudo Vandermonde matrices for 2D Hermite_e polynomials. This is useful for polynomial fitting and interpolation problems in two dimensions.