Page - 90 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

90 3 Computing Integrals a) Write a functionrectangle(f, a, b, n, height=’left’) for computing an integral Rb a f.x/dx by the rectanglemethodwithheight computedbasedon thevalueofheight,which is eitherleft,right,ormid. b) Write three test functions for the three unit test procedures described in Sect.3.4.2.Makesureyoutest forheightequal toleft,right, andmid.You maycall themidpoint functionforchecking the resultwhenheight=mid. Hint Edittest_trapezoidal.py. Filename:rectangle_methods.py. Exercise3.9:Adaptive integration Supposewewant touse the trapezoidalormidpointmethod tocomputean integralRb a f.x/dxwithanerror less thanaprescribed tolerance .What is theappropriate sizeofn? Toanswer thisquestion,wemayenteran iterativeprocedurewherewecompare the results producedbyn and2n intervals, and if the difference is smaller than , the value corresponding to 2n is returned. Otherwise, we halven and repeat the procedure. Hint Itmaybeagood idea to organizeyourcode so that the functionadaptive_ integrationcanbeusedeasily in futureprogramsyouwrite. a) Write a function adaptive_integration(f, a, b, eps, method=midpoint) that implements the ideaabove(epscorrespondsto the tolerance , andmethod canbemidpointortrapezoidal). b) Test themethod on R2 0 x2dx and R2 0 p xdx for D 10 1;10 10 andwrite out theexacterror. c) Makeaplot ofnversus 2 Œ10 1;10 10 forR2 0 p xdx. Use logarithmic scale for . Filename:adaptive_integration.py. Remarks The typeofmethodexplored in this exercise is calledadaptive, because it tries toadapt thevalueofn tomeetagivenerrorcriterion. Thetrueerrorcanvery seldombecomputed (sincewedonotknowtheexact answer to the computational problem),soonehas tofindother indicatorsof theerror,suchas theoneherewhere the changes in the integral value, as thenumberof intervals is doubled, is taken to reflect theerror. Exercise3.10: Integratingxraised tox Consider the integral I D 4Z 0 xxdx: