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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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    \frametitle{Outline}
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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}