Generate a Vandermonde matrix of the Laguerre polynomial with complex array of points in Python

To generate a pseudo Vandermonde matrix of the Laguerre polynomial with complex points, use the laguerre.lagvander() function from NumPy. This function returns a matrix where each row contains the evaluated Laguerre polynomials of different degrees at a specific point.

Syntax

numpy.polynomial.laguerre.lagvander(x, deg)

Parameters

The function accepts the following parameters ?

• x ? Array of points. The dtype is converted to float64 or complex128 depending on whether any elements are complex
• deg ? Degree of the resulting matrix

Return Value

Returns a pseudo-Vandermonde matrix with shape x.shape + (deg + 1,). The last index corresponds to the degree of the Laguerre polynomial.

Example

Let's create a Vandermonde matrix using complex array points ?

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

# Create a complex array
x = np.array([-2.+2.j, -1.+2.j, 0.+2.j, 1.+2.j, 2.+2.j])

# Display the array
print("Our Array...\n", x)

# Check the Dimensions
print("\nDimensions of our Array...\n", x.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n", x.dtype)

# Get the Shape
print("\nShape of our Array object...\n", x.shape)

# Generate the Vandermonde matrix of degree 2
print("\nVandermonde Matrix...\n", L.lagvander(x, 2))

Our Array...
 [-2.+2.j -1.+2.j  0.+2.j  1.+2.j  2.+2.j]

Dimensions of our Array...
 1

Datatype of our Array object...
 complex128

Shape of our Array object...
 (5,)

Vandermonde Matrix...
 [[ 1. +0.j  3. -2.j  5. -8.j]
  [ 1. +0.j  2. -2.j  1.5-6.j]
  [ 1. +0.j  1. -2.j -1. -4.j]
  [ 1. +0.j  0. -2.j -2.5-2.j]
  [ 1. +0.j -1. -2.j -3. +0.j]]

How It Works

Each row in the matrix represents the evaluation of Laguerre polynomials L?(x), L?(x), and L?(x) at each complex point. The first column contains L?(x) = 1, the second column contains L?(x) = 1-x, and the third column contains L?(x) = (2-4x+x²)/2.

Conclusion

The lagvander() function efficiently generates Vandermonde matrices for Laguerre polynomials with complex arrays. The resulting matrix has dimensions matching the input array plus one additional dimension for polynomial degrees.