Use Romberg Integration to compute R3,3{\displaystyle R_{3,3}} for ∫01x2e−xdx{\displaystyle \int _{0}^{1}x^{2}e^{-x}\,dx}
Solution:
R1,1=h12[f(0)+f(1)]{\displaystyle R_{1,1}={\frac {h_{1}}{2}}[f(0)+f(1)]}R1,1=12[0+1e]{\displaystyle R_{1,1}={\frac {1}{2}}[0+{\frac {1}{e}}]}R1,1=.1839397206{\displaystyle R_{1,1}=.1839397206}
R2,1=(12)[R1,1+h1f(a+h2)]{\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[R_{1,1}+h_{1}f(a+h_{2})]}R2,1=.1379547904{\displaystyle R_{2,1}=.1379547904}
R3,1=(12)[R2,1+h2(f(a+h3)+f(a+3h3))]{\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]}R3,1=.1475727039{\displaystyle R_{3,1}=.1475727039}
R2,2=R2,1+R2,1−R1,14−1{\displaystyle R_{2,2}=R_{2,1}+{\frac {R_{2,1}-R_{1,1}}{4-1}}}R2,2=.1226264803{\displaystyle R_{2,2}=.1226264803}
R3,2=R3,1+R3,1−R2,14−1{\displaystyle R_{3,2}=R_{3,1}+{\frac {R_{3,1}-R_{2,1}}{4-1}}}R3,2=.1507786751{\displaystyle R_{3,2}=.1507786751}
R3,3=R3,2+R3,2−R2,216−1{\displaystyle R_{3,3}=R_{3,2}+{\frac {R_{3,2}-R_{2,2}}{16-1}}}R3,3=.1526554881{\displaystyle R_{3,3}=.1526554881}