Chapter 23, Problem 1P

Compute forward and backward difference approximations of $O\left(h\right)$ and $O\left(h^2\right)$, and central difference approximations of $O\left(h^4\right)$ for the first derivative of $y=\cos x$ at $x=\pi /4$ using a value of $h=\pi /12$. Estimate the true percent relative error ${\epsilon }_t$ for each approximation.

To determine

To calculate: The forward and backward difference approximations of $O\left(h\right)$ and $O\left(h^2\right)$, and central difference approximations of $O\left(h^2\right)$ and $O\left(h^4\right)$ for the first derivative of $y=\cos x$ at $x=\frac{\pi }{4}$ using a value of $h=\frac{\pi }{12}$. Also find the true percent error $\epsilon$, for each approximation.

Answer to Problem 1P

Solution:

Forward difference approximation of $O\left(h\right)$ is $-0.79108963$ with true percent error $\epsilon =-11.877\%$

Forward difference approximation of $O\left(h^2\right)$ is $-0.72601275$ with true percent error $\epsilon =-2.674\%$

Backward difference approximation of $O\left(h\right)$ is $-0.60702442$ with true percent error $\epsilon =14.154\%$

Backward difference approximation of $O\left(h^2\right)$ is $-0.71974088$ with true percent error $\epsilon =-1.787\%$

Central difference approximation of $O\left(h^2\right)$ is $-0.69905703$ with true percent error $\epsilon =1.138\%$

Central difference approximation of $O\left(h^4\right)$ is $-0.70699696$ with true percent error $\epsilon =0.016\%$

Explanation of Solution

Given information:

Function, $y=\cos x$

Step size, $h=\frac{\pi }{12}$

The initial value of x, $x=\frac{\pi }{4}$

Formula used:

Forward difference approximation of $O\left(h\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{f\left(x_{i+1}\right)-f\left(x_i\right)}{h}$

Forward difference approximation of $O\left(h^2\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{-f\left(x_{i+2}\right)+4f\left(x_{i+1}\right)-3f\left(x_i\right)}{2h}$

Backward difference approximation of $O\left(h\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{f\left(x_i\right)-f\left(x_{i-1}\right)}{h}$

Backward difference approximation of $O\left(h^2\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{3f\left(x_i\right)-4f\left(x_{i-1}\right)+f\left(x_{i-2}\right)}{2h}$

Central difference approximation of $O\left(h^2\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{f\left(x_{i+1}\right)-f\left(x_{i-1}\right)}{2h}$

Central difference approximation of $O\left(h^4\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{-f\left(x_{i+2}\right)+8f\left(x_{i+1}\right)-8f\left(x_{i-1}\right)+f\left(x_{i-2}\right)}{12h}$

$x_{i+1}=x_i+h$ Here $i=0,1,2,\mathrm{...}$

True percent relative error is,

$\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100$

Calculation:

Consider the function,

$f\left(x\right)=\cos x$

First derivation of the function is,

$f^{\prime }\left(x\right)=-\sin x$

Thus, the true value of the first derivative of $y=\cos x$ at $x=\frac{\pi }{4}$ is,

$\begin{array}{c}{f}^{\prime }\left(\frac{\pi }{4}\right)=-\sin \left(\frac{\pi }{4}\right)\\ =-0.70710678\end{array}$

The value of $x_{i-2}$ is,

$\begin{array}{c}{x}_{i-2}=x_i-2h\\ =\frac{\pi }{4}-2\left(\frac{\pi }{12}\right)\\ =\frac{\pi }{12}\\ =0.261799388\end{array}$

The value of the function at $x_{i-2}=0.261799388$ is,

$\begin{array}{c}f\left(0.261799388\right)=\cos \left(0.261799388\right)\\ =0.965925826\end{array}$

The value of $x_{i-1}$ is,

$\begin{array}{c}{x}_{i-1}=x_i-h\\ =\frac{\pi }{4}-\left(\frac{\pi }{12}\right)\\ =0.523598776\end{array}$

The value of the function at $x_{i-1}=0.523598776$ is,

$\begin{array}{c}f\left(0.523598776\right)=\cos \left(0.523598776\right)\\ =0.866025404\end{array}$

The value of $x_i$ is,

$\begin{array}{c}{x}_i=\frac{\pi }{4}\\ =0.785398163\end{array}$

The value of the function at $x_i=0.785398163$ is,

$\begin{array}{c}f\left(0.785398163\right)=\cos \left(0.785398163\right)\\ =0.707106781\end{array}$

The value of $x_{i+1}$ is,

$\begin{array}{c}{x}_{i+1}=x_i+h\\ =\frac{\pi }{4}+\frac{\pi }{12}\\ =1.047197551\end{array}$

The value of the function at $x_{i+1}=1.047197551$ is,

$\begin{array}{c}f\left(1.047197551\right)=\cos \left(1.047197551\right)\\ =0.5\end{array}$

The value of $x_{i+2}$ is,

$\begin{array}{c}{x}_{i+2}=x_i+2h\\ =\frac{\pi }{4}+2\left(\frac{\pi }{12}\right)\\ =1.308996936\end{array}$

The value of the function at $x_{i+2}=1.308996936$ is,

$\begin{array}{c}f\left(1.308996936\right)=\cos \left(1.308996936\right)\\ =0.258819045\end{array}$

