Page - 148 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

148 6 ComputingIntegralsandTestingCode 6.4.3 SpeedupGainedwithVectorization Nowthatwehavecreatedfaster,vectorizedversionsof thefunctions, it isof interest tomeasurehowmuchfaster theyare.Restrictingourselves to themidpointmethod, wemightproceedas: import timeit from integration_methods_vec import midpoint as midpoint_vec from midpoint import midpoint from numpy import exp v = lambda t: 3*t**2*exp(t**3) t = timeit.Timer(’midpoint(v, 0, 1, 1000000)’, \ setup=’from __main__ import midpoint, v’) time_midpoint = t.timeit(10) print(’Time, midpoint: {:g} seconds’.format(time_midpoint)) # Vectorized version t = timeit.Timer(’midpoint_vec(v, 0, 1, 1000000)’, \ setup=’from __main__ import midpoint_vec, v’) time_midpoint_vec = t.timeit(10) print(’Time, midpoint vec: {:g} seconds’.format(time_midpoint_vec)) print(’Efficiency factor: {:g}’.format(time_midpoint/time_midpoint_vec)) Running theprogramgives Time, midpoint: 19.6083 seconds Time, midpoint vec: 0.868379 seconds Efficiency factor: 22.5804 We see that the vectorized version is about 20 times faster. The results for the trapezoidal method are very similar, and the factor of about 20 is independent of thenumberof intervals. 6.5 RateofConvergence We have seen that the numerical integration error drops when the number of sub- intervalsn is increased (causing each sub-interval to become smaller). This is fine and in linewithourexpectations,but some importantdetails shouldbe added. Asymptotic Behavior of the Integration Error It is known that, if only the size hof thesub-intervals is small enough,numerical integrationmethodstypicallygive anerror E=Khr , (6.21) whereK is anunknownconstant,while theconvergencerate r is aknownconstant that depends on the method. When a method has convergence rate r, it is known as an r-th order method. A large r is beneficial, sinceE then drops quicker when h→0.