\documentclass[12pt]{article}
\title{Matrices and Solution of Equations}
\author{FOSSEE}
\begin{document}
\date{}
\vspace{-1in}
\begin{center}
\LARGE{Interpolation, Differentiation and Integration}\\
\large{FOSSEE}
\end{center}
\section{}
Loading a data file
\begin{verbatim}
In [2]: x, y = loadtxt('points.txt', unpack = True)
#load data file directly into Arrays.
\end{verbatim}
\section{}
Interploate
\begin{verbatim}
In []: from scipy.interpolate import splrep
In []: tck = splrep(x,y) #get spline curve representation for x,y.
In []: from scipy.interpolate import splev
#To evaluate spline and it's derivatives.
In []: Xnew = arange(0.01,3,0.02)
#missing set of points
In []: Ynew = splev(Xnew, tck)
#Value of function at Xnew
In []: plot(Xnew, Ynew)
\end{verbatim}

\section{Differentiation}
Taylor series - finite difference approximations
$f(x+h)=f(x)+hf^{'}(x)$ Forward
\begin{verbatim}
In []: x = linspace(0, 2*pi, 100)
In []: y = sin(x)
In []: deltax = x[1] - x[0]
In []: fD = (y[1:] - y[:-1]) / deltax 
#fD is the required forward difference
\end{verbatim}

\section{Quadrature}
$\int_0^1(sin(x) + x^2)$ 
In []: def f(x):
           return sin(x)+x**2
In []: from scipy.integrate import quad
In []: quad(f, 0, 1)
\end{document}

