Python offers an extremely versatile programming environment, allowing for the inclusion of analytical studies in a numerical program. Here we show an example code with the trapezoidal rule using SymPy to evaluate an integral and compute the absolute error with respect to the numerically evaluated one of the integral 4\int_0^1 dx/(1+x^2) = \pi :
xxxxxxxxxxfrom math import *from sympy import *def Trapez(a,b,f,n): h = (b-a)/float(n) s = 0 x = a for i in range(1,n,1): x = x+h s = s+ f(x) s = 0.5*(f(a)+f(b)) +s return h*s# function to compute pidef function(x): return 4.0/(1+x*x)a = 0.0; b = 1.0; n = 100result = Trapez(a,b,function,n)print "Trapezoidal rule=", result# define x as a symbol to be used by sympyx = Symbol('x')exact = integrate(function(x), (x, 0.0, 1.0))print "Sympy integration=", exact# Find relative errorprint "Relative error", abs((exact-result)/exact)