Remove small trailing coefficients from Chebyshev polynomial in Python

To remove small trailing coefficients from Chebyshev polynomial, use the chebyshev.chebtrim() method in Python NumPy. This 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. "Trailing" means highest order coefficient(s). For example, in [0, 1, 1, 0, 0] (which represents 0 + x + x² + 0×x³ + 0×x?), both the 3rd and 4th order coefficients would be "trimmed."

Syntax

numpy.polynomial.chebyshev.chebtrim(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)

Basic Example

Let's start with a simple example to understand how chebtrim() works ?

import numpy as np
from numpy.polynomial import chebyshev as C

# Create an array with trailing zeros
coeffs = np.array([0, 5, 0, 0, 9, 0])
print("Original coefficients:", coeffs)

# Remove trailing zeros
trimmed = C.chebtrim(coeffs)
print("After trimming:", trimmed)

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

Using Tolerance Parameter

The tol parameter allows you to remove coefficients that are small but not exactly zero ?

import numpy as np
from numpy.polynomial import chebyshev as C

# Array with small trailing values
coeffs = np.array([1.0, 2.5, 3.2, 0.001, 0.0001])
print("Original coefficients:", coeffs)

# Trim with tolerance
trimmed = C.chebtrim(coeffs, tol=0.01)
print("After trimming (tol=0.01):", trimmed)

# Trim with smaller tolerance
trimmed_strict = C.chebtrim(coeffs, tol=0.0001)
print("After trimming (tol=0.0001):", trimmed_strict)

Original coefficients: [1.     2.5    3.2    0.001  0.0001]
After trimming (tol=0.01): [1.  2.5 3.2]
After trimming (tol=0.0001): [1.     2.5    3.2    0.001 ]

Detailed Example with Array Information

import numpy as np
from numpy.polynomial import chebyshev as C

# Create an array of coefficients
c = np.array([0, 5, 0, 0, 9, 0])

# Display array information
print("Our Array:", c)
print("Dimensions:", c.ndim)
print("Datatype:", c.dtype)
print("Shape:", c.shape)

# Remove trailing coefficients
result = C.chebtrim(c)
print("Result after trimming:", result)
print("New shape:", result.shape)

Our Array: [0 5 0 0 9 0]
Dimensions: 1
Datatype: int64
Shape: (6,)
Result after trimming: [0. 5. 0. 0. 9.]
New shape: (5,)

Key Points

• Only trailing (highest order) coefficients are removed
• Interior zeros are preserved in the polynomial
• The method returns a copy, not a view of the original array
• If all coefficients would be trimmed, returns [0.]

Conclusion

The chebyshev.chebtrim() method efficiently removes small trailing coefficients from Chebyshev polynomials. Use the tol parameter to control the threshold for what constitutes a "small" coefficient.