Evaluate a Laguerre series at array of points x in Python

To evaluate a Laguerre series at array of points x, use the polynomial.laguerre.lagval() method in Python NumPy. This method evaluates a Laguerre polynomial series at given points using the coefficients and evaluation points you provide.

Syntax

The lagval() method has the following syntax:

numpy.polynomial.laguerre.lagval(x, c, tensor=True)

Parameters

• x − Array of points at which to evaluate the polynomial. If x is a list or tuple, it is converted to an ndarray. Elements must support addition and multiplication with coefficient elements.
• c − Array of coefficients ordered so that coefficients for terms of degree n are in c[n]. For multidimensional arrays, remaining indices enumerate multiple polynomials.
• tensor − If True (default), extends coefficient array shape for broadcasting. If False, x is broadcast over columns of c.

Example

Let's evaluate a Laguerre series with coefficients [1, 2, 3] at various points:

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

# Create an array of coefficients
c = np.array([1, 2, 3])

# Display the coefficient array
print("Coefficient Array:")
print(c)

# Check array properties
print("\nArray Properties:")
print("Dimensions:", c.ndim)
print("Datatype:", c.dtype)
print("Shape:", c.shape)

# Evaluate Laguerre series at points
x = np.array([[1, 2], [3, 4]])
print("\nEvaluation points:")
print(x)

result = L.lagval(x, c)
print("\nLaguerre series evaluation result:")
print(result)

Coefficient Array:
[1 2 3]

Array Properties:
Dimensions: 1
Datatype: int64
Shape: (3,)

Evaluation points:
[[1 2]
 [3 4]]

Laguerre series evaluation result:
[[-0.5 -4. ]
 [-4.5 -2. ]]

How It Works

The Laguerre polynomial series is evaluated using the formula:

L?(x) = 1, L?(x) = 1 - x, L?(x) = (2 - 4x + x²)/2

For coefficients [1, 2, 3], the series becomes: 1×L?(x) + 2×L?(x) + 3×L?(x)

Different Evaluation Points

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

# Same coefficients
c = np.array([1, 2, 3])

# Single point evaluation
x_single = 2.0
result_single = L.lagval(x_single, c)
print(f"At x = {x_single}: {result_single}")

# Multiple points in 1D array
x_multiple = np.array([0, 1, 2, 3])
result_multiple = L.lagval(x_multiple, c)
print(f"At x = {x_multiple}: {result_multiple}")

At x = 2.0: -4.0
At x = [0 1 2 3]: [ 6.  -0.5 -4.  -4.5]

Conclusion

The lagval() method efficiently evaluates Laguerre polynomial series at given points. Use it with coefficient arrays and evaluation points to compute Laguerre series values for mathematical and scientific applications.