Use Romberg Integration to compute R 3 , 3 {\displaystyle R_{3,3}} for the following integral ∫ 0 π 2 c o s x d x {\displaystyle \int _{0}^{\frac {\pi }{2}}cosx\,dx}
Solution:
R 1 , 1 = π 4 [ c o s ( 0 ) + c o s ( π 2 ) ] {\displaystyle R_{1,1}={\frac {\pi }{4}}[cos(0)+cos({\frac {\pi }{2}})]}
R 1 , 1 = π 4 {\displaystyle R_{1,1}={\frac {\pi }{4}}}
R 2 , 1 = ( 1 2 ) [ R 1 , 1 + h 1 f ( a + h 2 ) ] {\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[R_{1,1}+h_{1}f(a+h_{2})]}
R 2 , 1 = ( 1 2 ) [ π 4 + π 2 c o s ( π 4 ) ] {\displaystyle 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 {\displaystyle R_{2,1}=1.178023457}
R 3 , 1 = ( 1 2 ) [ R 2 , 1 + h 2 ( f ( a + h 3 ) + f ( a + 3 h 3 ) ) ] {\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]} R 3 , 1 = ( 1 2 ) [ 1.178023457 + π 4 ( c o s ( π 8 ) + c o s ( 3 π 8 ) ] {\displaystyle 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 {\displaystyle R_{3,1}=1.374317658}
R 2 , 2 = R 2 , 1 + R 2 , 1 − R 1 , 1 4 − 1 {\displaystyle R_{2,2}=R_{2,1}+{\frac {R_{2,1}-R_{1,1}}{4-1}}} R 2 , 2 = 1.178023457 + .3926252936 3 {\displaystyle R_{2,2}=1.178023457+{\frac {.3926252936}{3}}} R 2 , 2 = 1.308898555 {\displaystyle R_{2,2}=1.308898555}
R 3 , 2 = R 3 , 1 + R 3 , 1 − R 2 , 1 4 − 1 {\displaystyle R_{3,2}=R_{3,1}+{\frac {R_{3,1}-R_{2,1}}{4-1}}} R 3 , 2 = 1.374317658 + .196294201 3 {\displaystyle R_{3,2}=1.374317658+{\frac {.196294201}{3}}} R 3 , 2 = 1.439749058 {\displaystyle R_{3,2}=1.439749058}
R 3 , 3 = R 3 , 2 + R 3 , 2 − R 2 , 2 16 − 1 {\displaystyle R_{3,3}=R_{3,2}+{\frac {R_{3,2}-R_{2,2}}{16-1}}} R 3 , 3 = 1.439749058 + .1308505033 15 {\displaystyle R_{3,3}=1.439749058+{\frac {.1308505033}{15}}} R 3 , 3 = 1.448472425 {\displaystyle R_{3,3}=1.448472425}
for the following integral
![{\displaystyle R_{1,1}={\frac {\pi }{4}}[cos(0)+cos({\frac {\pi }{2}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3f975de7d0b4c1f8d1bbddc5f3858dbd850477)
![{\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[R_{1,1}+h_{1}f(a+h_{2})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0eb68efb498a0c1bcc0c41162ad941d37272440)
![{\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[{\frac {\pi }{4}}+{\frac {\pi }{2}}cos\left({\frac {\pi }{4}}\right)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3aad4fb08bc8652cf1af74e318a614f6ca1006)
![{\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2a9d2c2f22ba9a362f98530065acde7717ad4f)
![{\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[1.178023457+{\frac {\pi }{4}}(cos({\frac {\pi }{8}})+cos({\frac {3\pi }{8}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/368c23b4dd27aefdab30c3c7856ccfd44d4c5419)
