Evaluate a Laguerre series at points x when coefficients are multi-dimensional in Python

To evaluate a Laguerre series at points x with multi-dimensional coefficients, use the polynomial.laguerre.lagval() method in Python NumPy. This method allows evaluation of multiple polynomials simultaneously when coefficients are arranged in a multi-dimensional array.

Parameters

The lagval() method takes three main parameters:

• x: The evaluation points. Can be a scalar, list, or array
• c: Array of coefficients where coefficients for degree n are in c[n]. For multi-dimensional arrays, additional indices represent multiple polynomials
• tensor: Boolean parameter (default True) controlling how x and c interact during evaluation

Basic Example with Multi-dimensional Coefficients

Let's create a 2D coefficient array and evaluate the Laguerre series at multiple points ?

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

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2,2)
print("Coefficient Array:")
print(c)
print(f"Shape: {c.shape}")

# Evaluate at points [1, 2]
result = L.lagval([1, 2], c)
print("\nResult:")
print(result)

Coefficient Array:
[[0 1]
 [2 3]]
Shape: (2, 2)

Result:
[[ 0. -2.]
 [ 1. -2.]]

Understanding the Coefficient Structure

In the 2D case, each column represents a separate polynomial. The coefficient array structure is ?

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

c = np.array([[0, 1],   # degree 0 coefficients for poly1, poly2
              [2, 3]])  # degree 1 coefficients for poly1, poly2

print("Polynomial 1 coefficients: [0, 2] (degrees 0, 1)")
print("Polynomial 2 coefficients: [1, 3] (degrees 0, 1)")

# Evaluate both polynomials at x=0.5
result = L.lagval(0.5, c)
print(f"\nEvaluation at x=0.5: {result}")

Polynomial 1 coefficients: [0, 2] (degrees 0, 1)
Polynomial 2 coefficients: [1, 3] (degrees 0, 1)

Evaluation at x=0.5: [1.  2.5]

Effect of Tensor Parameter

The tensor parameter controls how evaluation points and coefficients interact ?

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

c = np.arange(6).reshape(3,2)
x = [0, 1]

print("Coefficients:")
print(c)

# tensor=True (default): each x evaluates all polynomials
result_true = L.lagval(x, c, tensor=True)
print(f"\nWith tensor=True, shape: {result_true.shape}")
print(result_true)

# tensor=False: broadcasting behavior
result_false = L.lagval(x, c, tensor=False)
print(f"\nWith tensor=False, shape: {result_false.shape}")
print(result_false)

Coefficients:
[[0 1]
 [2 3]
 [4 5]]

With tensor=True, shape: (2, 2)
[[ 0.  1.]
 [-4. -7.]]

With tensor=False, shape: (2,)
[ 0. -7.]

Conclusion

The lagval() method efficiently evaluates multiple Laguerre polynomials simultaneously using multi-dimensional coefficient arrays. Use tensor=True to evaluate all polynomials at all points, or tensor=False for element-wise evaluation.