Question

Interpolate the Runge function of Example \(10.6\) at Chebyshev points for \(n\) from 10 to 170 in increments of \(10 .\) Calculate the maximum interpolation error on the uniform evaluation mesh \(\mathrm{x}=-1: .001: 1\) and plot the error vs. polynomial degree as in Figure \(10.8\) using semilogy. Observe spectral accuracy.

Short answer

#Summary# In this exercise, we interpolate the Runge function at Chebyshev points for different polynomial degrees (n = 10, 20, ..., 170) and calculate the maximum interpolation error on the uniform evaluation mesh x = -1:0.001:1. We follow these steps: 1. Define the Runge function (\(f(x) = \frac{1}{1+25x^2}\)). 2. Generate Chebyshev points using the formula (\(x_i = cos(\frac{(2i-1)\pi}{2n})\)). 3. Interpolate the Runge function using scipy.interpolate.BarycentricInterpolator function with Chebyshev points as input. 4. Calculate the maximum interpolation error (\(error_n = \max_{x\in[-1,1]} |f(x) - P_n(x)|\)). 5. Create a semilogy plot of the error vs polynomial degree (n = 10, 20, ..., 170) to observe spectral accuracy.

Step-by-step solution

Define the Runge function

The given Runge function in Example 10.6 is \(f(x) = \frac{1}{1+25x^2}\).

Generate Chebyshev points

To generate the n-th order Chebyshev points in the interval [-1, 1], we use the formula: \(x_i = cos(\frac{(2i-1)\pi}{2n})\) for i=1,...,n.

Interpolate Runge function using Chebyshev points

Perform the interpolation using the scipy.interpolate.BarycentricInterpolator function with the Chebyshev points as input.

Calculate the maximum interpolation error

Evaluate the interpolation function and the original Runge function in the uniform evaluation mesh x = -1:0.001:1. Then, calculate the maximum absolute difference between the original and interpolated function values: \(error_n = \max_{x\in[-1,1]} |f(x) - P_n(x)|\), where \(P_n(x)\) is the interpolation function.

Plot error vs polynomial degree using semilogy

Create a semilogy plot of the maximum interpolation error against polynomial degree n (10, 20, ..., 170). Observe the spectral accuracy by analyzing the relationship between the error values and polynomial degrees in the plot.

Key concepts

Chebyshev Points

Chebyshev points play a crucial role in polynomial interpolation. They are a specific set of points used to minimize the error caused by polynomial interpolation, particularly the notorious Runge's phenomenon.
In the context of an interval, such as [-1,1], the Chebyshev points of the first kind are calculated using the formula: \[ x_i = \cos\left(\frac{(2i-1)\pi}{2n}\right) \] where \(i\) ranges from 1 to \(n\). These points are essentially the x-coordinates of the Chebyshev polynomial roots, spread non-uniformly across the interval. This non-linear distribution tends to cluster more points at the edges of the interval, reducing the oscillations common with higher-degree polynomials known as Runge's phenomenon.- By concentrating more points at the ends of the interval, Chebyshev points work effectively to produce minimal polynomial oscillations during interpolation, which improves the overall approximation quality.

Interpolation Error

Interpolation error concerns the difference between the exact function and its polynomial approximation. When using techniques like Chebyshev interpolation, we aim to minimize this error across an interval, such as [-1,1].
In mathematical terms, if \(f(x)\) is the original function and \(P_n(x)\) is the interpolating polynomial of degree \(n\), the error can be expressed as:\[ error_n = \max_{x\in[-1,1]} |f(x) - P_n(x)| \]### Understanding Interpolation Error- The higher the difference we observe between \(f(x)\) and \(P_n(x)\), the larger the interpolation error.- Chebyshev interpolation, by leveraging Chebyshev points, helps minimize this error compared to naive, equispaced points that often lead to high errors near interval edges.- Plotting error versus polynomial degree can visually depict how error reduces as the polynomial degree increases, provided the points chosen are optimal, like Chebyshev points.This reduction in error is crucial for accurate function approximation and is a key step in achieving spectral accuracy.

Spectral Accuracy

Spectral accuracy refers to the rapid decrease in the interpolation error as the polynomial degree \(n\) increases. Spectral methods, including interpolation with Chebyshev points, typically achieve this when optimally chosen points lead to exponential decay of error.
With spectral accuracy, the accuracy of approximation improves across the interval without requiring extremely high-degree polynomials.
### Key Characteristics of Spectral Accuracy- It often implies that the method is not just accurate at a specific set of discrete points but across the entire interval.- The plot of interpolation error versus polynomial degree, typically plotted using a semilogy graph, demonstrates this rapidly decreasing error trend. As the polynomial degree increases from, say, 10 to 170, the error should decrease almost exponentially across specific orders.- Achieving spectral accuracy is particularly noticeable in problems like the Runge function interpolation with Chebyshev points as it avoids the large oscillations typically caused by higher-degree polynomials.This demonstrates the powerful capability of spectral methods in providing highly accurate solutions efficiently, with fewer data points.