Page - 145 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.3 TheCompositeMidpointMethod 145 6.3.3 ComparingtheTrapezoidalandtheMidpointMethods Thenextexampleshowshoweasyit is tocombinethetrapezoidalandmidpoint functions to make a comparison of the two methods. The coding is given in compare_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}’.format(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 32 0.8820933014203766 0.8820575578012112 64 0.8820843709743319 0.8820754296107942 128 0.8820821359746071 0.8820799002925637 256 0.8820815770754198 0.8820810181335849 512 0.8820814373412922 0.8820812976045025 1024 0.8820814024071774 0.8820813674728968 2048 0.8820813936736116 0.8820813849400392 4096 0.8820813914902204 0.8820813893068272 8192 0.8820813909443684 0.8820813903985197 16384 0.8820813908079066 0.8820813906714446 32768 0.8820813907737911 0.8820813907396778 65536 0.8820813907652575 0.8820813907567422 131072 0.8820813907631487 0.8820813907610036 262144 0.8820813907625702 0.8820813907620528 524288 0.8820813907624605 0.8820813907623183 1048576 0.8820813907624268 0.8820813907623890 A visual inspection of the numbers shows how fast the digits stabilize in both methods. It appears that 13digitshavestabilized in the last tworows.