EXERCISES

$3.$ Modify the M$-$file euler.m to implement the algorithm for Improved Euler. Call the new file

impeuler.m $($include the file in your report$).$

Test your code for the IVP y$=0\mathrm{.}5$ y$,$ y$(0)=4,$ in the interval $[0,2\mathrm{.}25],$ using N $=9$ steps.

First enter the function f $($t$,$ y$)=0\mathrm{.}5$ y as anonymous function and then enter the following:

$>>[$ t$9,$ y$9]=$ impeuler $($f $,[0,2\mathrm{.}25],-4,9)$; $\%$ use @ f if defined in separate m $-$ file

$>>[$ t$9,$ y$9]$

ans $=$

$0-4\mathrm{.}0000$

$0\mathrm{.}2500-4\mathrm{.}5312$

$0\mathrm{.}5000-5\mathrm{.}1331$

$0\mathrm{.}7500-5\mathrm{.}8148$

$1\mathrm{.}0000-6\mathrm{.}5871$

$1\mathrm{.}2500-7\mathrm{.}4619$

$1\mathrm{.}5000-8\mathrm{.}4529$

$1\mathrm{.}7500-9\mathrm{.}5756$

$2\mathrm{.}0000-10\mathrm{.}8474$

$2\mathrm{.}2500-12\mathrm{.}2880$

Compare your output with the one above. You should obtain the same values