On the quad or trapz'd in ChemE heaven

| categories: integration, python | tags: | View Comments

Matlab post

What is the difference between quad and trapz? The short answer is that quad integrates functions (via a function handle) using numerical quadrature, and trapz performs integration of arrays of data using the trapezoid method.

Let us look at some examples. We consider the example of computing \(\int_0^2 x^3 dx\). the analytical integral is \(1/4 x^4\), so we know the integral evaluates to 16/4 = 4. This will be our benchmark for comparison to the numerical methods.

We use the scipy.integrate.quad command to evaluate this \(\int_0^2 x^3 dx\).

from scipy.integrate import quad

ans, err = quad(lambda x: x**3, 0, 2)
print ans
4.0

you can also define a function for the integrand.

from scipy.integrate import quad

def integrand(x):
    return x**3

ans, err = quad(integrand, 0, 2)
print ans
4.0

1 Numerical data integration

if we had numerical data like this, we use trapz to integrate it

import numpy as np

x = np.array([0, 0.5, 1, 1.5, 2])
y = x**3

i2 = np.trapz(y, x)

error = (i2 - 4)/4

print i2, error
4.25 0.0625

Note the integral of these vectors is greater than 4! You can see why here.

import numpy as np
import matplotlib.pyplot as plt
x = np.array([0, 0.5, 1, 1.5, 2])
y = x**3

x2 = np.linspace(0, 2)
y2 = x2**3

plt.plot(x, y, label='5 points')
plt.plot(x2, y2, label='50 points')
plt.legend()
plt.savefig('images/quad-1.png')

The trapezoid method is overestimating the area significantly. With more points, we get much closer to the analytical value.

import numpy as np

x2 = np.linspace(0, 2, 100)
y2 = x2**3

print np.trapz(y2, x2)
4.00040812162

2 Combining numerical data with quad

You might want to combine numerical data with the quad function if you want to perform integrals easily. Let us say you are given this data:

x = [0 0.5 1 1.5 2]; y = [0 0.1250 1.0000 3.3750 8.0000];

and you want to integrate this from x = 0.25 to 1.75. We do not have data in those regions, so some interpolation is going to be needed. Here is one approach.

from scipy.interpolate import interp1d
from scipy.integrate import quad
import numpy as np

x = [0, 0.5, 1, 1.5, 2]
y = [0,    0.1250,    1.0000,    3.3750,    8.0000]

f = interp1d(x, y)

# numerical trapezoid method
xfine = np.linspace(0.25, 1.75)
yfine = f(xfine)
print np.trapz(yfine, xfine)

# quadrature with interpolation
ans, err = quad(f, 0.25, 1.75)
print ans
2.53199187838
2.53125

These approaches are very similar, and both rely on linear interpolation. The second approach is simpler, and uses fewer lines of code.

3 Summary

trapz and quad are functions for getting integrals. Both can be used with numerical data if interpolation is used. The syntax for the quad and trapz function is different in scipy than in Matlab.

Finally, see this post for an example of solving an integral equation using quad and fsolve.

Copyright (C) 2013 by John Kitchin. See the License for information about copying.

org-mode source

blog comments powered by Disqus