--- a/day1/session4.tex Tue Oct 27 19:25:25 2009 +0530
+++ b/day1/session4.tex Tue Oct 27 19:25:54 2009 +0530
@@ -74,7 +74,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Title page
-\title[Basic Python]{Matrices, Solution of equations and Integration\\}
+\title[Basic Python]{Matrices, Solution of equations}
\author[FOSSEE] {FOSSEE}
@@ -124,54 +124,14 @@
% \pausesections
\end{frame}
-\section{Solving linear equations}
-\begin{frame}[fragile]
-\frametitle{Solution of equations}
-Consider,
- \begin{align*}
- 3x + 2y - z & = 1 \\
- 2x - 2y + 4z & = -2 \\
- -x + \frac{1}{2}y -z & = 0
- \end{align*}
-Solution:
- \begin{align*}
- x & = 1 \\
- y & = -2 \\
- z & = -2
- \end{align*}
+\section{Matrices}
+
+\begin{frame}
+\frametitle{Matrices: Introduction}
+We looked at the Van der Monde matrix in the previous session,\\
+let us now look at matrices in a little more detail.
\end{frame}
-\begin{frame}[fragile]
-\frametitle{Solving using Matrices}
-Let us now look at how to solve this using \kwrd{matrices}
- \begin{lstlisting}
- In []: A = matrix([[3,2,-1],[2,-2,4],[-1, 0.5, -1]])
- In []: b = matrix([[1], [-2], [0]])
- In []: x = linalg.solve(A, b)
- In []: Ax = dot(A, x)
- In []: allclose(Ax, b)
- Out[]: True
- \end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solution:}
-\begin{lstlisting}
-In []: x
-Out[]:
-array([[ 1.],
- [-2.],
- [-2.]])
-
-In []: Ax
-Out[]:
-matrix([[ 1.00000000e+00],
- [ -2.00000000e+00],
- [ 2.22044605e-16]])
-\end{lstlisting}
-\end{frame}
-
-\section{Matrices}
\subsection{Initializing}
\begin{frame}[fragile]
\frametitle{Matrices: Initializing}
@@ -237,173 +197,51 @@
\end{small}
\end{frame}
-
-\section{Integration}
-
-\subsection{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}
-\small{\typ{In []: from scipy.integrate import quad}}
-\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}
-\alert{\typ{error:}}
-\typ{First argument must be a callable function.}
-\end{frame}
+\section{Solving linear equations}
\begin{frame}[fragile]
-\frametitle{Functions - Definition}
-\begin{lstlisting}
-In []: def f(x):
- return sin(x)+x**2
-In []: quad(f, 0, 1)
-\end{lstlisting}
-\begin{itemize}
-\item \typ{def}
-\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}
-\end{frame}
-
-
-\begin{frame}[fragile]
-\frametitle{Functions - Default Arguments}
-\begin{lstlisting}
-In []: def f(x=1):
- return sin(x)+x**2
-In []: f(10)
-Out[]: 99.455978889110625
-In []: f(1)
-Out[]: 1.8414709848078965
-In []: f()
-Out[]: 1.8414709848078965
-\end{lstlisting}
+\frametitle{Solution of equations}
+Consider,
+ \begin{align*}
+ 3x + 2y - z & = 1 \\
+ 2x - 2y + 4z & = -2 \\
+ -x + \frac{1}{2}y -z & = 0
+ \end{align*}
+Solution:
+ \begin{align*}
+ x & = 1 \\
+ y & = -2 \\
+ z & = -2
+ \end{align*}
\end{frame}
\begin{frame}[fragile]
-\frametitle{Functions - Keyword Arguments}
-\begin{lstlisting}
-In []: def f(x=1, y=pi):
- return sin(y)+x**2
-In []: f()
-Out[]: 1.0000000000000002
-In []: f(2)
-Out[]: 4.0
-In []: f(y=2)
-Out[]: 1.9092974268256817
-In []: f(y=pi/2,x=0)
-Out[]: 1.0
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
- \frametitle{More on functions}
- \begin{itemize}
- \item Scope of variables in the function is local
- \item Mutable items are \alert{passed by reference}
- \item First line after definition may be a documentation string
- (\alert{recommended!})
- \item Function definition and execution defines a name bound to the
- function
- \item You \emph{can} assign a variable to a function!
- \end{itemize}
+\frametitle{Solving using Matrices}
+Let us now look at how to solve this using \kwrd{matrices}
+ \begin{lstlisting}
+ In []: A = matrix([[3,2,-1],[2,-2,4],[-1, 0.5, -1]])
+ In []: b = matrix([[1], [-2], [0]])
+ In []: x = linalg.solve(A, b)
+ In []: Ax = dot(A, x)
+ In []: allclose(Ax, b)
+ Out[]: True
+ \end{lstlisting}
\end{frame}
\begin{frame}[fragile]
-\frametitle{Quadrature \ldots}
+\frametitle{Solution:}
\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 Use \typ{dblquad} for Double integrals
-\item Use \typ{tplquad} for Triple integrals
-\end{itemize}
-\end{frame}
-
-\subsection{ODEs}
-
-\begin{frame}[fragile]
-\frametitle{ODE Integration}
-We shall use the simple ODE of a simple pendulum.
-\begin{equation*}
-\ddot{\theta} = -\frac{g}{L}sin(\theta)
-\end{equation*}
-\begin{itemize}
-\item This equation can be written as a system of two first order ODEs
-\end{itemize}
-\begin{align}
-\dot{\theta} &= \omega \\
-\dot{\omega} &= -\frac{g}{L}sin(\theta) \\
- \text{At}\ t &= 0 : \nonumber \\
- \theta = \theta_0\quad & \&\quad \omega = 0 \nonumber
-\end{align}
-\end{frame}
+In []: x
+Out[]:
+array([[ 1.],
+ [-2.],
+ [-2.]])
-\begin{frame}[fragile]
-\frametitle{Solving ODEs using SciPy}
-\begin{itemize}
-\item We use the \typ{odeint} function from scipy to do the integration
-\item Define a function as below
-\end{itemize}
-\begin{lstlisting}
-In []: def pend_int(unknown, t, p):
- .... theta, omega = unknown
- .... g, L = p
- .... f=[omega, -(g/L)*sin(theta)]
- .... return f
- ....
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solving ODEs using SciPy \ldots}
-\begin{itemize}
-\item \typ{t} is the time variable \\
-\item \typ{p} has the constants \\
-\item \typ{initial} has the initial values
-\end{itemize}
-\begin{lstlisting}
-In []: t = linspace(0, 10, 101)
-In []: p=(-9.81, 0.2)
-In []: initial = [10*2*pi/360, 0]
-\end{lstlisting}
-\end{frame}
-
-\begin{frame}[fragile]
-\frametitle{Solving ODEs using SciPy \ldots}
-
-\small{\typ{In []: from scipy.integrate import odeint}}
-\begin{lstlisting}
-In []: pend_sol = odeint(pend_int,
- initial,t,
- args=(p,))
+In []: Ax
+Out[]:
+matrix([[ 1.00000000e+00],
+ [ -2.00000000e+00],
+ [ 2.22044605e-16]])
\end{lstlisting}
\end{frame}
@@ -412,18 +250,6 @@
\begin{itemize}
\item
\item
- \item Functions
- \begin{itemize}
- \item Definition
- \item Calling
- \item Default Arguments
- \item Keyword Arguments
- \end{itemize}
- \item Integration
- \begin{itemize}
- \item Quadrature
- \item ODEs
- \end{itemize}
\end{itemize}
\end{frame}