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