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

To evaluate a Legendre series at points x with multi-dimensional coefficients, use the polynomial.legendre.legval() method in Python NumPy. This method handles arrays of coefficients where each column represents a separate polynomial.

Syntax

numpy.polynomial.legendre.legval(x, c, tensor=True)

Parameters

The method accepts three parameters ?

• x − Points at which to evaluate the series. Can be scalar, list, or array
• c − Array of coefficients. For multi-dimensional arrays, columns represent different polynomials
• tensor − If True (default), evaluates every column for every point in x

Example

Let's create a multi-dimensional coefficient array and evaluate the Legendre series ?

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

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2,2)

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

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

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

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

# Evaluate Legendre series at points x=[1,2]
print("\nResult...\n",L.legval([1,2],c))

Our Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
2

Datatype of our Array object...
int64

Shape of our Array object...
(2, 2)

Result...
 [[2. 4.]
 [4. 7.]]

How It Works

The coefficient array c has shape (2, 2), representing two polynomials:

• First polynomial: coefficients [0, 2] − represents 0 + 2x
• Second polynomial: coefficients [1, 3] − represents 1 + 3x

For Legendre polynomials, the evaluation uses the basis functions L?(x) = 1 and L?(x) = x.

Understanding Multi-dimensional Results

The output shape is (2, 2) because we evaluate 2 polynomials at 2 points ?

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

# Same coefficient array
c = np.array([[0, 1], [2, 3]])

# Evaluate at single point
result_single = L.legval(1, c)
print("At x=1:", result_single)

# Evaluate at multiple points
result_multi = L.legval([0, 1, 2], c)
print("At x=[0,1,2]:")
print(result_multi)

At x=1: [2. 4.]
At x=[0,1,2]:
 [[0. 1.]
 [2. 4.]
 [4. 7.]]

Conclusion

Use polynomial.legendre.legval() to evaluate Legendre series with multi-dimensional coefficients. Each column in the coefficient array represents a separate polynomial, and the method efficiently evaluates all polynomials at the specified points.