Numerical Analysis/Romberg Example

Use Romberg Integration to compute $$ R_{3,3} $$ for the following integral $$ \int_{0}^{\frac{\pi}{2}}cos x \,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})]

$$

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

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

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

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

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