Generate a Pseudo Vandermonde matrix of the Laguerre polynomial and x, y, z floating array of points in Python

To generate a pseudo Vandermonde matrix of the Laguerre polynomial with x, y, z sample points, use the laguerre.lagvander3d() in Python NumPy. The parameters x, y, z are arrays of points with the same shape. The parameter deg is a list of maximum degrees of the form [x_deg, y_deg, z_deg].

Syntax

numpy.polynomial.laguerre.lagvander3d(x, y, z, deg)

Parameters

The function accepts the following parameters ?

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

Example

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

import numpy as np
from numpy.polynomial import laguerre as L

# 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])

# Display the arrays
print("Array x:", x)
print("Array y:", y)
print("Array z:", z)

# Display array properties
print("\nArray datatypes:")
print("x dtype:", x.dtype)
print("y dtype:", y.dtype)
print("z dtype:", z.dtype)

print("\nArray shapes:")
print("x shape:", x.shape)
print("y shape:", y.shape)
print("z shape:", z.shape)

# Generate pseudo Vandermonde matrix
x_deg, y_deg, z_deg = 2, 3, 4
result = L.lagvander3d(x, y, z, [x_deg, y_deg, z_deg])
print("\nPseudo Vandermonde matrix shape:", result.shape)
print("Result matrix:")
print(result)

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

Array datatypes:
x dtype: float64
y dtype: float64
z dtype: float64

Array shapes:
x shape: (2,)
y shape: (2,)
z shape: (2,)

Pseudo Vandermonde matrix shape: (2, 60)
Result matrix:
[[ 1.          -4.3          4.445        2.42216667 -2.30432917
  -2.7         11.61       -12.0015      -6.53985     6.22168875
   0.445       -1.9135       1.978025     1.07786417 -1.02542648
   1.99283333  -8.56918333   8.85814417   4.82697447 -4.59214397
  -0.5          2.15        -2.2225      -1.21108333  1.15216458
   1.35        -5.805        6.00075      3.269925   -3.11084437
  -0.2225       0.95675     -0.9890125   -0.53893208  0.51271324
  -0.99641667   4.28459167  -4.42907208  -2.41348724  2.29607199
  -0.875        3.7625      -3.889375    -2.11939583  2.01628802
   2.3625     -10.15875     10.5013125    5.72236875 -5.44397766
  -0.389375     1.6743125   -1.73077188  -0.94313115  0.89724817
  -1.74372917   7.49803542  -7.75087615  -4.22360266  4.01812598]
 [ 1.          -5.6          9.58        -1.376      -7.3226
  -3.4         19.04       -32.572        4.6784     24.89684
   1.88       -10.528       18.0104      -2.58688   -13.766488
   2.64266667 -14.79893333  25.31674667  -3.63630933 -19.35119093
  -1.3          7.28       -12.454        1.7888      9.51938
   4.42       -24.752       42.3436      -6.08192   -32.365892
  -2.444       13.6864     -23.41352      3.362944   17.8964344
  -3.43546667  19.23861333 -32.91177067   4.72720213  25.15654821
  -0.955        5.348       -9.1489       1.31408     6.993083
   3.247      -18.1832      31.10626     -4.467872  -23.7764822
  -1.7954      10.05424    -17.199932     2.4704704   13.14699604
  -2.52374667  14.13298133 -24.17749307   3.47267541  18.48038734]]

How It Works

The lagvander3d() function creates a 3D pseudo Vandermonde matrix where each row corresponds to a point (x[i], y[i], z[i]) and each column represents a combination of Laguerre polynomial basis functions. The matrix has shape (n_points, (x_deg+1)*(y_deg+1)*(z_deg+1)), where n_points is the number of coordinate points.

Conclusion

Use numpy.polynomial.laguerre.lagvander3d() to generate pseudo Vandermonde matrices for 3D Laguerre polynomial fitting. The function requires coordinate arrays of equal shape and degree specifications for each dimension.