$\%$ Clear command window, close all graphs, clear workspace

clc$,$ close all, clear

$\%========================================================================$

fprintf$("============================================================$

$")$

fprintf$("$Project $1-$ Round$-$off vs Truncation Error

$")$

fprintf$("$Insert Your Name Here

$")$

display$($date$())$

fprintf$("============================================================$

$")$

$\%========================================================================$

format short e

$\%$ Define the function f$($x$)$

f $=$ @$($x$)$ ; $\%$ Insert Function f$($x$)$

df $=$ @$($x$)$ ; $\%$ Insert Exact First Derivative df$($x$)$

$\%$ Select x value to evaluate derivative

x $=$ ;

$\%\%$ Double$-$Precision

$\%$ loop through the exponents for each delta x value and evaluate the derivative

for i $=1$:$20$

$\%$ calculate the delta x

del$($i$,1)=10^(-$i$)$;

$\%$ Backward Finite Difference

backward$($i$,1)=$ ;

$\%$ Forward Finite Difference

forward$($i$,1)=$ ;

$\%$ Central Finite Difference

central$($i$,1)=$ ;

$\%$ calculate the errors compared to exact $($analytical$)$ derivative

backward$\_$error$($i$,1)=$ ;

forward$\_$error$($i$,1)=$ ;

central$\_$error$($i$,1)=$ ;

end

$\%$ Plot the error vs del x for each method on the log scale

figure

hold on

$\%$ Scale of log for axis

set$($gca$,$ 'XScale', 'log', 'YScale', 'log'$)$

$\%$ the loglog$()$ function is the same as plot$()$ but on log scale for both axes

loglog$()$

loglog$()$

loglog$()$

xlabel$(\text{'}\backslash$Delta$\text{'}),$ ylabel$(\text{'}$Error$\text{'})$

title$(\text{'}$Error on Fordward, Backward, and Central'$)$

legend$(\text{'}$Forward$\text{'},$'Backward','Central'$)$

grid on$,$ hold off

$\%$ assemble the results into a table $($actually a matrix$)$

Table $=[]$;

$\%$ print to screen

fprintf$("===========$ Double Precision Results $===========$

$")$

disp$($Table$)$

$\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%$

$\%\%$ Repeat everything, but in single precision

$\%$ Select x value to evaluate derivative

x $=$ single$($x$)$;

for i $=$ single$(1$:$20)$

end

$\%$ Plot the error vs del x for each method on the log scale

figure

hold on

$\%$ Scale of log for axis

set$($gca$,$ 'XScale', 'log', 'YScale', 'log'$)$

$\%$ the loglog$()$ function is the same as plot$()$ but on log scale for both axes

loglog$()$

loglog$()$

loglog$()$

xlabel$(\text{'}\backslash$Delta$\text{'}),$ ylabel$(\text{'}$Error$\text{'})$

title$(\text{'}$Error on Fordward, Backward, and Central'$)$

legend$(\text{'}$Forward$\text{'},$'Backward','Central'$)$

grid on

$\%$ assemble the results into a table $($actually a matrix$)$

fprintf$("$

$")$

Table $=[]$;

$\%$ print to screen

fprintf$("===========$ Single Precision Results $===========$

$")$

disp$($Table$)$;

$\%\%$ Machine epsilon