Page - 139 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.2 TheCompositeTrapezoidalRule 139 function.Thus, in the presentcase, we put the statements (otherwiseplaced a main program)insidea functionnamedapplication. To achieve flexibility, we proceed to modifytrapezoidal.py, so that it has a test blockand functiondefinitionsoftrapezoidalandapplication: def trapezoidal(f, a, b, n): h = (b-a)/n f_sum = 0 for i in range(1, n, 1): x = a + i*h f_sum = f_sum + f(x) return h*(0.5*f(a) + f_sum + 0.5*f(b)) def application(): from math import exp v = lambda t: 3*(t**2)*exp(t**3) n = int(input(’n: ’)) numerical = trapezoidal(v, 0, 1, n) # Compare with exact result V = lambda t: exp(t**3) exact = V(1) - V(0) error = abs(exact - numerical) print(’n={:d}: {:.16f}, error: {:g}’.format(n, numerical, error)) if __name__ == ’__main__’: application() With our newly gained knowledge about the making of modules, we understand that theif test becomes true when the module file,trapezoidal.py, is run as a program,and false when the module (or part of it) is imported in anotherprogram. Consequently, with an import like from trapezoidal import trapezoidal, the test fails and application() is not called. On the other hand, if we run trapezoidal.pyas a program, the test condition is positiveandapplication() is called. A call toapplication implies that our special problem gets computed. The main program now gets very small, being just a single function call to application. Running theprogram,e.g.,withn=4gives theoutput n=4: 1.9227167504675762, error: 0.204435 Clearly, with a module like the one shown here, thetrapezoidal function alone (i.e., withoutapplication)can easily be importedby other programs to compute other integrals. 6.2.3 ASpecificImplementation:What’stheProblem? Let us illustrate the implementation implied by alternative 1 in the Programmer’s dilemma box in Sect.6.2.2. That is, we make a special-purpose code, where we adapt the general formula (6.17) to the specific problem ∫1 0 3t 2et 3 dt, in which the integrandis avelocity functionv(t).