Remove small trailing coefficients from Legendre polynomial in Python

To remove small trailing coefficients from Legendre polynomial, use the legendre.legtrim() method in Python NumPy. The method returns a 1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.

The small means "small in absolute value" and is controlled by the parameter tol. The trailing means highest order coefficient(s), e.g., in [0, 1, 1, 0, 0] (which represents 0 + x + x**2 + 0*x**3 + 0*x**4) both the 3rd and 4th order coefficients would be trimmed.

Syntax

numpy.polynomial.legendre.legtrim(c, tol=0)

Parameters

• c ? 1-d array of coefficients, ordered from lowest order to highest
• tol ? Trailing elements with absolute value less than or equal to tol are removed (default: 0)

Example

Let's create a Legendre polynomial with trailing zeros and remove them ?

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

# Create an array with trailing zeros
coefficients = np.array([0, 5, 0, 0, 9, 0])

print("Original coefficients:")
print(coefficients)

print("\nDimensions:", coefficients.ndim)
print("Datatype:", coefficients.dtype)
print("Shape:", coefficients.shape)

# Remove trailing coefficients
trimmed = L.legtrim(coefficients)
print("\nAfter trimming:")
print(trimmed)

Original coefficients:
[0 5 0 0 9 0]

Dimensions: 1
Datatype: int64
Shape: (6,)

After trimming:
[0. 5. 0. 0. 9.]

Using Custom Tolerance

You can specify a tolerance value to remove small coefficients that are close to zero ?

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

# Array with small trailing coefficients
coefficients = np.array([1.0, 2.5, 3.0, 0.001, 0.0001])

print("Original coefficients:")
print(coefficients)

# Remove coefficients smaller than 0.01
trimmed = L.legtrim(coefficients, tol=0.01)
print("\nAfter trimming (tol=0.01):")
print(trimmed)

Original coefficients:
[1.     2.5    3.     0.001  0.0001]

After trimming (tol=0.01):
[1.  2.5 3. ]

Edge Case - All Zeros

When all coefficients are zeros or below tolerance, the function returns a single zero ?

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

# Array of all zeros
all_zeros = np.array([0, 0, 0, 0])

print("All zeros array:")
print(all_zeros)

trimmed = L.legtrim(all_zeros)
print("\nAfter trimming:")
print(trimmed)

All zeros array:
[0 0 0 0]

After trimming:
[0.]

Conclusion

The legendre.legtrim() method efficiently removes trailing coefficients from Legendre polynomials. Use the tol parameter to control the threshold for small coefficients. This is essential for maintaining clean polynomial representations in numerical computations.