Use Romberg Integration to compute R3,3{\displaystyle R_{3,3}} for the following integral∫0π2cosxdx{\displaystyle \int _{0}^{\frac {\pi }{2}}cosx\,dx}
Solution:
R1,1=π4[cos(0)+cos(π2)]{\displaystyle R_{1,1}={\frac {\pi }{4}}[cos(0)+cos({\frac {\pi }{2}})]}
R1,1=π4{\displaystyle R_{1,1}={\frac {\pi }{4}}}
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=(12)[π4+π2cos(π4)]{\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[{\frac {\pi }{4}}+{\frac {\pi }{2}}cos\left({\frac {\pi }{4}}\right)]}R2,1=1.178023457{\displaystyle R_{2,1}=1.178023457}
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=(12)[1.178023457+π4(cos(π8)+cos(3π8)]{\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[1.178023457+{\frac {\pi }{4}}(cos({\frac {\pi }{8}})+cos({\frac {3\pi }{8}})]}R3,1=1.374317658{\displaystyle R_{3,1}=1.374317658}
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=1.178023457+.39262529363{\displaystyle R_{2,2}=1.178023457+{\frac {.3926252936}{3}}}R2,2=1.308898555{\displaystyle R_{2,2}=1.308898555}
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=1.374317658+.1962942013{\displaystyle R_{3,2}=1.374317658+{\frac {.196294201}{3}}}R3,2=1.439749058{\displaystyle R_{3,2}=1.439749058}
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=1.439749058+.130850503315{\displaystyle R_{3,3}=1.439749058+{\frac {.1308505033}{15}}}R3,3=1.448472425{\displaystyle R_{3,3}=1.448472425}