2.4 Recursive Rules and Romberg Integration
2.4 Recursive Rules and Romberg Integration
Theorem 7. 4(Successive Trapezoidal rules). Suppose that J> 1 and the points k=a+ kh subdivide [a, b into 2=2M subintervals of equal width h=(b-a)/2. The trapezoidal rules T(, h)and T(, 2h)obey the relationship T(, 2h) M +h∑f(a2-1
Definition 2. 3(Sequence of Trapezoidal Rules). Define T(0)=(h/2)(f(a)+ f(b)), which is the trapezoidal rule with step size h= b. Then for each J21 define T()=T(, h ), where T(, h)is the trapezoidal rule with step size h=(b-a)/
Corollary 7. 4 (Recursive Trapezoidal Rule). Start with T(0)=(h/2)(f(a)+ f(b)). Then a sequence of trapezoidal rules T()) is generated by the recur- SIve formula (=。-+b∑f(2x-)forJ=1,2,…,(246) where h=(b-a)/2and ck=a+hhI
Example 2. 11. Use the sequential trapezoidal rule to compute the approxi mations T(O), T(1), T(2), and T 3) for the integral fi d =In(5)-In(1) 1.609437912
Table 2. 4 The Nine Points Used to Compute T(3) and the Midpoints Required to Compute T(1), T(2), and T(3 x|f(x)=1(0)T(1)(2)T(3) 1.01.0000001.000000 1.50.666667 0.666667 2.00.500000 0.500000 2.50.400000 0.400000 3.00.333333 0.33333 3.50.285714 0.285714 4.00.250000 0.250000 4.50.22222 0.222222 500.2000000.200000
Theorem 2.5(Recursive Simpson Rules). Suppose that T()) is the sequence of trapezoidal rules generated by Corollary 2. 4. If J> 1 and S() is Simpson's rule for 2, subintervals of [a, b], then S() and the trapezoidal rules T(J-1)and T( ) obey the relationship S()=4 T(J-1 for =1.2 (251
Example 2. 12. Use the sequential Simpson rule to compute the approxi mations $(1), S(2), and S(3)for the integral of Example 2.11
Theorem 2.6(Recursive Boole Rules). Suppose that S( )) is the se. quence of Simpson's rules generated by Theorem 2.5. If 2 and B()is Boole's rule for 2'subintervals of a, b], then B()and Simpson's rules S(J-1) and S() obey the relationship B(n) 16S(J)-S(-1 for J=2, 3 (2.59)
Example 2. 13. Use the sequential Boole rule to compute the approxima tions B(2)and B(3)for the integral of Example 2.11
