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%Tutorial slides on Python.

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% Author: FOSSEE 

% Copyright (c) 2009, FOSSEE, IIT Bombay

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% Title page

\title[Calculus]{Python for Science and Engg: Interpolation, Differentiation and Integration}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\Day 1, Session 5}

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\begin{document}

\begin{frame}

  \titlepage

\end{frame}

\begin{frame}

  \frametitle{Outline}

  \tableofcontents

%  \pausesections

\end{frame}

\section{Interpolation}

\begin{frame}[fragile]

\frametitle{Interpolation}

\begin{itemize}

\item Let us begin with interpolation

\item Let's use the L and T arrays and interpolate this data to obtain data at new points

\end{itemize}

\begin{lstlisting}

In []: L = []

In []: T = []

In []: for line in open('pendulum.txt'):

           l, t = line.split()

           L.append(float(l))

           T.append(float(t))

In []: L = array(L)

In []: T = array(T)

\end{lstlisting}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Interpolation \ldots}

%% \begin{small}

%%   \typ{In []: from scipy.interpolate import interp1d}

%% \end{small}

%% \begin{itemize}

%% \item The \typ{interp1d} function returns a function

%% \begin{lstlisting}

%%   In []: f = interp1d(L, T)

%% \end{lstlisting}

%% \item Functions can be assigned to variables 

%% \item This function interpolates between known data values to obtain unknown

%% \end{itemize}

%% \end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Interpolation \ldots}

%% \begin{lstlisting}

%% In []: Ln = arange(0.1,0.99,0.005)

%% # Interpolating! 

%% # The new values in range of old data

%% In []: plot(L, T, 'o', Ln, f(Ln), '-')

%% In []: f = interp1d(L, T, kind='cubic')

%% # When kind not specified, it's linear

%% # Others are ...

%% # 'nearest', 'zero', 

%% # 'slinear', 'quadratic'

%% \end{lstlisting}

%% \end{frame}

\begin{frame}[fragile]

\frametitle{Spline Interpolation}

\begin{small}

\begin{lstlisting}

In []: from scipy.interpolate import splrep

In []: from scipy.interpolate import splev

\end{lstlisting}

\end{small}

\begin{itemize}

\item Involves two steps

  \begin{enumerate}

  \item Find out the spline curve, coefficients

  \item Evaluate the spline at new points

  \end{enumerate}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{splrep}}

To find the B-spline representation

\begin{lstlisting}

In []: tck = splrep(L, T)

\end{lstlisting}

Returns 

\begin{enumerate}

\item the vector of knots, 

\item the B-spline coefficients 

\item the degree of the spline (default=3)

\end{enumerate}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{splev}}

To Evaluate a B-spline and it's derivatives

\begin{lstlisting}

In []: Lnew = arange(0.1,1,0.005)

In []: Tnew = splev(Lnew, tck)

#To obtain derivatives of the spline

#use der=1, 2,.. for 1st, 2nd,.. order

In []: Tnew = splev(Lnew, tck, der=1)

\end{lstlisting}

\end{frame}

%% \begin{frame}[fragile]

%% \frametitle{Interpolation \ldots}

%% \begin{itemize}

%% \item 

%% \end{itemize}

%% \end{frame}

\section{Differentiation}

\begin{frame}[fragile]

\frametitle{Numerical Differentiation}

\begin{itemize}

\item Given function $f(x)$ or data points $y=f(x)$

\item We wish to calculate $f^{'}(x)$ at points $x$

\item Taylor series - finite difference approximations

\end{itemize}

\begin{center}

\begin{tabular}{l l}

$f(x+h)=f(x)+h.f^{'}(x)$ &Forward \\

$f(x-h)=f(x)-h.f^{'}(x)$ &Backward

\end{tabular}

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{Forward Difference}

\begin{lstlisting}

In []: x = linspace(0, 2*pi, 100)

In []: y = sin(x)

In []: deltax = x[1] - x[0]

