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