Add one Chebyshev series to another in Python

To add one Chebyshev series to another, use the polynomial.chebyshev.chebadd() method in Python NumPy. The method returns an array representing the Chebyshev series of their sum. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.

Syntax

numpy.polynomial.chebyshev.chebadd(c1, c2)

Parameters

c1, c2 ? 1-D arrays of Chebyshev series coefficients ordered from low to high.

Basic Example

Let's start with a simple example of adding two Chebyshev series ?

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

# Create 1-D arrays of Chebyshev series coefficients
c1 = np.array([1, 2, 3])
c2 = np.array([3, 2, 1])

print("First series coefficients:", c1)
print("Second series coefficients:", c2)

# Add the two Chebyshev series
result = C.chebadd(c1, c2)
print("Sum of series:", result)

First series coefficients: [1 2 3]
Second series coefficients: [3 2 1]
Sum of series: [4. 4. 4.]

Adding Series of Different Lengths

The chebadd() method can handle Chebyshev series of different lengths ?

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

# Series with different lengths
c1 = np.array([1, 2])
c2 = np.array([3, 2, 1, 4])

print("Series 1 (length 2):", c1)
print("Series 2 (length 4):", c2)

result = C.chebadd(c1, c2)
print("Sum:", result)

Series 1 (length 2): [1 2]
Series 2 (length 4): [3 2 1 4]
Sum: [4. 4. 1. 4.]

Understanding the Addition Process

Chebyshev series addition works coefficient-wise. For the series c1 = [1, 2, 3] representing 1*T_0 + 2*T_1 + 3*T_2 and c2 = [3, 2, 1] representing 3*T_0 + 2*T_1 + 1*T_2, the sum is [4, 4, 4] representing 4*T_0 + 4*T_1 + 4*T_2.

Complete Example with Array Properties

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

# Create Chebyshev series coefficients
c1 = np.array([1, 2, 3])
c2 = np.array([3, 2, 1])

print("Array 1:", c1)
print("Array 2:", c2)
print("\nArray 1 datatype:", c1.dtype)
print("Array 2 datatype:", c2.dtype)
print("\nDimensions of Array 1:", c1.ndim)
print("Dimensions of Array 2:", c2.ndim)
print("\nShape of Array 1:", c1.shape)
print("Shape of Array 2:", c2.shape)

# Add the Chebyshev series
result = C.chebadd(c1, c2)
print("\nResult (sum):", result)

Array 1: [1 2 3]
Array 2: [3 2 1]

Array 1 datatype: int64
Array 2 datatype: int64

Dimensions of Array 1: 1
Dimensions of Array 2: 1

Shape of Array 1: (3,)
Shape of Array 2: (3,)

Result (sum): [4. 4. 4.]

Conclusion

The numpy.polynomial.chebyshev.chebadd() method efficiently adds two Chebyshev series by summing their coefficients. It handles series of different lengths automatically and returns the result as a NumPy array.