Page - 170 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

170 6 ComputingIntegralsandTestingCode Fig. 6.4 Illustration of the rectangle method withevaluating the rectangle height byeither the left or right point Exercise6.7:WriteTest Functions for ∫4 0 √ xdx We want to test how the trapezoidal function works for the integral ∫4 0 √ xdx. Two of the tests in test_trapezoidal.py are meaningful for this integral. Compute by hand the result of using two or three trapezoids and modify the test_trapezoidal_one_exact_result function accordingly. Then modify test_trapezoidal_conv_rate tohandle thesquare root integral. Filename:test_trapezoidal3.py. Remarks The convergence rate test fails. Printing out r shows that the actual convergencerate for this integral is1.5andnot2.Thereason is that theerror in the trapezoidal method6 is −(b−a)3n−2f ′′(ξ) for some (unknown) ξ ∈ [a,b]. With f(x)=√x,f ′′(ξ)→−∞asξ →0,pointingtoapotentialproblemin thesizeof theerror.Runninga testwitha>0, say ∫4 0.1 √ xdx showsthat theconvergencerate is indeedrestored to2. Exercise6.8:RectangleMethods Themidpointmethoddividestheintervalofintegrationintoequal-sizedsubintervals andapproximatestheintegralineachsubintervalbyarectanglewhoseheightequals the function value at the midpoint of the subinterval. Instead, one might use either the left or right end of the subinterval as illustrated in Fig.6.4. This defines a rectangle method of integration. The height of the rectangle can be based on the leftor rightendor themidpoint. a) Write a functionrectangle(f, a, b, n, height=’left’) for computing an integral ∫b a f(x)dx by the rectangle method with height computed based on thevalueofheight,which is eitherleft,right, ormid. b) Write three test functions for the three unit test procedures described in Sect.6.6.2. Make sure you test for height equal to left, right, and mid. Youmaycall themidpoint functionforcheckingtheresultwhenheight=mid. 6 http://en.wikipedia.org/wiki/Trapezoidal_rule#Error_analysis.