Question

Let $$ \begin{aligned} &g=\frac{1}{h^{2}}\left(f\left(x_{0}-h\right)-2 f\left(x_{0}\right)+f\left(x_{0}+h\right)\right) \\ &\hat{g}=\frac{1}{4 h^{2}}\left(f\left(x_{0}-2 h\right)-2 f\left(x_{0}\right)+f\left(x_{0}+2 h\right)\right) \end{aligned} $$ Show that if \(f\) has three bounded derivatives in a neighborhood of \(x_{0}\) that includes \(\left[x_{0}-\right.\) \(\left.2 h, x_{0}+2 h\right]\), then the computable expression $$ \hat{e}=\frac{g-\hat{g}}{3} $$ provides an estimation for the error in \(f^{\prime \prime}\left(x_{0}\right)-g\) accurate up to \(\mathcal{O}(h)\). Note that here \(g\) is the actual approximation used, and \(\hat{g}\) is an auxiliary expression employed in computing the error estimation for \(g\).

Short answer

Question: Show that the error estimation for the second derivative approximation error is accurate up to 𝑂(ℎ), given that the function \(f\) has three bounded derivatives in the neighborhood of \(x_0\) that includes \([x_0 - 2h, x_0 + 2h]\). Answer: The error estimation for the second derivative approximation error is given by \(\hat{e} = -\frac{1}{6}h f'''(x_0) + \mathcal{O}(h^2)\), which shows that the error estimation is accurate up to \(\mathcal{O}(h)\).

Step-by-step solution

Expand the Taylor series of \(f\) up to \(\mathcal{O}(h^3)\)

Expand the Taylor series of the function \(f\) around \(x_0\) up to the third derivative term: $$ f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{6}f'''(x_0)(x-x_0)^3 + \mathcal{O}(h^4). $$

Substitute the Taylor series expansions into the expressions for \(g\) and \(\hat{g}\)

Using the Taylor series expansions, substitute \(x=x_0-h\) and \(x=x_0+h\) into the equations for \(g\) and \(\hat{g}\) and simplify: $$ \begin{aligned} g &= \frac{1}{h^2}\left(f(x_0-h)-2f(x_0)+f(x_0+h)\right) \\ &= \frac{1}{h^2}\left(\left( -h^2f''(x_0) - \frac{1}{2}h^3f'''(x_0) + \mathcal{O}(h^4) \right)- 2f(x_0) + \left(h^2f''(x_0) - \frac{1}{2}h^3f'''(x_0) + \mathcal{O}(h^4) \right) \right) \\ &= f''(x_0) - \frac{1}{3}h f'''(x_0) + \mathcal{O}(h^2) \end{aligned} $$ and $$ \begin{aligned} \hat{g} &=\frac{1}{4 h^{2}}\left(f\left(x_{0}-2 h\right)-2 f\left(x_{0}\right)+f\left(x_{0}+2 h\right)\right) \\ &=\frac{1}{4 h^2}\left(\left(4h^2f''(x_0)-\frac{4}{3}h^3f'''(x_0)+\mathcal{O}(h^4) \right) - 2f(x_0) + \left(4h^2f''(x_0)+\frac{4}{3}h^3f'''(x_0) + \mathcal{O}(h^4) \right)\right) \\ &= f''(x_0) + \frac{1}{6}h f'''(x_0) + \mathcal{O}(h^2). \end{aligned} $$

Formulate the error estimation \(\hat{e}\)

Using the expressions for \(g\) and \(\hat{g}\) obtained in Step 2, compute the error estimation \(\hat{e}\): $$ \begin{aligned} \hat{e} &= \frac{g-\hat{g}}{3} \\ &= \frac{\left(f''(x_0) - \frac{1}{3}h f'''(x_0) + \mathcal{O}(h^2) \right)-\left(f''(x_0) + \frac{1}{6}h f'''(x_0) + \mathcal{O}(h^2)\right)}{3} \\ &= \frac{-\frac{1}{2}h f'''(x_0) + \mathcal{O}(h^2)}{3} \end{aligned} $$

Verify that the error estimation is accurate up to \(\mathcal{O}(h)\)

Looking at the expression for \(\hat{e}\), we can see that the error estimation is given by: $$ \hat{e} = -\frac{1}{6}h f'''(x_0) + \mathcal{O}(h^2). $$ Since the leading term in \(\hat{e}\) has order of \(h\), we can conclude that the error estimation for \(f''(x_0)-g\) is accurate up to \(\mathcal{O}(h)\).

Key concepts

Taylor Series Expansion

Understanding the Taylor series expansion is crucial for analyzing numerical differentiation methods. Simplified, a Taylor series represents a function as an infinite sum of terms that are calculated from the values of its derivatives at a single point. It allows us to express the value of a function at one point in terms of the function's value and its derivatives at another point.

In the context of numerical differentiation, we use the Taylor series to approximate the values of a function using a polynomial constructed from the function's derivatives. This is particularly useful when the exact expression of a function is complex or unknown, and we only know its values at discrete points.

The formula for a Taylor series expansion of a function f around a point x₀ up to the third derivative is given by:
\[f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{6}f'''(x_0)(x-x_0)^3 + \mathcal{O}(h^4)\].
This expansion includes higher-order terms represented by the big O notation \(\mathcal{O}(h^4)\), which indicate the remainder of the series and its contribution to the overall error in the approximation.

Second Derivative Approximation

When we approximate the second derivative, we often employ finite difference methods. The exercise provided uses a central difference approach to estimate the second derivative of a function f at a point x₀. The approximation g is given by the formula:
\[g=\frac{1}{h^{2}}(f(x_{0}-h)-2f(x_{0})+f(x_{0}+h))\].
On the other hand, \hat{g} is a slightly altered approximation where the step size is doubled.
As the step size h becomes smaller, the accuracy of the approximation for the second derivative increases. However, taking h too small can introduce numerical errors due to limitations in computer arithmetic. Thus, a balance must be struck in choosing a suitable h that avoids such errors while still providing a good approximation of f''(x_{0}).

It’s important to note that the accuracy of the second derivative approximation also depends on the smoothness of the function being differentiated. Functions with continuous higher derivatives are more suited for such approximations, as they align with the assumptions made in the Taylor series expansion.

Error Analysis in Numerical Methods

Error analysis is a prominent part of numerical methods which involves assessing the accuracy and stability of numerical approximations. Numerical errors can generally be categorized into truncation errors and round-off errors.

Truncation errors arise when we approximate a mathematically precise procedure with a simpler one that doesn't capture the full complexity of the original. The term 'truncation' often refers to the discarding of the tail of the Taylor series. In our case, when approximating the second derivative, the truncation error is concerned with the terms in the Taylor series that are omitted.

Round-off errors occur due to the limited precision with which computers represent real numbers. Calculations with these approximated numbers consequently lead to errors which could potentially accumulate.

The exercise demonstrates a method to estimate the error of a second derivative approximation by utilizing an auxiliary expression \(\hat{g}\). The computable error estimation \hat{e} gives us a way to quantify the truncation error. The formulation of \(\hat{e}\) takes advantage of the differing truncation errors within g and \hat{g} to create an estimate that cancels out the leading order of error, thereby providing a measure which is 'accurate up to \(\mathcal{O}(h)\)'. This concept emphasizes that the error in the approximation does not disappear but is reduced to a magnitude that is dependent on the size of the step h taken.