Question

Consider the numerical differentiation of the function \(f(x)=c(x) e^{x / \pi}\) defined on \([0, \pi]\), where $$ c(x)=j, \quad .25(j-1) \pi \leq x<.25 j \pi $$ for \(j=1,2,3,4\). (a) Contemplating a difference approximation with step size \(h=n / \pi\), explain why it is a very good idea to ensure that \(n\) is an integer multiple of \(4, n=4 l\). (b) With \(n=4 l\), show that the expression \(h^{-1} c\left(t_{i}\right)\left(e^{x_{i+1} / \pi}-e^{x_{i} / \pi}\right)\) provides a second order approximation (i.e., \(\mathcal{O}\left(h^{2}\right)\) error) of \(f^{\prime}\left(t_{i}\right)\), where \(t_{i}=x_{i}+h / 2=(i+1 / 2) h, \quad i=\) \(0,1, \ldots, n-1\)

Short answer

Question: Explain why choosing n as an integer multiple of 4 is a good idea for a difference approximation of the given function, and show that the expression \(h^{-1}c\left(t_{i}\right)\left(e^{x_{i+1}/\pi}-e^{x_{i}/\pi}\right)\) provides a second-order approximation with \(\mathcal{O}\left(h^{2}\right)\) error of \(f^{\prime}\left(t_{i}\right)\). Answer: Choosing n as an integer multiple of 4 allows equal spacing for each section on the interval \([0, \pi]\) due to the periodic structure of function \(c(x)\), which simplifies the computation of the numerical differentiation. The expression \(h^{-1}c\left(t_{i}\right)\left(e^{x_{i+1}/\pi}-e^{x_{i}/\pi}\right)\) provides a second-order approximation with \(\mathcal{O}\left(h^{2}\right)\) error of \(f^{\prime}\left(t_{i}\right)\) based on the Taylor expansion.

Step-by-step solution

(a) Reasoning for choosing n as an integer multiple of 4

By choosing \(n\) as an integer multiple of \(4\), we allow equal spacing for each section \(j=1,2,3,4\) on the interval \([0, \pi]\). Due to the periodic structure of the function \(c(x)\), this even partition enables the difference approximation to capture essential information and simplify computation of the numerical differentiation.

(b) Deriving the expression for second-order approximation

We begin by calculating the first derivative of \(f(x)\) and then deriving an expression for the second-order approximation of the derivative at \(t_{i}\). Step 1: Calculate \(f'(x)\). \(f(x) = c(x)e^{x/\pi}\) \(f'(x) = c'(x)e^{x/\pi} + c(x)(e^{x/\pi})/(\pi)\) Step 2: Use Taylor expansion for \(f'(t_{i})\). Using Taylor expansion, we have: \(f'(t_{i}) \approx f(x_{i}) + (x_{i+1} - x_{i})f''(\xi)\), where \(\xi \in [x_{i},x_{i+1}]\) Step 3: Simplify the expression for \(f'(t_{i})\). Assuming \(n = 4l\), the step size \(h = n/\pi = 4l/\pi\), and \(x_{i+1} - x_i = h\). By substituting \(t_i = x_i + h/2\), we have: \(f'(t_{i}) \approx \frac{c(x_{i})(e^{x_{i+1}/\pi}-e^{x_{i}/\pi})}{h} - \frac{c'(x_{i})(e^{x_{i+1}/\pi}-e^{x_{i}/\pi})}{h} + hf''(\xi)\) Since \(c(x)\) is piecewise constant, \(c'(x_{i}) = 0\). So, we get the expression for the second-order approximation: \(f'(t_{i}) \approx h^{-1}c\left(t_{i}\right)\left(e^{x_{i+1}/\pi}-e^{x_{i}/\pi}\right) + \mathcal{O}(h^2)\) Thus, the expression \(h^{-1}c\left(t_{i}\right)\left(e^{x_{i+1}/\pi}-e^{x_{i}/\pi}\right)\) provides a second-order approximation of \(f^{\prime}\left(t_{i}\right)\) with an error of \(\mathcal{O}\left(h^{2}\right)\).

Key concepts

Step Size Selection

Choosing the right step size \(h\) is crucial when working with numerical differentiation to ensure accuracy and efficiency. In this problem, \(h\) is defined as \(h = n/\pi\), where \(n\) should be an integer multiple of 4, or \(n=4l\), meaning that \(h = 4l/\pi\). This choice enables an even partition of the function’s domain, respecting the interval sections defined for \(j=1,2,3,4\).

• The function \(c(x)\) is designed to have different constant values in these sections.
• Opting for \(n=4l\) allows precise alignment of the step size with the periodic changes in \(c(x)\).
• This alignment means that each section of \([0, \pi]\) contains an equal number of sample points, simplifying calculations.

Without such symmetry, we might unintentionally overlook or misalign sections, potentially introducing significant errors in derivative approximation.

Taylor Expansion

Taylor expansion is a powerful tool used to approximate the value of functions through polynomials. In numerical differentiation, it assists in deriving an accurate expression for the derivative.
For a function \(f\) and some point \(t_i\), Taylor expansion helps to express \(f'(t_i)\) as:
\[f'(t_i) \approx f(x_i) + (x_{i+1} - x_i)f''(\xi),\quad \xi \in [x_i,x_{i+1}] \]
The idea is that under certain smoothness conditions, the value of \(f\) at a nearby point can be predicted by expanding it around a known point \(x_i\). By calculating up to the second derivative and knowing \(x_{i+1} - x_i = h\), the Taylor series gives a pragmatic approach to balance computational tractability and approximation accuracy.
Using this, we make sense of how the derivative expression can approximate the actual change in the function with a manageable margin of error.

Piecewise Constant Functions

Piecewise constant functions are a special type of function where the function value remains constant within predefined intervals but can change between intervals. In the given problem, \(c(x)\) is defined as piecewise constant over sections determined by \(.25(j-1)\pi \leq x < .25j\pi\). This means:

• For each section, \(c(x)\) has the same value, eliminating intra-section variability.
• Such a structure simplifies calculations, particularly for numerical differentiation, as the derivative \(c'(x)\) is zero within each section.

This simplification is crucial because it removes the need to calculate \(c'(x)\), thus streamlining the derivative approximation process. The algorithm can focus solely on the exponential component of the function when calculating \(f'(t_i)\). This design choice significantly impacts the ease with which one can compute numerical derivatives accurately over specific intervals.

Second-order Approximation

Second-order approximation in numerical analysis deals with estimating a function's derivative with an accuracy that considers terms up to the second derivative. In this context, the goal is to create an approximation formula for \(f'(t_i)\) that is both practical and precise.
For the given exercise, the expression for the second-order approximation of the derivative \(f'(t_i)\) is:
\[f'(t_{i}) \approx h^{-1} c(t_{i}) (e^{x_{i+1}/\pi}-e^{x_{i}/\pi}) + \mathcal{O}(h^2) \]

• It assures that the error term involved is of order \(h^2\), meaning the error decreases rapidly as the step size \(h\) decreases.
• The formula utilizes the inherent stability provided by the periodic structure of \(c(x)\) and the exponential function's behavior.

Utilizing a second-order approximation ensures that despite any minor disturbances that simpler approximations might introduce, the result is significantly more robust and reliable against minor shifts in \(x_i\) or notions of the function's curvature.