Polynomial 1-D Interpolation with SciPy

Polynomial 1-D Interpolation is a numerical technique used to estimate unknown values between known data points by fitting a polynomial function through them. This method is widely employed in scientific computing, data analysis, and numerical modeling.

What is Polynomial 1-D Interpolation?

Polynomial interpolation involves constructing a polynomial that passes exactly through a given set of data points. The resulting polynomial function can then be used to estimate values at intermediate points.

SciPy offers specialized functions for efficient and stable polynomial interpolation in one dimension:

• BarycentricInterpolator: An efficient and stable method, particularly effective for larger datasets.
• KroghInterpolator: Supports interpolation using function values and their derivatives, making it suitable for smooth functions where derivative information is available.

It's important to note that using high-degree polynomials can lead to instability, especially near the data points' boundaries. This phenomenon is commonly known as Runge's phenomenon.

Key Characteristics of Polynomial 1-D Interpolation in SciPy

Understanding Polynomial Degree

The degree of a polynomial refers to the highest power of the variable present in its expression. This degree significantly influences the complexity and shape of the resulting curve.

Types of Polynomials by Degree

Polynomial Interpolation Using SciPy: Code Example

Barycentric Interpolation Example

This example demonstrates how to use BarycentricInterpolator to fit a polynomial through a set of points and then interpolate new values.

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import BarycentricInterpolator

# Sample data points
x = np.array([0, 1, 2, 3, 4])
y = np.array([1, 2, 0, 2, 1])

# Create the interpolator object
interpolator = BarycentricInterpolator(x, y)

# Generate new x values for interpolation
x_new = np.linspace(0, 4, 100)

# Perform interpolation
y_new = interpolator(x_new)

# Plot the results
plt.scatter(x, y, color='red', label='Data Points')
plt.plot(x_new, y_new, color='blue', label='Interpolated Curve')
plt.title('Polynomial Interpolation using BarycentricInterpolator')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.legend()
plt.show()

SciPy Functions for Polynomial Interpolation

BarycentricInterpolator

The BarycentricInterpolator offers an efficient and numerically stable method for polynomial interpolation. It's particularly advantageous for large datasets as it reduces computational overhead and minimizes numerical errors by employing barycentric coordinates.

from scipy.interpolate import BarycentricInterpolator

# Example using BarycentricInterpolator
x_data = np.array([0, 1, 2, 3, 4, 5])
y_data = np.sin(x_data) # Example data from a sine function

# Create the interpolator
interpolator = BarycentricInterpolator(x_data, y_data)

# Generate new x values for interpolation
x_new = np.linspace(0, 5, 100)

# Perform interpolation
y_new = interpolator(x_new)

# Note: Plotting code would follow to visualize the interpolation

KroghInterpolator

The KroghInterpolator is suitable for situations where not only function values but also their derivatives are known. This makes it an excellent choice for interpolating smooth functions, yielding more accurate results when derivative information is available. It utilizes divided differences for efficient computation.

from scipy.interpolate import KroghInterpolator

# Example using KroghInterpolator
x_data = np.array([0, 1, 2, 3])
y_data = np.array([1, 2, 0, 1]) # Sample y values

# Create the interpolator
interpolator = KroghInterpolator(x_data, y_data)

# Generate new x values for interpolation
x_new = np.linspace(0, 3, 100)

# Perform interpolation
y_new = interpolator(x_new)

# Note: Plotting code would follow to visualize the interpolation

Limitations of Polynomial Interpolation

While polynomial interpolation is a powerful tool, it comes with several limitations that users should be aware of:

• Runge’s Phenomenon: As the degree of the interpolating polynomial increases, it can exhibit large oscillations, particularly near the endpoints of the data interval. This can lead to inaccurate estimations in these regions.
• Overfitting: When interpolating a dataset with many points, or noisy data, the resulting high-degree polynomial may fit the noise rather than the underlying trend, leading to poor generalization on new data.
• Computational Cost: The computational resources required to construct and evaluate a polynomial generally increase significantly with its degree.
• Global Sensitivity: Each point on the interpolated curve is influenced by all the input data points. Consequently, a small change or error in a single data point can affect the entire polynomial curve.

Summary

Polynomial 1-D Interpolation is a valuable technique for approximating data and modeling functions. SciPy provides robust implementations through BarycentricInterpolator and KroghInterpolator for efficient and stable interpolation. While these methods are effective for small, smooth datasets, users should exercise caution when employing high-degree polynomials due to the potential for instability (Runge's phenomenon) and overfitting.