Forward difference approximation of $O\left(h\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{f\left(x_{i+1}\right)-f\left(x_i\right)}{h}$

For $y=\cos x$ forward difference approximation is,

$\begin{array}{c}{f}^{\prime }\left(x_i\right)=\frac{0.5-0.707106781}{\left(\frac{\pi }{12}\right)}\\ =-\frac{0.207106781}{0.261799388}\\ =-0.79108963\end{array}$

True percent error is,

$\begin{array}{c}\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100\\ =\left(\frac{\left(-0.70710678\right)-\left(-0.79108963\right)}{\left(-0.70710678\right)}\right)\times 100\\ =-11.877\%\end{array}$

Forward difference approximation of $O\left(h^2\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{-f\left(x_{i+2}\right)+4f\left(x_{i+1}\right)-3f\left(x_i\right)}{2h}$

For $y=\cos x$ forward difference approximation is,

$\begin{array}{c}{f}^{\prime }\left(x_i\right)=\frac{-0.258819045+4\left(0.5\right)-3\left(0.707106781\right)}{2\left(\frac{\pi }{12}\right)}\\ =-\frac{0.380139388}{\left(\frac{\pi }{6}\right)}\\ =-0.72601275\end{array}$

True percent error is,

$\begin{array}{c}\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100\\ =\left(\frac{\left(-0.70710678\right)-\left(-0.72601275\right)}{\left(-0.70710678\right)}\right)\times 100\\ =-2.674\%\end{array}$

Backward difference approximation of $O\left(h\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{f\left(x_i\right)-f\left(x_{i-1}\right)}{h}$

For $y=\cos x$ backward difference approximation is,

$\begin{array}{c}{f}^{\prime }\left(x_i\right)=\frac{0.707106781-0.866025404}{\left(\frac{\pi }{12}\right)}\\ =-\frac{0.15891623}{0.261799388}\\ =-0.60702442\end{array}$

True percent error is,

$\begin{array}{c}\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100\\ =\left(\frac{\left(-0.70710678\right)-\left(-0.60702442\right)}{\left(-0.70710678\right)}\right)\times 100\\ =14.154\%\end{array}$

Backward difference approximation of $O\left(h^2\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{3f\left(x_i\right)-4f\left(x_{i-1}\right)+f\left(x_{i-2}\right)}{2h}$

For $y=\cos x$ backward difference approximation is,

$\begin{array}{c}{f}^{\prime }\left(x_i\right)=\frac{3\left(0.707106781\right)-4\left(0.866025404\right)+0.965925826}{2\left(\frac{\pi }{12}\right)}\\ =-\frac{0.376855447}{\left(\frac{\pi }{6}\right)}\\ =-0.71974088\end{array}$

True percent error is,

$\begin{array}{c}\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100\\ =\left(\frac{\left(-0.70710678\right)-\left(-0.71974088\right)}{\left(-0.70710678\right)}\right)\times 100\\ =-1.787\%\end{array}$

Central difference approximation of $O\left(h^2\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{f\left(x_{i+1}\right)-f\left(x_{i-1}\right)}{2h}$

For $y=\cos x$ central difference approximation is,

$\begin{array}{c}{f}^{\prime }\left(x_i\right)=\frac{0.5-0.866025404}{2\left(\frac{\pi }{12}\right)}\\ =-\frac{0.366025404}{\left(\frac{\pi }{6}\right)}\\ =-0.69905703\end{array}$

True percent error is,

$\begin{array}{c}\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100\\ =\left(\frac{\left(-0.70710678\right)-\left(-0.69905703\right)}{\left(-0.70710678\right)}\right)\times 100\\ =1.138\%\end{array}$

Central difference approximation of $O\left(h^4\right)$ is,

$f^{\prime }\left(x_i\right)=\frac{-f\left(x_{i+2}\right)+8f\left(x_{i+1}\right)-8f\left(x_{i-1}\right)+f\left(x_{i-2}\right)}{12h}$

For $y=\cos x$ central difference approximation is,

$\begin{array}{c}{f}^{\prime }\left(x_i\right)=\frac{-\left(0.258819045\right)+8\left(0.5\right)-8\left(0.866025404\right)+0.965925826}{12\left(\frac{\pi }{12}\right)}\\ =-\frac{2.221096451}{\pi }\\ =-0.70699696\end{array}$

True percent error is,

$\begin{array}{c}\epsilon =\left(\frac{\text{exact value }-\text{ numerical value}}{\text{exact value}}\right)\times 100\\ =\left(\frac{\left(-0.70710678\right)-\left(-0.70699696\right)}{\left(-0.70710678\right)}\right)\times 100\\ =0.016\%\end{array}$

Therefore, Forward difference approximation of $O\left(h\right)$ is $-0.79108963$ with true percent error $\epsilon =-11.877\%$

Forward difference approximation of $O\left(h^2\right)$ is $-0.72601275$ with true percent error $\epsilon =-2.674\%$

Backward difference approximation of $O\left(h\right)$ is $-0.60702442$ with true percent error $\epsilon =14.154\%$

Backward difference approximation of $O\left(h^2\right)$ is $-0.71974088$ with true percent error $\epsilon =-1.787\%$

Central difference approximation of $O\left(h^2\right)$ is $-0.69905703$ with true percent error $\epsilon =1.138\%$

Central difference approximation of $O\left(h^4\right)$ is $-0.70699696$ with true percent error $\epsilon =0.016\%$