Generate a Pseudo Vandermonde matrix of the Chebyshev and Legendre polynomial in Python

In this article, we will generate a Pseudo Vandermonde matrix of the Chebyshev and Legendre polynomial x, y, and z floating array of points in Python.

Example 1

Generating a Pseudo Vandermonde matrix of Legendre..legvander3d() function

We use the legendre.legvander3d() method present in the Numpy module of python to produce a Vandermonde matrix of the Legendre polynomial with x, y, and z as the sample points. The pseudo-Vandermonde matrices of degrees ‘deg’ and sample points  (x, y, z) is returned as a result. The x, y, and z parameters are all arrays of point coordinates with a similar shape. Depending on whether any of the elements are complex, the dtypes will be changed to float64 or complex128. If any scalar is present then it will be converted to a one-dimensional array. ‘deg’ parameter is a list of maximum degrees mentioned in the form of  [x deg, y deg, z deg].

Syntax : legendre.legvander3d(x, y, z, deg)

Parameter:

• x,y,z = These are the arrays of the same-shaped point coordinates.
• deg = It is the list of maximum degrees of the form [x_deg, y_deg, z_deg].

Return : It returns a Vandermonde matrix.

Python3

# import numpy and legendre libraries
import numpy as np
from numpy.polynomial import legendre

# Create arrays of point coordinates x, y and z
x = np.array([2.4, 1.2])
y = np.array([7.1, 2.7])
z = np.array([3.4, 2.2])

# Assign degrees to x, y and z
deg_of_x = 3
deg_of_y = 1
deg_of_z = 2

# use chebvander3d() function to generate a
# pseudo Vandermonde matrix of the
# Chebyshev polynomial and x, y, z sample points
print(legendre.legvander3d(x, y, z, [deg_of_x, deg_of_y, deg_of_z]))

Output :

[[1.00000000e+00 3.40000000e+00 1.68400000e+01 7.10000000e+00

  2.41400000e+01 1.19564000e+02 2.40000000e+00 8.16000000e+00

  4.04160000e+01 1.70400000e+01 5.79360000e+01 2.86953600e+02

  8.14000000e+00 2.76760000e+01 1.37077600e+02 5.77940000e+01

  1.96499600e+02 9.73250960e+02 3.09600000e+01 1.05264000e+02

  5.21366400e+02 2.19816000e+02 7.47374400e+02 3.70170144e+03]

 [1.00000000e+00 2.20000000e+00 6.76000000e+00 2.70000000e+00

  5.94000000e+00 1.82520000e+01 1.20000000e+00 2.64000000e+00

  8.11200000e+00 3.24000000e+00 7.12800000e+00 2.19024000e+01

  1.66000000e+00 3.65200000e+00 1.12216000e+01 4.48200000e+00

  9.86040000e+00 3.02983200e+01 2.52000000e+00 5.54400000e+00

  1.70352000e+01 6.80400000e+00 1.49688000e+01 4.59950400e+01]]

Example 2: 

Generating a Pseudo Vandermonde matrix of the Chebyshev.chebvander() function

We use the chebyshev.chebvander() function present in the Numpy module of python to construct a pseudo Vandermonde matrix of the Chebyshev polynomial and x, y, and z as the sample points. The pseudo-Vandermonde matrix of degrees ‘deg’ and sample points  (x, y, z) is returned by this function.

The x, y, and z parameters are all arrays of point coordinates with the same shape. Depending on whether any of the elements are complex, the dtypes will be changed to float64 or complex128. If any scalar is present then it will be converted to a one-dimensional array. Also, the ‘deg’ parameter is a list of maximum degrees with the format [x deg, y deg, z deg].

Syntax : chebyshev.chebvander(x, y, z, [deg_of_x, deg_of_y, deg_of_z])

Parameter:

• x,y,z = These are the arrays of the same-shaped point coordinates.
• deg = It is the list of maximum degrees of the form [x_deg, y_deg, z_deg].

Return : It returns ‘vander3d’ which is an ndarray.

Python3

# import numpy and chebyshev libraries
import numpy as np
from numpy.polynomial import chebyshev as C

# Create arrays of point coordinates x, y and z
x = [1.2, 2.1]
y = [2.3, 3.2]
z = [3.1, 1.3]

# Assign degrees to x, y and z
deg_of_x = 2
deg_of_y = 1
deg_of_z = 3

# use chebvander3d() function to generate a
# pseudo Vandermonde matrix of the
# Chebyshev polynomial and x, y, z sample points
print(C.chebvander3d(x, y, z, [deg_of_x, deg_of_y, deg_of_z]))

Output :

[[  1.         3.1       18.22     109.864      2.3        7.13

   41.906    252.6872     1.2        3.72      21.864    131.8368

    2.76       8.556     50.2872   303.22464    1.88       5.828

   34.2536   206.54432    4.324     13.4044    78.78328  475.051936]

 [  1.         1.3        2.38       4.888      3.2        4.16

    7.616     15.6416     2.1        2.73       4.998     10.2648

    6.72       8.736     15.9936    32.84736    7.82      10.166

   18.6116    38.22416   25.024     32.5312    59.55712  122.317312]]