\end{lstlisting}

Obtain the finite forward difference of y

\end{frame}

\begin{frame}[fragile]

\frametitle{Forward Difference \ldots}

\begin{lstlisting}

In []: fD = (y[1:] - y[:-1]) / deltax

In []: plot(x, y, x[:-1], fD)

\end{lstlisting}

\begin{center}

  \includegraphics[height=2in, interpolate=true]{data/fwdDiff}

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example}

\begin{itemize}

\item Given x, y positions of a particle in \typ{pos.txt}

\item Find velocity \& acceleration in x, y directions

\end{itemize}

\small{

\begin{center}

\begin{tabular}{| c | c | c |}

\hline

$X$ & $Y$ \\ \hline

0.     &  0.\\ \hline

0.25   &  0.47775\\ \hline

0.5    &  0.931\\ \hline

0.75   &  1.35975\\ \hline

1.     &  1.764\\ \hline

1.25   &  2.14375\\ \hline

\vdots & \vdots\\ \hline

\end{tabular}

\end{center}}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example \ldots}

\begin{itemize}

\item Read the file

\item Obtain an array of x, y

\item Obtain velocity and acceleration

\item use \typ{deltaT = 0.05}

\end{itemize}

\begin{lstlisting}

In []: X = []

In []: Y = []

In []: for line in open('location.txt'):

  ....     points = line.split()

  ....     X.append(float(points[0]))

  ....     Y.append(float(points[1]))

In []: S = array([X, Y])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Example \ldots}

\begin{itemize}

\item use \typ{deltaT = 0.05}

\end{itemize}

\begin{lstlisting}

In []: deltaT = 0.05

In []: v = (S[:,1:]-S[:,:-1])/deltaT

In []: a = (v[:,1:]-v[:,:-1])/deltaT

\end{lstlisting}

Try Plotting the position, velocity \& acceleration.

\end{frame}

\section{Quadrature}

\begin{frame}[fragile]

\frametitle{Quadrature}

\begin{itemize}

\item We wish to find area under a curve

\item Area under $(sin(x) + x^2)$ in $(0,1)$

\item scipy has functions to do that

\end{itemize}

\begin{small}

  \typ{In []: from scipy.integrate import quad}

\end{small}

\begin{itemize}

\item Inputs - function to integrate, limits

\end{itemize}

\begin{lstlisting}

In []: x = 0

In []: quad(sin(x)+x**2, 0, 1)

\end{lstlisting}

\begin{small}

\alert{\typ{error:}}

\typ{First argument must be a callable function.}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Definition}

We have been using them all along. Now let's see how to define them.

\begin{lstlisting}

In []: def f(x):

           return sin(x)+x**2

In []: quad(f, 0, 1)

\end{lstlisting}

\begin{itemize}

\item \typ{def}

\item name

\item arguments

\item \typ{return}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Calling them}

\begin{lstlisting}

In [15]: f()

---------------------------------------

\end{lstlisting}

\alert{\typ{TypeError:}}\typ{f() takes exactly 1 argument}

\typ{(0 given)}

\begin{lstlisting}

In []: f(0)

Out[]: 0.0

In []: f(1)

Out[]: 1.8414709848078965

\end{lstlisting}

More on Functions later \ldots

\end{frame}

\begin{frame}[fragile]

\frametitle{Quadrature \ldots}

\begin{lstlisting}

In []: quad(f, 0, 1)

\end{lstlisting}

Returns the integral and an estimate of the absolute error in the result.

\begin{itemize}

\item Look at \typ{dblquad} for Double integrals

\item Use \typ{tplquad} for Triple integrals

\end{itemize}

\end{frame}

\begin{frame}

  \frametitle{Things we have learned}

  \begin{itemize}

  \item Interpolation

  \item Differentiation

  \item Functions

    \begin{itemize}

    \item Definition

    \item Calling

    \item Default Arguments

    \item Keyword Arguments

    \end{itemize}

  \item Quadrature

  \end{itemize}

\end{frame}

\end{document}