Chebyshev Polynomial Interpolation for Function Approximation

Chebyshev Polynomials

• They have the property of being orthogonal when evaluated at specific points called Chebyshev points of the first kind.
• These are special polynomials used for approximating functions.

Function of chebinterpolate

• This Chebyshev series represents the approximating polynomial.
• Using these evaluations and the properties of Chebyshev polynomials, it calculates the coefficients of a Chebyshev series that interpolates the function at those points.
• It evaluates the input function (func) at these Chebyshev points.
• It finds the Chebyshev points of the first kind within this [0, 1] interval.
• The function operates on the interval [-1, 1]. Internally, it converts this interval to the domain [0, 1] for calculations.
• It takes a function (func) as input, along with the degree (deg) of the desired interpolating polynomial.

Key Points

• The chebinterpolate function returns the coefficients of the interpolating Chebyshev series as a NumPy array. These coefficients are ordered from lowest degree to highest degree.
• The resulting Chebyshev series tends to be a good approximation to the original function, especially if the function is continuous within the [-1, 1] interval. This approximation minimizes a concept called Runge's phenomenon, which can cause issues with polynomial interpolation.

• This function was introduced in a newer version of NumPy (1.14 or later).

---

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

# Define the function to interpolate
def func(x):
  return np.sin(2*np.pi*x)

# Degree of the interpolating polynomial
degree = 5

# Sample points within the interval [-1, 1]
x = np.linspace(-1, 1, 10)

# Use chebinterpolate to get coefficients
cheb_coeff = chebinterpolate(func, degree)

# Print the coefficients
print("Coefficients of the interpolating Chebyshev series:", cheb_coeff)

# (Optional) Evaluate the interpolating polynomial at new points
new_x = np.linspace(-0.8, 0.8, 20)
interpolated_y = np.polynomial.chebyshev.chebval(new_x, cheb_coeff)

# Plot the original function and the interpolated polynomial
import matplotlib.pyplot as plt

plt.plot(x, func(x), label="Original Function")
plt.plot(new_x, interpolated_y, label="Interpolated Polynomial")
plt.legend()
plt.xlabel("x")
plt.ylabel("y")
plt.title("Interpolation using Chebyshev Polynomials")
plt.show()

---

Lagrange Polynomial Interpolation

This classical method constructs a polynomial that passes exactly through a set of specified data points. You can create these data points by sampling your function at desired locations within the interval [-1, 1]. Then, you can use libraries like SymPy or construct the Lagrange polynomial yourself to achieve interpolation.

Vandermonde Matrix-based Interpolation

Similar to Lagrange interpolation, this method involves creating a Vandermonde matrix containing the function evaluations at chosen points. By solving a linear system involving this matrix, you can find the coefficients of the interpolating polynomial. Libraries like NumPy offer functionalities for working with Vandermonde matrices.

Spline Interpolation

Spline interpolation uses piecewise polynomials to approximate a function. There are different spline types, such as cubic splines, that provide smooth interpolation across the entire interval. Libraries like SciPy offer spline interpolation functionalities.

• Spline Interpolation
  Provides smooth interpolation across the entire domain, but might not be as accurate for highly oscillatory functions.
• Vandermonde Interpolation
  Similar to Lagrange but more computationally expensive for larger datasets.
• Lagrange Interpolation
  Exact interpolation at specified points, but can suffer from oscillations for higher degrees.
• Chebyshev Interpolation
  Offers good approximation with minimized Runge's phenomenon, especially for smooth functions. Works well within the [-1, 1] interval.