\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}