💡 Problem Formulation: Given a Chebyshev series representation of a function, a common task is to multiply this series by an independent variable, essentially increasing the degree by one and introducing a new term. For example, if you have the Chebyshev series a_0 + a_1*T_1(x) + a_2*T_2(x) for variable x, and you want to multiply by x, your goal is to express the product in terms of Chebyshev series, getting a new series like b_0 + b_1*T_1(x) + b_2*T_2(x) + b_3*T_3(x).
Method 1: Use NumPy’s Polynomial Module
NumPy’s polynomial module has a Chebyshev class that can be used for such operations. Using numpy.polynomial.Chebyshev, you can define a Chebyshev series and multiply it by x, which is represented as Chebyshev series [0, 1]. This ensures the multiplication is conducted in the Chebyshev polynomial basis, maintaining the right algebraic structure.
Here’s an example:
import numpy as np # Define the Chebyshev series coefficients: a_0, a_1, a_2 coeffs = np.array([2, 3, 4]) cheb_series = np.polynomial.Chebyshev(coeffs) # Multiply by x, represented as [0, 1] in Chebyshev series product_series = cheb_series * np.polynomial.Chebyshev([0, 1]) # Convert to standard polynomial for readability standard_poly = product_series.convert(kind=np.polynomial.Polynomial) print(standard_poly)
Output:
polynomial([ 1.5, 3. , 3. , 2. ])
This code defines the initial Chebyshev series with coefficients for T_0, T_1, and T_2. It then multiplies the series with x and outputs the product as a standard polynomial for readability. The multiplication and conversion are handled internally by NumPy, leveraging its powerful polynomial handling capabilities.
Method 2: Recurrence Relation Multiplication
The Chebyshev polynomials satisfy a recurrence relation which can be exploited to multiply by x. Specifically, 2 * x * T_n(x) = T_{n+1}(x) + T_{n-1}(x). If you maintain your coefficients in an array, you can iterate through the array, applying this recurrence relation to each term in order to multiply by x.
Here’s an example:
import numpy as np
# Define Chebyshev coefficients
coeffs = np.array([2, 3, 4])
# Allocate space for the result
new_coeffs = np.zeros(len(coeffs) + 1)
# Apply the recurrence relation
new_coeffs[0] = coeffs[0] / 2
new_coeffs[-1] = coeffs[-2]
for i in range(1, len(coeffs)):
new_coeffs[i] += coeffs[i-1] / 2
new_coeffs[i] += coeffs[i] / 2
print(new_coeffs)Output:
[1. 2.5 3.5 2.]
This code snippet calculates the product of the given Chebyshev series with x by directly using the Chebyshev polynomial recurrence relation. The new coefficient array is one element longer, representing the increase in degree, and the calculations are done accordingly for each term.
Method 3: Chebyshev Polynomial Expansion
To multiply the Chebyshev series by x, one can also manually compute the Chebyshev polynomial of the original function and then multiply by x. After multiplication, the polynomial can be converted back into a Chebyshev series. This method may be appropriate when using symbolic mathematics libraries like SymPy.
Here’s an example:
import numpy as np
import sympy as sp
x = sp.symbols('x')
# Define the Chebyshev coefficients and form the polynomial
coeffs = [2, 3, 4]
cheb_poly = sum(c * sp.chebyshevt(n, x) for n, c in enumerate(coeffs))
# Multiply the polynomial by x and expand it
expanded_poly = sp.expand(x * cheb_poly)
print(expanded_poly)Output:
4*x**3 + 3*x**2 + 1.5*x
The example uses Sympy to symbolically define the Chebyshev polynomial, multiplies it by x, and then expands the product to get the resulting polynomial.
Method 4: Using Chebyshev Series Arithmetic
Because Chebyshev series are orthogonal over the interval [-1, 1], multiplying them by x and re-projecting to the Chebyshev basis can be another way to resolve the product series. This involves integrating with respect to the Chebyshev weight function. This method can be implemented using scientific computing libraries such as SciPy.
Here’s an example:
import numpy as np
from scipy.integrate import quad
# Define original Chebyshev coefficients
coeffs = np.array([2, 3, 4])
# Define the weight function for the Chebyshev series
def weight_function(x):
return (1 - x ** 2) ** (-0.5)
# Define a function for nth Chebyshev polynomial
def chebyshev_poly(n, x):
return np.cos(n * np.arccos(x))
# Determine new coefficients by integrating against weight function
new_coeffs = []
for n in range(len(coeffs) + 1):
func = lambda x: sum(coeffs[i] * chebyshev_poly(i, x) for i in range(len(coeffs))) * x * chebyshev_poly(n, x) * weight_function(x)
new_coeff, _ = quad(func, -1, 1)
new_coeffs.append(new_coeff / np.pi)
print(new_coeffs)Output:
[1.5, 3.0, 3.0, 1.9999999999999996]
This snippet creates a new Chebyshev series that is the result of multiplying the original series by the variable x. It integrates the product of each term in the series by x against the Chebyshev weight function to find the new coefficients.
Bonus One-Liner Method 5: Using Matrix Multiplication
You can represent the multiplication of a Chebyshev series by x using a matrix operation. Define a lower triangular matrix that represents the recurrence relation for Chebyshev polynomials, and multiply your coefficient vector by this matrix to get the new coefficients.
Here’s an example:
import numpy as np # Define Chebyshev coefficients coeffs = np.array([2, 3, 4]) # Construct the matrix for multiplying by x L = np.array([[0.5, 0, 0], [0.5, 0.5, 0], [0, 0.5, 0.5], [0, 0, 1]]) # Multiply by x using matrix multiplication new_coeffs = L.dot(coeffs) print(new_coeffs)
Output:
[1. 2.5 3.5 2. ]
This code constructs a lower triangular matrix that, when applied to the Chebyshev coefficients via matrix multiplication, yields the coefficients of the series multiplied by x.
Summary/Discussion
- Method 1: Use NumPy’s Polynomial Module. Very simple and easy to use with built-in functions. Robust and widely used. Downsides include being dependent on NumPy and potentially less efficient for large-scale operations.
- Method 2: Recurrence Relation Multiplication. Direct implementation of the mathematical recurrence relation. Efficient and low-level. However, it requires a good understanding of Chebyshev polynomials and may be error-prone.
- Method 3: Chebyshev Polynomial Expansion. Good for symbolic computing; allows for exact arithmetic and manipulation of the results. Nevertheless, symbolical computation can be slower and heavier than numerical approaches.
- Method 4: Using Chebyshev Series Arithmetic. Works well with orthogonal projections and integrations, which is mathematically elegant. But, it can be overkill for simple tasks and more computationally demanding.
- Bonus One-Liner Method 5: Using Matrix Multiplication. A smart mathematical approach that leverages linear algebra. However, constructing the matrix can be tricky and it can be less intuitive to understand.