## Finding roots¶

Demonstration of two methods:

• brentrq(), for a root known to lie an interval [a,b], where a and b have opposite signs
• newton(), for a root known to be near some point a

Let's find the points where $\tan x = x$.

In [1]:
import scipy as sp
from scipy.optimize import brentq, newton

import matplotlib as mpl       # As of July 2017 Bucknell computers use v. 2.x
import matplotlib.pyplot as plt

# Following is an Ipython magic command that puts figures in the  notebook.
# For figures in separate windows, comment out following line and uncomment
# the next line
# Must come before defaults are changed.
%matplotlib notebook
#%matplotlib

# As of Aug. 2017 reverting to 1.x defaults.
# In 2.x text.ustex requires dvipng, texlive-latex-extra, and texlive-fonts-recommended,
# which don't seem to be universal
# See https://stackoverflow.com/questions/38906356/error-running-matplotlib-in-latex-type1cm?
mpl.style.use('classic')

# M.L. modifications of matplotlib defaults using syntax of v.2.0
# Changes can also be put in matplotlibrc file, or effected using mpl.rcParams[]
plt.rc('figure', figsize = (6, 4.5))            # Reduces overall size of figures
plt.rc('axes', labelsize=16, titlesize=14)
plt.rc('figure', autolayout = True)             # Adjusts supblot parameters for new size

In [2]:
def f(x):
return sp.tan(x) - x

In [3]:
plt.figure(1)
x = sp.linspace(0,10,201)
y = sp.tan(x)
plt.ylim(-10,10)
plt.plot(x,y)
plt.plot(x,x);

In [4]:
# Using interactive plot features helps here
root1 = sp.optimize.newton(f, 4.5)
root2 = sp.optimize.brentq(f,2,4.6)

In [5]:
root1, root2

Out[5]:
(4.4934094579093378, 4.4934094579089185)

### Version Information¶

version_information is from J.R. Johansson (jrjohansson at gmail.com)
See Introduction to scientific computing with Python:
http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb

If version_information has been installed system wide (as it has been on linuxremotes), continue with next cell as written. If not, comment out top line in next cell and uncomment the second line.

In [6]:
%load_ext version_information

#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py

Loading extensions from ~/.ipython/extensions is deprecated. We recommend managing extensions like any other Python packages, in site-packages.

In [7]:
version_information scipy, matplotlib

Out[7]:
SoftwareVersion
Python3.6.1 64bit [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)]
IPython6.1.0
OSLinux 3.10.0 327.36.3.el7.x86_64 x86_64 with redhat 7.2 Maipo
scipy0.19.1
matplotlib2.0.2
Tue Aug 01 11:26:55 2017 EDT