Numerical Analysis/Romberg Excercise

Use Romberg Integration to compute $$R_{3,3} $$ for $$ \int_0^1 x^2 e^{-x} \,dx$$ Solution: $$ R_{1,1} = \frac{h_{1}}{2}[f(0)+f(1)] $$ $$ R_{1,1} = \frac{1}{2}[0+\frac{1}{e}] $$ $$ R_{1,1} = .1839397206 $$

$$ R_{2,1} = \left(\frac{1}{2}\right)[R_{1,1}+h_{1}f(a+h_{2})] $$ $$ R_{2,1} = .1379547904 $$

$$ R_{3,1} = \left(\frac{1}{2}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))] $$ $$ R_{3,1} = .1475727039 $$

$$ R_{2,2} = R_{2,1}+ \frac{R_{2,1}-R_{1,1}}{4-1} $$ $$ R_{2,2} = .1226264803 $$

$$ R_{3,2} = R_{3,1} + \frac{R_{3,1}-R_{2,1}}{4-1} $$ $$ R_{3,2} = .1507786751 $$

$$ R_{3,3} = R_{3,2} + \frac{R_{3,2}-R_{2,2}}{16-1} $$ $$ R_{3,3} = .1526554881 $$