Question

In Evercises \(2.5 .19-2.5 .22, \Delta\) is the forwand difference operator with spacing \(h>0\). Let \(f^{\prime \prime \prime}\) be bounded on an open interval containing \(x_{0}\) and \(x_{0}+2 h .\) Find a constant \(k\) such that the magnitude of the error in the approximation $$ f^{\prime}\left(x_{0}\right) \approx \frac{\Delta f\left(x_{0}\right)}{h}+k \frac{\Delta^{2} f\left(x_{0}\right)}{h^{2}} $$ is not greater than \(M h^{2}\), where \(M=\sup \left\\{\left|f^{\prime \prime \prime}(c)\right||| x_{0}

Short answer

#tag_title# The constant k #tag_content# The constant factor \(k\) that yields a good approximation of the first order derivative of the function \(f\) using forward differences and meets the given error threshold requirement is: $$ k = \frac{3}{2}. $$

Step-by-step solution

Taylor expansion of \(f(x_0 + h)\)

Using Taylor's theorem with the remainder term, we have: $$ f(x_0 + h) = f(x_0) + hf'(x_0) + \frac{h^2}{2!}f''(x_0) + \frac{h^3}{3!}f'''(c_1), $$ where \(c_1\) lies between \(x_0\) and \(x_0 + h\). #Step 2: Express given function using Taylor expansion# Similarly, we will expand the function \(f(x)\) using the Taylor series with a remainder term around \(x_0 + h\).

Taylor expansion of \(f(x_0 + 2h)\)

Using Taylor's theorem with the remainder term, we have: $$ f(x_0 + 2h) = f(x_0 + h) + hf'(x_0 + h) + \frac{h^2}{2!}f''(x_0 + h) + \frac{h^3}{3!}f'''(c_2), $$ where \(c_2\) lies between \(x_0 + h\) and \(x_0 + 2h\). #Step 3: Calculate the forward difference operator# Now, let's find the forward difference expressions for the function \(f(x)\).

Calculate the first order forward difference

The first order forward difference operator, \(\Delta f(x_0) = f(x_0 + h) - f(x_0)\). Using the Taylor expansion from Step 1, we get: $$ \Delta f(x_0) = hf'(x_0) + \frac{h^2}{2} f''(x_0) + \frac{h^3}{6} f'''(c_1). $$

Calculate the second order forward difference

Next, we find the second order forward difference operator, \(\Delta^2 f(x_0) = \Delta f(x_0 + h) - \Delta f(x_0)\). Using the Taylor expansions from Step 1 and Step 2, we get: $$ \Delta^2 f(x_0) = h^2 f''(x_{0}) + \frac{h^3}{2} [f'''(c_{2}) - f'''(c_{1})]. $$ #Step 4: Find the error term and the constant \(k\)# Now we will find the error term and manipulate the given expression for the approximation to find the constant \(k\).

Solve for the error term

Solving the expression for \(f'(x_0)\) in terms of \(\Delta f(x_0)\) and \(\Delta^2 f(x_0)\), we get: $$ f'(x_0) \approx \frac{\Delta f(x_0)}{h} + k \frac{\Delta^2 f(x_0)}{h^2}, $$ and $$ f'(x_0) \approx \frac{1}{h} (hf'(x_0) + \frac{h^2}{2} f''(x_0) + \frac{h^3}{6} f'''(c_1)) + k \frac{1}{h^2}(h^2 f''(x_{0}) + \frac{h^3}{2} [f'''(c_{2}) - f'''(c_{1})]). $$ Now, let's find the error term of the approximation: $$ \text{Error} = f'(x_0) - \left(\frac{\Delta f(x_0)}{h} + k\frac{\Delta^2 f(x_0)}{h^2}\right) = \frac{h^2}{6}(2k[f'''(c_2) - f'''(c_1)] - 3f'''(c_1)). $$

Find the value of \(k\) for the given requirements

With the given error threshold \(Mh^2\), we have: $$ \left|\frac{h^2}{6}(2k[f'''(c_2) - f'''(c_1)] - 3f'''(c_1))\right| \leq Mh^2. $$ Since the third derivative is bounded, we can define \(M = \sup\{|f'''(c)|\,:\, x_0 < c < x_0 + 2h\}\). Then, to find the \(k\) meeting the requirement, we get: $$ k = \frac{3}{2}. $$ Thus, the constant \(k\) should be \(\frac{3}{2}\) in order to meet the given error threshold requirement.

Key concepts

Taylor Expansion

In numerical differentiation, Taylor Expansion is a powerful tool that helps approximate the behavior of a function around a specific point. By using the Taylor series, we can represent a function in terms of its derivatives at a certain point. For instance, if we expand a function at point \(x_0\), the Taylor expansion up to the third derivative is:

• \( f(x_0 + h) = f(x_0) + hf'(x_0) + \frac{h^2}{2!}f''(x_0) + \frac{h^3}{3!}f'''(c_1) \)

Here, \(c_1\) is a point between \(x_0\) and \(x_0 + h\). This expansion includes a remainder term \(\frac{h^3}{3!}f'''(c_1)\) which indicates the error in approximation as we include more terms, we become more accurate in representing \(f(x)\) near \(x_0\).
In our problem, using successive Taylor series expansions, we approximate functions at \(x_0 + h\) and \(x_0 + 2h\), involving higher-order derivatives to refine our calculations and improve accuracy in numerical differentiation.

Forward Difference Operator

The Forward Difference Operator is used to approximate derivatives, making it a key tool in numerical methods, particularly when dealing with discrete data. The first order forward difference is calculated by:

• \(\Delta f(x_0) = f(x_0 + h) - f(x_0)\)

With the Taylor expansion formula, this becomes:

• \(hf'(x_0) + \frac{h^2}{2} f''(x_0) + \frac{h^3}{6} f'''(c_1)\)

This expression shows that the higher-order terms give you insights into how accurate the approximation is.
The second order forward difference, \(\Delta^2 f(x_0)\), further refines this by:

• \(h^2 f''(x_{0}) + \frac{h^3}{2} \left[f'''(c_{2}) - f'''(c_{1})\right]\)

These tools, combined, help approximate derivatives by offsetting errors through detailed difference calculations, a crucial step for precise approximation in numerical differentiation.

Error Analysis

Error Analysis in numerical differentiation provides insight into how the error accumulates and influences the final estimation. By analyzing the error introduced through approximation methods, we understand how close our computational approximation is to the actual derivative.When approximating \(f'(x_0)\), the error is not just from substituting \(\Delta f(x_0)\) and \(\Delta^2 f(x_0)\), but also from inherent method limitations. Here, the error term derived is:

• \(\text{Error} = \frac{h^2}{6}(2k[f'''(c_2) - f'''(c_1)] - 3f'''(c_1))\)

The task is to ensure this error doesn't surpass \(Mh^2\). With \(M\) as the bound on the third derivative, choosing \(k = \frac{3}{2}\) ensures that the error is within acceptable limits.
This set error threshold highlights the importance of correctly choosing parameters to minimize the error, a vital part of effective numerical approximation.