Page - 72 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

72 3 Computing Integrals Ifwecannotmake tests like0.1 + 0.2 == 0.3,whatshouldwe thendo?The answer is thatwemust accept some small inaccuracyandmakea testwith a toler- ance. Here is the recipe: >>> a = 0.1; b = 0.2; expected = 0.3 >>> computed = a + b >>> diff = abs(expected - computed) >>> tol = 1E-15 >>> diff < tol True Herewe have set the tolerance for comparison to10 15, but calculating0.3 - (0.1 + 0.2) shows that it equals-5.55e-17, soa lower tolerancecouldbeused in this particular example. However, inother calculationswehave little ideaabout howaccurate theanswer is (therecouldbeaccumulationofroundingerrors inmore complicated algorithms), so 10 15 or 10 14 are robust values. Aswedemonstrate below,these tolerancesdependonthemagnitudeof thenumbersin thecalculations. Doing an experiment with 10k C0:3 .10k C0:1C0:2/ for k D 1;:: :;10 shows that the answer (which should be zero) is around 1016 k. Thismeans that the tolerancemust be larger ifwe computewith larger numbers. Setting a proper tolerance therefore requires some experiments to see what level of accuracy one canexpect. Awayoutof thisdifficulty is toworkwith relative insteadofabsolute differences. Ina relativedifferencewedividebyoneof theoperands, e.g., aD10k C0:3; bD .10k C0:1C0:2/; cD a b a : Computing this c for variousk shows a value around10 16. A safer procedure is thus touse relative differences. 3.4.4 ConstructingUnitTestsandWritingTestFunctions Pythonhasseveralframeworksforautomaticallyrunningandcheckingapotentially very largenumber of tests for parts of your software byone command. This is an extremely useful feature during program development: whenever you have done somechanges tooneormorefiles, launchthe test commandandmakesurenothing isbrokenbecauseofyouredits. The test frameworks nose and py.test are particularly attractive as they are veryeasy touse. Tests are placed in special test functions that the frameworkscan recognizeandrun foryou.The requirements toa test functionaresimple: thenamemust startwithtest_ the test functioncannothaveanyarguments the tests inside test functionsmustbebooleanexpressions a boolean expression bmust be testedwith assert b, msg, where msg is an optionalobject (stringornumber) tobewrittenoutwhenb is false