Multiply one Chebyshev series to another in Python

To multiply one Chebyshev series to another, use the polynomial.chebyshev.chebmul() method in Python. This method returns an array of Chebyshev series coefficients representing their product. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.

Syntax

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

Parameters

The parameters are:

• c1, c2 − 1-D arrays of Chebyshev series coefficients ordered from low to high degree

Example

Let's multiply two Chebyshev series using chebmul() ?

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])

# Display the arrays of coefficients
print("Array1...\n", c1)
print("\nArray2...\n", c2)

# Display the datatype
print("\nArray1 datatype...\n", c1.dtype)
print("\nArray2 datatype...\n", c2.dtype)

# Check the Dimensions of both the arrays
print("\nDimensions of Array1...\n", c1.ndim)
print("\nDimensions of Array2...\n", c2.ndim)

# Check the Shape of both the arrays
print("\nShape of Array1...\n", c1.shape)
print("\nShape of Array2...\n", c2.shape)

# Multiply the two Chebyshev series
result = C.chebmul(c1, c2)
print("\nResult (multiplication)....\n", result)

Array1...
 [1 2 3]

Array2...
 [3 2 1]

Array1 datatype...
 int64

Array2 datatype...
 int64

Dimensions of Array1...
 1

Dimensions of Array2...
 1

Shape of Array1...
 (3,)

Shape of Array2...
 (3,)

Result (multiplication)....
 [ 6.5 12.  12.   4.   1.5]

How It Works

The chebmul() function performs the multiplication of two Chebyshev series by convolving their coefficients according to Chebyshev polynomial multiplication rules. The resulting array has a degree equal to the sum of the degrees of the input series.

Simple Example

Here's a simpler example with basic coefficients ?

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

# Simple Chebyshev series: T_0 + T_1 and T_0 + 2*T_1
series1 = np.array([1, 1])  # 1 + T_1
series2 = np.array([1, 2])  # 1 + 2*T_1

result = C.chebmul(series1, series2)
print("Series 1:", series1)
print("Series 2:", series2) 
print("Product:", result)

Series 1: [1 1]
Series 2: [1 2]
Product: [1.5 3.  1. ]

Conclusion

The chebmul() function efficiently multiplies Chebyshev series by handling the polynomial multiplication according to Chebyshev algebra rules. The result is a new series with coefficients representing the product of the input series.