Seite - 156 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

156 6 ComputingIntegralsandTestingCode error problems explained in Sect.6.6.3, and we must use a test with tolerance instead: def test_add(): expected = 0.3 computed = add(0.1, 0.2) tol = 1E-14 diff = abs(expected - computed) assert diff < tol, ’diff={:g}’.format(diff) Below we shall write test functions for each of the three test procedures we suggested: comparison with hand calculations, checking problems that can be exactly solved, and checking convergencerates. We stick to testing the trapezoidal integration code and collect all test functions in one common file by the name test_trapezoidal.py. Hand-Computed Numerical Results Our previous hand calculations for two trapezoidscanbe utilized in a test function like this: from trapezoidal import trapezoidal def test_trapezoidal_one_exact_result(): """Compare one hand-computed result.""" from math import exp v = lambda t: 3*(t**2)*exp(t**3) n = 2 computed = trapezoidal(v, 0, 1, n) expected = 2.463642041244344 error = abs(expected - computed) tol = 1E-14 success = error < tol msg = ’error={:g} > tol={:g}’.format(error, tol) assert success, msg Note the importanceofcheckingcomputedagainstexpectedwith a tolerance: roundingerrors from the arithmetics insidetrapezoidalwill not make the result exactly like thehand-computedone. SolvingaProblemWithoutNumericalErrors Weknowthat the trapezoidalrule isexactfor linear integrands.Choosingtheintegral ∫4.4 1.2 (6x−4)dxasatestcase, the correspondingtest functioncould, forexample,checkwith threedifferentnvalues, andmay look like def test_trapezoidal_linear(): """Check that linear functions are integrated exactly.""" f = lambda x: 6*x - 4 F = lambda x: 3*x**2 - 4*x # Anti-derivative a = 1.2; b = 4.4 expected = F(b) - F(a) tol = 1E-14 for n in 2, 20, 21: computed = trapezoidal(f, a, b, n) error = abs(expected - computed) success = error < tol msg = ’n={:d}, err={:g}’.format(n, error) assert success, msg