#!/usr/bin/env python
# coding: utf-8

# ## Exercise for the course [Python for MATLAB users](http://sese.nu/python-for-matlab-users-ht15/), by Olivier Verdier

# In[1]:


get_ipython().run_line_magic('pylab', '')
get_ipython().run_line_magic('matplotlib', 'inline')


# A  continuous functions $f$ which changes its sign in an interval $[a,b]$, i.e. $f(a)f(b)< 0$, has at least one root in this interval. Such a root can be found by the *bisection method*. 
# 
# This method starts from the given interval. Then it investigates the sign changes in the subintervals $[a,\frac{a+b}{2}]$ 
# and $[\frac{a+b}{2},b]$. If the sign changes in the first subinterval $ b$ is redefined to be 
# $b:=\frac{a+b}{2}$
# otherwise $a$ is redefined in the same manner to 
# $a:=\frac{a+b}{2}$,
# and the process is repeated until the $b-a$ is less than a given tolerance.
# 

# In[2]:


def bisect(f, a, b):
    tol = 1e-9
    for i in range(100):
        if (b-a) < tol:
            break
        c = (a+b)/2
        if f(a)*f(c) <= 0:
            b = c
        else:
            a = c
    return c


# Implement the function `bisect` above until the following does not complain:

# In[3]:


def lin(x):
    return x
assert allclose(bisect(sin, 3., 4.), pi)
assert allclose(bisect(lin, -1,1), 0)


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