Raise a Chebyshev series to a power in Python

To raise a Chebyshev series to a power, use the chebyshev.chebpow() method in Python NumPy. This function returns the Chebyshev series c raised to the specified power. The argument c is a sequence of coefficients ordered from low to high, where [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.

Syntax

numpy.polynomial.chebyshev.chebpow(c, pow, maxpower=16)

Parameters

• c − 1-D array of Chebyshev series coefficients ordered from low to high
• pow − Power to which the series will be raised
• maxpower − Maximum power allowed (default is 16) to limit series growth

Example

Let's create a Chebyshev series and raise it to different powers ?

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

# Create 1-D array of Chebyshev series coefficients
c = np.array([1, 2, 3])

print("Original coefficient array:", c)
print("Dimensions:", c.ndim)
print("Shape:", c.shape)
print("Datatype:", c.dtype)

# Raise to power 2
result_pow2 = C.chebpow(c, 2)
print("\nRaised to power 2:", result_pow2)

# Raise to power 3
result_pow3 = C.chebpow(c, 3)
print("Raised to power 3:", result_pow3)

Original coefficient array: [1 2 3]
Dimensions: 1
Shape: (3,)
Datatype: int64

Raised to power 2: [ 7.5 12.  18.5 12.   4.5]
Raised to power 3: [ 22.   48.   85.5  78.   58.5  27.   13.5]

Understanding the Result

When a Chebyshev series is raised to a power, the resulting coefficients represent the new series. The number of coefficients increases as the degree of the polynomial increases with the power operation ?

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

# Simple example with fewer coefficients
c_simple = np.array([1, 1])  # T_0 + T_1

print("Original series [1, 1] represents: T_0 + T_1")
print("Power 1:", C.chebpow(c_simple, 1))
print("Power 2:", C.chebpow(c_simple, 2))
print("Power 3:", C.chebpow(c_simple, 3))

Original series [1, 1] represents: T_0 + T_1
Power 1: [1. 1.]
Power 2: [1.5 2.  0.5]
Power 3: [2.5 3.  1.5 0.5]

Using maxpower Parameter

The maxpower parameter limits the maximum power to prevent unmanageable series growth ?

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

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

# Default maxpower (16)
result_default = C.chebpow(c, 5)
print("Power 5 (default maxpower):", len(result_default), "coefficients")

# Custom maxpower
try:
    result_custom = C.chebpow(c, 5, maxpower=3)
    print("Power 5 (maxpower=3):", result_custom)
except ValueError as e:
    print("Error with maxpower=3:", str(e))

Power 5 (default maxpower): 9 coefficients
Error with maxpower=3: power too large

Conclusion

The chebpow() function efficiently raises Chebyshev series to any power while maintaining polynomial properties. Use the maxpower parameter to control computational complexity for large powers.