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 ak=a+kh subdivide [ a, b] into 2=2M subintervals of equal width h=(6-a)/2. The trapezoidal rules T(f, h) and T(, 2h)obey the relationshi T(, 2h) +b∑(2-1 2.45 k=1
Definition 2. 3(Sequence of Trapezoidal Rules). Define T(0)=(h /2)(f(a)+ f(6)), which is the trapezoidal rule with step size h=b. Then for each 21 define T()=T(,h), where T(, h)is the trapezoidal rule with step size h=(6 -a)/0
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 T(J-1) +b∑f(x2-1)forJ=1,2,…,(2.4) where h=(b-a)/2and h=a+kh)
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 5i d. c/=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) xfx)=T0)(1)T(2)(3) 101.0000001.000000 1.50.6667 0.666667 2.00.50000 0.500000 250.400000 0.400000 300.33333 0.333333 3.50.285714 0.285714 4.00.250000 0.250000 450.22222 0.222222 5.00.2000000.200000
heorem 2.5(Recursive Simpson Rules). Suppose that T()) is the sequence of trapezoidal rules generated by Corollary 2.4. If J> 1 and S(J) is Simpson's rule for 2 subintervals of a, bl, then S()and the trapezoid rules T(J-1) and T()obey the relationship 9147(J)-T(-1) for =1.2 2.51
Example 2.12. Use the sequential Simpson rule to compute the approxi mations S(1), $(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 B ooes rue for 2 subintervals of [a, bl, then B()and Simpsons rules S(J-1) and S( obey the relationship B() 16S(J)-S(J-1 for J=2, 3
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