Page - 63 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3.2 TheCompositeTrapezoidalRule 63 3.2.4 AlternativeFlatSpecial-PurposeImplementation Let us illustrate the implementation implied by alternative 1 in theProgrammer’s dilemma box in Sect. 3.2.2. That is, wemake a special-purpose codewhere we adapt thegeneral formula (3.17) to the specificproblem R1 0 3t2et 3 dt. Basically,we use afor loop to compute the sum. Each termwithf.x/ in the formula (3.17) is replaced by 3t2et 3 ,x by t, andh by t 1. Afirst try atwriting aplain,flat programdoing thespecial calculation is from math import exp a = 0.0; b = 1.0 n = input(’n: ’) dt = float(b - a)/n # Integral by the trapezoidal method numerical = 0.5*3*(a**2)*exp(a**3) + 0.5*3*(b**2)*exp(b**3) for i in range(1, n): numerical += 3*((a + i*dt)**2)*exp((a + i*dt)**3) numerical *= dt exact_value = exp(1**3) - exp(0**3) error = abs(exact_value - numerical) rel_error = (error/exact_value)*100 print ’n=%d: %.16f, error: %g’ % (n, numerical, error) Theproblemwith theabovecode is at least three-fold: 1. Weneed to reformulate (3.17) forourspecialproblemwithadifferentnotation. 2. The integrand3t2et 3 is insertedmany times in thecode,whichquickly leads to errors. 3. A lot of edits are necessary to use the code to compute a different integral – theseedits are likely to introduceerrors. The potential errors involved in point 2 serve to illustrate how important it is to use Python functions asmathematical functions. Herewe have chosen to use the lambdafunction todefine the integrandas thevariablev: from math import exp v = lambda t: 3*(t**2)*exp(t**3) # Function to be integrated a = 0.0; b = 1.0 n = input(’n: ’) dt = float(b - a)/n 1Replacing h by t is not strictly required asmany use h as interval also along the time axis. Nevertheless, t is an even more popular notation for a small time interval, so we adopt that commonnotation.