Generate a Vandermonde matrix of the Laguerre polynomial in Python

To generate a pseudo Vandermonde matrix of the Laguerre polynomial, use the laguerre.lagvander() function in Python NumPy. The method returns the pseudo-Vandermonde matrix with shape x.shape + (deg + 1,), where the last index corresponds to the degree of the Laguerre polynomial. The dtype will match the converted x array.

The parameter x is an array of points, converted to float64 or complex128 depending on whether elements are complex. If x is scalar, it converts to a 1-D array. The parameter deg specifies the degree of the resulting matrix.

Syntax

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

Parameters

The function accepts the following parameters ?

• x ? Array of points (scalar converted to 1-D array)
• deg ? Degree of the resulting matrix (non-negative integer)

Creating the Vandermonde Matrix

Let's create a complete example showing how to generate the Laguerre Vandermonde matrix ?

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

# Create an array of points
x = np.array([0, 1, -1, 2])

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

# Check array properties
print("\nDimensions:", x.ndim)
print("Datatype:", x.dtype)  
print("Shape:", x.shape)

# Generate Vandermonde matrix of degree 2
result = L.lagvander(x, 2)
print("\nVandermonde Matrix (degree 2):")
print(result)

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

Dimensions: 1
Datatype: int64
Shape: (4,)

Vandermonde Matrix (degree 2):
[[ 1.   1.   1. ]
 [ 1.   0.  -0.5]
 [ 1.   2.   3.5]
 [ 1.  -1.  -1. ]]

Understanding the Output

Each row corresponds to evaluating Laguerre polynomials L?(x), L?(x), L?(x) at each point ?

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

x = np.array([0, 1, -1, 2])

# Generate degree 3 matrix for more examples
result = L.lagvander(x, 3)
print("Vandermonde Matrix (degree 3):")
print(result)

print(f"\nMatrix shape: {result.shape}")
print(f"Each row evaluates L?, L?, L?, L? at x values")

Vandermonde Matrix (degree 3):
[[ 1.    1.    1.    1.  ]
 [ 1.    0.   -0.5  -0.5 ]
 [ 1.    2.    3.5   4.5 ]
 [ 1.   -1.   -1.    0.  ]]

Matrix shape: (4, 4)
Each row evaluates L?, L?, L?, L? at x values

Different Input Types

The function works with different array types and shapes ?

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

# Float array
x_float = np.array([0.5, 1.5, 2.5])
result_float = L.lagvander(x_float, 2)
print("Float input result:")
print(result_float)

# 2D array input
x_2d = np.array([[0, 1], [2, 3]])
result_2d = L.lagvander(x_2d, 1)
print(f"\n2D input shape: {x_2d.shape}")
print(f"Result shape: {result_2d.shape}")
print("Result:")
print(result_2d)

Float input result:
[[ 1.    0.5  -0.125]
 [ 1.   -0.5  -0.625]
 [ 1.   -1.5  -0.625]]

2D input shape: (2, 2)
Result shape: (2, 2, 2)
Result:
[[[ 1.  1.]
  [ 1.  0.]]

 [[ 1.  1.]
  [ 1. -2.]]]

Conclusion

The lagvander() function generates pseudo-Vandermonde matrices for Laguerre polynomials, useful in polynomial fitting and numerical analysis. The resulting matrix evaluates Laguerre polynomials L? through L_deg at each input point.