How to find the intersection of two functions

Previously we have seen how to find roots of a function with fsolve, in this example we use fsolve to find an intersection between two functions, sin(x) and cos(x):

from scipy.optimize import fsolve
import pylab
import numpy

def findIntersection(fun1,fun2,x0):
 return fsolve(lambda x : fun1(x) - fun2(x),x0)

result = findIntersection(numpy.sin,numpy.cos,0.0)
x = numpy.linspace(-2,2,50)
pylab.plot(x,numpy.sin(x),x,numpy.cos(x),result,numpy.sin(result),'ro')
pylab.show()

In the graph we can see sin(x) (blue), cos(x) (green) and the intersection found (red dot) starting from x = 0.

How to find the roots of a function with fsolve

The function fsolve provided by numpy return the roots of the (non-linear) equations defined by func(x) = 0 given a starting estimate. We will see how to use fsolve to find the root of the function

from scipy.optimize import fsolve
import pylab
import numpy

pow3 = lambda x : x**3

result = fsolve(pow3,10) # starting from x = 10
print result
x = numpy.linspace(-1,1,50)
pylab.plot(x,pow3(x),result,pow3(result),'ro')
pylab.grid(b=1)
pylab.show()

In the following graph we can see f(x) (blue curve) and the solution found (red dot)

How to use string's Template

Templates provide simple string substitutions mechanism, here's a simple demonstration:

from string import Template

template = Template('The president of $state is $name')
message = template.substitute(state='USA', name='Obama')
print '1.',message
message = template.substitute(state='France', name='Sarkozy')
print '2.',message

try:
 # will raise an exception
 message = template.substitute(state='England')
except Exception as e:
 print 'I cannot fill the placeholder',e
 #  original name placeholder will be used
 message = template.safe_substitute(state='England') 

print '3.',message

The output of this script will be

1. The president of USA is Obama
2. The president of France is Sarkozy
I cannot fill the placeholder 'name'
3. The president of England is $name