In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from scipy import integrate
The trapezoidal rule generates a numerical approximation to the 1d integral:
$$ I(a,b) = \int_a^b f(x) dx $$by dividing the interval $[a,b]$ into $N$ subdivisions of length $h$:
$$ h = (b-a)/N $$Note that this means the function will be evaluated at $N+1$ points on $[a,b]$. The main idea of the trapezoidal rule is that the function is approximated by a straight line between each of these points.
Write a function trapz(f, a, b, N) that performs trapezoidal rule on the function f over the interval $[a,b]$ with N subdivisions (N+1 points).
In [2]:
def trapz(f, a, b, N):
"""Integrate the function f(x) over the range [a,b] with N points."""
h=(b-a)/N
k=np.arange(1,N)
return h*(0.5*f(a)+0.5*f(b)+f(a+k*h).sum())
In [3]:
f = lambda x: x**2
g = lambda x: np.sin(x)
In [4]:
I = trapz(f, 0, 1, 1000)
assert np.allclose(I, 0.33333349999999995)
J = trapz(g, 0, np.pi, 1000)
assert np.allclose(J, 1.9999983550656628)
Now use scipy.integrate.quad to integrate the f and g functions and see how the result compares with your trapz function. Print the results and errors.
In [18]:
x,err1=integrate.quad(f,0.0,1.0)
y,err2=integrate.quad(g,0.0,np.pi)
print("Integral result for f: ", x)
print("Trapezoidal result for f: ", I)
print("Error for integral of f: ", err1)
print("Integral result for g: ", y)
print("Trapezoidal result for g: ", J)
print("Error for integral of g: ", err2)
In [17]:
assert True # leave this cell to grade the previous one