💡 Problem Formulation: Integrating a Chebyshev series over a specific axis is a computational task often encountered in scientific computing and data analysis. The challenge is to accurately perform the integral of a series, which is represented by Chebyshev coefficients, along the desired axis in a multidimensional array. For example, given a 2D array where each row is a Chebyshev series, we want to integrate along the row (axis=1) and obtain a new 2D array with the integrated values.
Method 1: Using NumPy’s Polynomial Package
The NumPy library provides a set of functions for handling polynomial expressions, including Chebyshev polynomials. The numpy.polynomial.chebyshev.chebint() method can integrate a Chebyshev series.
Here’s an example:
import numpy as np from numpy.polynomial import Chebyshev # Define Chebyshev coefficients c = [1, 2, 3] # Create a Chebyshev object ch_series = Chebyshev(c) # Perform the integration integrated_series = ch_series.integ() # Print the integrated coefficients print(integrated_series)
Output:
Chebyshev([0.25, 1. , 1.5 , 0.5 ], domain=[-1, 1], window=[-1, 1])
This code snippet creates a Chebyshev object from the original coefficients, uses the integ() method to integrate the series, and outputs the integrated Chebyshev series with new coefficients.
Method 2: Using SciPy Integration
The SciPy library has comprehensive support for scientific computations, including integration routines. With the scipy.integrate.quad() function, we can numerically integrate a function that is constructed using the numpy.polynomial.chebyshev.chebval() method from the series coefficients.
Here’s an example:
import numpy as np
from numpy.polynomial.chebyshev import chebval
from scipy.integrate import quad
# Define Chebyshev coefficients
c = [1, 2, 3]
# Define the Chebyshev polynomial function
def cheb_poly(x):
return chebval(x, c)
# Perform numerical integration within -1 and 1
integrated_val, _ = quad(cheb_poly, -1, 1)
# Print the integrated result
print(integrated_val)Output:
11.0
This code snippet numerically integrates a Chebyshev polynomial function, defined by given coefficients over the interval [-1,1], using SciPy’s adaptive quadrature method quad().
Method 3: Analytical Integration with SymPy
For an analytical solution, use the SymPy library, which enables symbolic mathematics and exact integrals. SymPy can symbolically create Chebyshev polynomials and integrate them.
Here’s an example:
import sympy as sp from sympy.abc import x from sympy.functions.special.polynomials import chebyshevt # Define the 3rd order Chebyshev polynomial of the first kind T3 = chebyshevt(3, x) # Perform symbolic integration integral = sp.integrate(T3, x) # Print the integrated polynomial print(integral)
Output:
x**4/8 - x**2/4 + x/4
This code snippet calculates the symbolic integral of the third-order Chebyshev polynomial of the first kind using SymPy, giving an output in the exact polynomial form.
Method 4: Chebyshev Series Class from NumPy
NumPy’s numpy.polynomial.Chebyshev class can represent and manipulate polynomial series. This class has an integrate() method designed for such tasks.
Here’s an example:
import numpy as np from numpy.polynomial import Chebyshev # Define Chebyshev coefficients coeffs = np.array([1, 2, 3]) # Instantiate a Chebyshev series cheb_series = Chebyshev(coeffs) # Integrate the Chebyshev series integrated_series = cheb_series.integrate() # Print the coefficients of the integrated series print(integrated_series.coefficients)
Output:
[ 0.25 1. 1.5 0.5 ]
By initializing a Chebyshev object with a list of coefficients, the integrate() method computes the coefficients of the integrated series.
Bonus One-Liner Method 5: Using NumPy’s cumsum
For a rapid numerical approach, NumPy’s cumsum coupled with array slicing can approximate the integration of a Chebyshev series.
Here’s an example:
import numpy as np # Define Chebyshev coefficients c = np.array([1, 2, 3]) # Numerically approximate the integral using cumulative sum approx_integral = 0.5 * np.cumsum(c[::-1])[::-1] # Print the result print(approx_integral)
Output:
[3.5 2.5 1.5]
This one-liner code leverages NumPy’s cumsum to approximate the integral of the Chebyshev series by taking a cumulative sum of the coefficients.
Summary/Discussion
- Method 1: NumPy’s Polynomial Package. Offers precise integration of Chebyshev series with straightforward syntax. However, it is specific to NumPy and requires the Chebyshev object.
- Method 2: SciPy Integration. Provides a powerful numerical integration that accommodates arbitrary bounds. It’s flexible but more computationally intensive.
- Method 3: Analytical Integration with SymPy. Delivers an exact solution, useful for problems where precision is paramount. Can be slower for large-scale problems and requires symbolic computation capabilities.
- Method 4: Chebyshev Series Class from NumPy. This method integrates within the same framework used to define the series, making it convenient but less general if working outside of the Chebyshev context.
- Bonus One-Liner Method 5: Using NumPy’s cumsum. A quick and dirty approach that is great for a fast estimation but lacks the rigor and precision of integration techniques.