00; 00. 2. CHAPTER 2. NUMERICAL INTEGRATION. BONDS. 58 ~ h2 - (b - a) max Ifl/(x)l, a~x~b 24 since Ifl/(ei,T) I ~ maxa~x~b Ifl/(x)l, for all i = 1 : n. 22) is therefore proven. 30). 26). Recall that any continuous function on a closed interval has a finite maximum, which is achieved at (at least) one point of the interval.
NUMERICAL INTEGRATION. BONDS. 21) for computing approximate values I:;, and I~ of the integral f;, l I ical integration method with n partition intervals. The method is convergent if and only if the approximations In converge to I as the number of intervals n goes to infinity (and therefore as h = b~a goes to 0); i. ; lim II - Inl = O. n-fOO The order of convergence of the numerical integration method is k only if >0 if and f(x) dx; and let I:;; I;') or I~ be the approximations of I given by the Midpoint; Trapezoidal; and Simpson;s rules corresponding to n partition intervals of size h = b-a.
A Primer for the Mathematics of Financial Engineering by Dan Stefanica