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

# > This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.
# 

# # 9.1. Finding the root of a mathematical function

# 1. Let's import NumPy, SciPy, scipy.optimize, and matplotlib.

# In[ ]:


import numpy as np
import scipy as sp
import scipy.optimize as opt
import matplotlib.pyplot as plt
get_ipython().run_line_magic('matplotlib', 'inline')


# 2. We define the mathematical function $f(x)=\cos(x)-x$ in Python. We will try to find a root of this function numerically, which corresponds to a fixed point of the cosine function.

# In[ ]:


f = lambda x: np.cos(x) - x


# 3. Let's plot this function on the interval $[-5, 5]$.

# In[ ]:


x = np.linspace(-5, 5, 1000)
y = f(x)
plt.figure(figsize=(5,3));
plt.plot(x, y);
plt.axhline(0, color='k');
plt.xlim(-5,5);


# 4. We see that this function has a unique root on this interval (this is because the function's sign changes on this interval). The scipy.optimize module contains a few root-finding functions that are adapted here. For example, the `bisect` function implements the **bisection method** (also called the **dichotomy method**). It takes as input the function and the interval to find the root in.

# In[ ]:


opt.bisect(f, -5, 5)


# Let's visualize the root on the plot.

# In[ ]:


plt.figure(figsize=(5,3));
plt.plot(x, y);
plt.axhline(0, color='k');
plt.scatter([_], [0], c='r', s=100);
plt.xlim(-5,5);


# 5. A faster and more powerful method is `brentq` (Brent's method). This algorithm also requires that $f$ is continuous and that $f(a)$ and $f(b)$ have different signs.

# In[ ]:


opt.brentq(f, -5, 5)


# The `brentq` method is faster than `bisect`. If the conditions are satisfied, it is a good idea to try Brent's method first.

# In[ ]:


get_ipython().run_line_magic('timeit', 'opt.bisect(f, -5, 5)')
get_ipython().run_line_magic('timeit', 'opt.brentq(f, -5, 5)')


# > You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).
# 
# > [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages).