Page - 67 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3.3 TheCompositeMidpointMethod 67 3.3.2 Implementation We follow the advice and lessons learned from the implementation of the trape- zoidalmethodandmakea functionmidpoint(f, a, b, n) (inafilemidpoint. py) for implementing thegeneral formula (3.21): def midpoint(f, a, b, n): h = float(b-a)/n result = 0 for i in range(n): result += f((a + h/2.0) + i*h) result *= h return result Wecan test the functionasweexplained for the similartrapezoidalmethod. Theerror inourparticularproblem R1 0 3t2et 3 dtwithfour intervals isnowabout0.1 in contrast to 0.2 for the trapezoidal rule. This is in fact not accidental: one can showmathematically that theerrorof themidpointmethod is abit smaller than for the trapezoidalmethod. The differences are seldom of any practical importance, andonalaptopwecaneasilyusenD106andget theanswerwithanerrorofabout 10 12 in acoupleof seconds. 3.3.3 ComparingtheTrapezoidalandtheMidpointMethods The next example shows how easy we can combine the trapezoidal and midpointfunctionstomakeacomparisonof the twomethodsin thefilecompare_ integration_methods.py: from trapezoidal import trapezoidal from midpoint import midpoint from math import exp g = lambda y: exp(-y**2) a = 0 b = 2 print ’ n midpoint trapezoidal’ for i in range(1, 21): n = 2**i m = midpoint(g, a, b, n) t = trapezoidal(g, a, b, n) print ’%7d %.16f %.16f’ % (n, m, t) Note theeffortsput intonice formatting– theoutputbecomes n midpoint trapezoidal 2 0.8842000076332692 0.8770372606158094 4 0.8827889485397279 0.8806186341245393 8 0.8822686991994210 0.8817037913321336 16 0.8821288703366458 0.8819862452657772