Question

Using the centered difference \(h^{-2}\left(f\left(x_{i+1}\right)-2 f\left(x_{i}\right)+f\left(x_{i-1}\right)\right)\), construct an \((n-2) \times n\) differentiation matrix, \(D^{2}\), for the second derivative of \(f(x)=e^{x} \sin (10 x)\) at the points \(x_{i}=\) \(i h, i=1,2, \ldots, n-1\), with \(h=\pi / n .\) Record the maximum absolute error in \(D^{2} \mathbf{f}\) for \(n=\) \(25,50,100\), and 200. You should observe \(\mathcal{O}\left(n^{-2}\right)\) improvement. Compare these results against those obtained using the Chebyshev differentiation matrix, as recorded in Figure \(14.5\).

Short answer

Answer: The process followed to find the maximum absolute error in the second derivative of the given function is to: 1. Define the centered difference formula for the second derivative: \(h^{-2}(f(x_{i+1}) - 2f(x_i) + f(x_{i-1}))\). 2. Calculate the second derivative of f(x) for different values of n (25, 50, 100, and 200) using the centered difference formula and points xi = i*pi/n for i = 1, 2, ..., n-1. 3. Compute the difference between the exact second derivative (obtained from differentiating f(x) twice) and the second derivative calculated using the centered difference formula for each point xi and each value of n. 4. Determine the maximum absolute error for each n value. 5. Observe the O(n^(-2)) improvement in the maximum absolute error as n increases.

Step-by-step solution

Define the centered difference formula and function

We have the centered difference formula to find the second derivative given by: \(h^{-2}(f(x_{i+1}) - 2f(x_i) + f(x_{i-1}))\) We will apply this formula to find the second derivative of the given function f(x) = e^x sin(10x).

Calculate the second derivative for different n values

We will apply the centered difference formula to find the second derivative of f(x) for different values of n (25, 50, 100, and 200). For each value of n, calculate the points xi = i*pi/n for i = 1, 2, ..., n-1.

Compute the maximum absolute error

Once we have the second derivative for each point xi and each value of n, find the difference between the exact second derivative (obtained from differentiating f(x) twice) and the second derivative calculated using the centered difference formula. Compute the maximum absolute error for each n value.

Observe the improvement

Look for the O(n^(-2)) improvement in the maximum absolute error as n increases. This means that the error should decrease as 1/n^2. If this improvement is observed, the centered difference formula is performing as expected.

Compare results with Chebyshev differentiation matrix

Finally, compare the results obtained using the centered difference formula to those obtained using the Chebyshev differentiation matrix as shown in Figure 14.5. Note any differences in accuracy or convergence.

Key concepts

Second Derivative Approximation

The second derivative of a function gives us the curvature or the rate at which the slope is changing. In numerical analysis, we approximate the second derivative using discrete points, because we may not have an analytical form of the function or the function may be too complex to differentiate symbolically. A common method for this approximation is the centered difference formula. It uses three consecutive points \( x_{i-1} \) , \( x_{i} \) , and \( x_{i+1} \) and their function values to estimate the second derivative at \( x_{i} \) .
The formula is given by \( h^{-2}(f(x_{i+1}) - 2f(x_i) + f(x_{i-1})) \) where \( h \) is the spacing between the points. This approximation is part of finite difference methods, and its accuracy depends on the choice of \( h \) and the nature of the function.

Understanding this approximation is crucial when dealing with functions like \( f(x) = e^{x} \sin (10 x) \) for which we need to estimate second derivatives at specific points to analyze their behavior, particularly in engineering and scientific computations where such measures are often needed.

Differentiation Matrix

In numerical analysis, a differentiation matrix is a powerful tool for finding derivatives of discretely sampled functions. It's essentially a matrix that, when multiplied by a vector of function values, returns the approximate derivatives at those points. This is similar to how a single derivative operation can be represented as a matrix-vector multiplication.

For the exercise involving the function \( f(x) = e^{x} \sin (10 x) \) and its second derivative, we construct an \( (n-2) \times n \) matrix \( D^{2} \) . Each row in \( D^{2} \) corresponds to a point \( x_i \) , and contains the coefficients that replicate the centered difference formula when multiplied by the vector of function values. This matrix representation is useful for systematic error analysis and is a basis for many numerical software implementations of differentiation.

Numerical Methods

Numerical methods encompass algorithms and techniques for approximating solutions to mathematical problems that cannot be solved exactly. They play a crucial role in engineering, physics, and computer science, especially when dealing with differential equations and functions that are difficult or impossible to manage analytically.

Numerical differentiation, as part of these methods, involves algorithms for calculating approximate derivatives of functions based on sampled points. The use of the centered difference formula for estimating second derivatives is one such method. It highlights how numerical methods can provide solutions where analytical ones are unattainable, yet it also underscores the importance of understanding the limitations and potential inaccuracies introduced by these approximations.

Error Analysis

Error analysis is integral to numerical methods, allowing us to gauge the reliability of our approximated solutions. In this context, we are often concerned with the maximum absolute error, which is the largest absolute value of the differences between the exact and approximated derivatives across all sampled points.

By calculating the maximum absolute error for different values of \( n \) (where \( n \) determines the number of intervals and hence the \( h \) value), we can observe the rate at which the error diminishes. Ideally, we expect an \( \mathcal{O}(n^{-2}) \) improvement in maximum absolute error with increased \( n \) for the centered difference method, confirming that the error is inversely proportional to the square of the number of intervals. Performing this error analysis provides confidence in our numerical methods and assists in optimizing calculations for accuracy and computational efficiency.

Chebyshev Differentiation Matrix

The Chebyshev differentiation matrix is a specialized form used in numerical analysis for approximating derivatives using Chebyshev polynomials. These polynomials form an orthogonal basis set, and sampling points at the Chebyshev nodes (the roots or extrema of the polynomials) minimizes numerical errors and instability, particularly in approximating higher derivatives.

For the given function \( f(x) = e^{x} \sin (10 x) \) , comparing the accuracy of second derivative approximations obtained via the centered difference formula and the Chebyshev differentiation matrix provides insight into the performance of different approaches. The Chebyshev method usually leads to superior accuracy and convergence properties, especially for functions with rapid oscillations or in cases where high-resolution approximations are required.