Numerical Analysis/Romberg Example

Use Romberg Integration to compute $R_{3,3}$ for the following integral
$\int _{0}^{\frac {\pi }{2}}cosx\,dx$

Solution:

$R_{1,1}={\frac {\pi }{4}}[cos(0)+cos({\frac {\pi }{2}})]$

$R_{1,1}={\frac {\pi }{4}}$

$R_{2,1}=\left({\frac {1}{2}}\right)[R_{1,1}+h_{1}f(a+h_{2})]$

$R_{2,1}=\left({\frac {1}{2}}\right)[{\frac {\pi }{4}}+{\frac {\pi }{2}}cos\left({\frac {\pi }{4}}\right)]$
$R_{2,1}=1.178023457$

$R_{3,1}=\left({\frac {1}{2}}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]$
$R_{3,1}=\left({\frac {1}{2}}\right)[1.178023457+{\frac {\pi }{4}}(cos({\frac {\pi }{8}})+cos({\frac {3\pi }{8}})]$
$R_{3,1}=1.374317658$

$R_{2,2}=R_{2,1}+{\frac {R_{2,1}-R_{1,1}}{4-1}}$
$R_{2,2}=1.178023457+{\frac {.3926252936}{3}}$
$R_{2,2}=1.308898555$

$R_{3,2}=R_{3,1}+{\frac {R_{3,1}-R_{2,1}}{4-1}}$
$R_{3,2}=1.374317658+{\frac {.196294201}{3}}$
$R_{3,2}=1.439749058$

$R_{3,3}=R_{3,2}+{\frac {R_{3,2}-R_{2,2}}{16-1}}$
$R_{3,3}=1.439749058+{\frac {.1308505033}{15}}$
$R_{3,3}=1.448472425$