Evaluate a Laguerre series at list of points x in Python

To evaluate a Laguerre series at points x, use the polynomial.laguerre.lagval() method in NumPy. This function takes evaluation points and coefficients to compute the series values.

Syntax

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

Parameters

The function accepts three parameters:

• x ? Array of points at which to evaluate the series. Can be a scalar, list, or array
• c ? Array of coefficients ordered from lowest to highest degree
• tensor ? Boolean controlling evaluation behavior for multidimensional arrays (default: True)

Example

Let's evaluate a Laguerre series with coefficients [1, 2, 3] at multiple 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 array
print("Coefficients:", c)

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

# Define evaluation points
x = [5, 10, 15]
print("Evaluation points:", x)

# Evaluate Laguerre series at points x
result = L.lagval(x, c)
print("Result:", result)

Coefficients: [1 2 3]
Dimensions: 1
Datatype: int64
Shape: (3,)
Evaluation points: [5, 10, 15]
Result: [  3.5  76.  223.5]

How It Works

The Laguerre series is evaluated using the formula:

L(x) = c[0]*L?(x) + c[1]*L?(x) + c[2]*L?(x) + ...

Where L?(x), L?(x), L?(x) are the Laguerre polynomials of degrees 0, 1, 2 respectively.

Single Point Evaluation

You can also evaluate at a single point ?

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

c = np.array([1, 2, 3])
x = 5

# Evaluate at single point
result = L.lagval(x, c)
print(f"Laguerre series at x={x}: {result}")

Laguerre series at x=5: 3.5

Conclusion

The lagval() function efficiently evaluates Laguerre series at specified points using coefficient arrays. It handles both single points and arrays of evaluation points seamlessly.