Generate Pseudo Vandermonde Matrix of Legendre Polynomial in Python

To generate a pseudo Vandermonde matrix of the Legendre polynomial, use the legendre.legvander2d() method in Python Numpy. The method returns the pseudo-Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where The last index is the degree of the corresponding Legendre polynomial. The dtype will be the same as the converted x.

The parameter, x, y is an array of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. The parameter, deg is a list of maximum degrees of the form [x_deg, y_deg].

Steps

At first, import the required library −

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

Create arrays of point coordinates, all of the same shape using the numpy.array() method −

x = np.array([1, 2])
y = np.array([3, 4])

Display the arrays −

print("Array1...\n",x)
print("\nArray2...\n",y)

Display the datatype −

print("\nArray1 datatype...\n",x.dtype)
print("\nArray2 datatype...\n",y.dtype)

Check the Dimensions of both the arrays −

print("\nDimensions of Array1...\n",x.ndim)
print("\nDimensions of Array2...\n",y.ndim)

Check the Shape of both the arrays −

print("\nShape of Array1...\n",x.shape)
print("\nShape of Array2...\n",y.shape)

To generate a pseudo Vandermonde matrix of the Legendre polynomial, use the legendre.legvander2d() method in Python Numpy −

x_deg, y_deg = 2, 3
print("\nResult...\n",L.legvander2d(x,y, [x_deg, y_deg]))

Example

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

# Create arrays of point coordinates, all of the same shape using the numpy.array() method
x = np.array([1, 2])
y = np.array([3, 4])

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

# Display the datatype
print("\nArray1 datatype...\n",x.dtype)
print("\nArray2 datatype...\n",y.dtype)

# Check the Dimensions of both the arrays
print("\nDimensions of Array1...\n",x.ndim)
print("\nDimensions of Array2...\n",y.ndim)

# Check the Shape of both the arrays
print("\nShape of Array1...\n",x.shape)
print("\nShape of Array2...\n",y.shape)

# To generate a pseudo Vandermonde matrix of the Legendre polynomial, use the legendre.legvander2d() method in Python Numpy

x_deg, y_deg = 2, 3
print("\nResult...\n",L.legvander2d(x,y, [x_deg, y_deg]))

Output

Array1...
   [1 2]

Array2...
   [3 4]

Array1 datatype...
int64

Array2 datatype...
int64

Dimensions of Array1...
1

Dimensions of Array2...
1

Shape of Array1...
(2,)

Shape of Array2...
(2,)

Result...
   [[ 1. 3. 13. 63. 1. 3. 13. 63. 1. 3.
     13. 63. ]
   [ 1. 4. 23.5 154. 2. 8. 47. 308. 5.5 22.
    129.25 847. ]]