SciPy – Integration

Scipy is the scientific computing module of Python providing in-built functions on a lot of well-known Mathematical functions. The scipy.integrate sub-package provides several integration techniques including an ordinary differential equation integrator. 

Finding Integration using scipy.integrate

Numerical Integration is the approximate computation of an integral using numerical techniques. Methods for Integrating function given function object:

• quad – General Purpose Integration
• dblquad – General Purpose Double Integration
• nquad – General Purpose n- fold Integration
• fixed_quad – Gaussian quadrature, order n
• quadrature – Gaussian quadrature to tolerance
• romberg – Romberg integration
• trapz – Trapezoidal rule
• cumtrapz – Trapezoidal rule to cumulatively compute integral
• simps – Simpson’s rule
• romb – Romberg integration
• polyint – Analytical polynomial integration (NumPy)

(1) quad :

The function quad is provided to integrate a function of one variable between two points. The points can be  +infinite or – infinite to indicate infinite limits.

Example:

Python3

from scipy.integrate import quad

def f(x):
  return 3.0*x*x + 1.0

I, err = quad(f, 0, 1)
print(I)
print(err)

Output : 

2.0

2.220446049250313e-14

(2) dblquad :

This performs Double Integration with 2 arguments. 

Example:

Python3

from scipy.integrate import dblquad

area = dblquad(lambda x, y: x*y, 0, 0.5,
               lambda x: 0, lambda x: 1-2*x)

print(area)

Output : 

(0.010416666666666668, 4.101620128472366e-16)

(3) nquad :

Performs integration of n variables

Example:

Python3

from scipy.integrate import nquad


def f(x, y, z):
    return x*y*z


I = nquad(f, [[0, 1], [0, 5], [0, 5]])
print(I)

Output : 

(78.12499999999999, 8.673617379884033e-13)

(4) fixed_quad :

With the help of scipy.integrate.fixed_quad() method, we can get the computation of a definite integral using fixed order gaussian quadrature

Example:

Python3

# import scipy.integrate
from scipy import integrate


def func(x): return 3*x**3


# using scipy.integrate.fixed_quad() method
# n is the order of integration
gfg = integrate.fixed_quad(func, 1.0, 2.0, n=2)

print(gfg)

Output:

(11.25, None)

(5) quadrature :

With the help of scipy.integrate.quadrature() method, we can get the computation of definite integral using fixed tolerance gaussian quadrature

Example:

Python3

# import scipy.integrate.
from scipy import integrate

def f(x): return 3*x**3

# using scipy.integrate.quadrature() method
g = integrate.quadrature(f, 0.0, 1.0)

print(g)

Output:

(0.7500000000000001, 2.220446049250313e-16)

(6) romberg :

With the help of scipy.integrate.romberg() method, we can get the romberg integration of a callable function from limit a to b 

Example:

Python3

# import numpy and scipy.integrate
import numpy as np
from scipy import integrate

f = lambda x: 3*(np.pi)*x**3

# using scipy.integrate.romberg()
g = integrate.romberg(f, 1, 2, show = True)

print(g)

Output:

Romberg integration of <function vectorize1.<locals>.vfunc at 0x0000003C1E212790> from [1, 2]

Steps  StepSize   Results

    1  1.000000 42.411501  

    2  0.500000 37.110063 35.342917  

    4  0.250000 35.784704 35.342917 35.342917  

The final result is 35.34291735288517 after 5 function evaluations.

35.34291735288517

(7) trapz :

numpy.trapz() function integrate along the given axis using the composite trapezoidal rule.

Ref : numpy.trapz() function | Python

Python3

# Python program explaining
# numpy.trapz() function

# importing numpy as geek
import numpy as np

b = [2, 4]
a = [6, 8]

f = np.trapz(b, a)

print(f)

Output:

6.0

(8) cumtraz:

With the help of scipy.integrate.cumtrapz() method, we can get the cumulative integrated value of y(x) using composite trapezoidal rule.

Example:

Python3

# import numpy and scipy.integrate.cumtrapz
import numpy as np
from scipy import integrate

a = np.arange(0, 5)
b = np.arange(0, 5)

# using scipy.integrate.cumtrapz() method
f = integrate.cumtrapz(b, a)

print(f)

Output:

[0.5 2.  4.5 8. ]

(9) simps:

With the help of scipy.integrate.simps() method, we can get the integration of y(x) using samples along the axis and composite simpson’s rule.

Example:

Python3

# import numpy and scipy.integrate
import numpy as np
from scipy import integrate

a = np.arange(0, 5)
b = np.arange(0, 5)

# using scipy.integrate.simps() method
f = integrate.simps(b, a)

print(f)

Output:

8.0

(10) romb:

With the help of scipy.integrate.romb() method, we can get the romberg integration using samples of a function from limit a to b 

Example:

Python3

# import numpy and scipy.integrate
import numpy as np
from scipy import integrate


x = np.arange(0,5)

# using scipy.integrate.romb() method
f = integrate.romb(x)

print(f)

Output:

8.0

(11) polyint:

numpy.polyint(p, m) : Evaluates the anti – derivative of a polynomial with the specified order.

Ref : numpy.polyint() in Python

Python3

# Python code explaining
# numpy.polyint()

# importing libraries
import numpy as np

# Constructing polynomial
p1 = np.poly1d([2,6])
p2 = np.poly1d([4,8])

a = np.polyint(p1, 1)
b = np.polyint(p2, 2)

print ("\n\nUsing polyint")
print ("p1 anti-derivative of order = 2 : \n", a)
print ("p2 anti-derivative of order = 2 : \n", b)

Output :

Using polyint

p1 anti-derivative of order = 2 :  

   2

1 x + 6 x

p2 anti-derivative of order = 2 :  

        3     2

0.6667 x + 4 x