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SF.net SVN: matplotlib: [4022] trunk/py4science/workbook
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From: <jd...@us...> - 2007-10-26 19:44:09
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Revision: 4022
http://matplotlib.svn.sourceforge.net/matplotlib/?rev=4022&view=rev
Author: jdh2358
Date: 2007-10-26 12:44:07 -0700 (Fri, 26 Oct 2007)
Log Message:
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added some more workbook files
Added Paths:
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trunk/py4science/workbook/glass_dots.tex
trunk/py4science/workbook/intro_linalg.tex
trunk/py4science/workbook/intro_sigproc.tex
Added: trunk/py4science/workbook/glass_dots.tex
===================================================================
--- trunk/py4science/workbook/glass_dots.tex (rev 0)
+++ trunk/py4science/workbook/glass_dots.tex 2007-10-26 19:44:07 UTC (rev 4022)
@@ -0,0 +1,52 @@
+\section{Glass Moir\'e Patterns}
+\label{sec:glass_patterns}
+
+When a random dot pattern is scaled, rotated, and superimposed over
+the original dots, interesting visual patterns known as Glass Patterns
+emerge\footnote{L. Glass. 'Moir\'e effect from random dots' Nature 223,
+ 578580 (1969).} In this exercise, we generate random dot fields
+using numpy's uniform distribution function, and apply
+transformations to the random dot field using a scale $\mathbf{S}$
+and rotation $\mathbf{R}$ matrix $\mathbf{X_2} = \mathbf{S} \mathbf{R}
+\mathbf{X_1}$.
+
+If the scale and rotation factors are small, the transformation is
+analogous to a single step in the numerical solution of a 2D ODE, and
+the plot of both $\mathbf{X_1}$ and $\mathbf{X_2}$ will reveal the
+structure of the vecotr field flow around the fixed point (the
+invariant under the transformation); see for example the
+\textit{stable focus}, aka \textit{spiral}, in
+Figure~\ref{fig:glass_dots1}.
+
+The eigenvalues of the tranformation matrix $\mathbf{M} =
+\mathbf{S}\mathbf{R}$ determine the type of fix point:
+\textit{center}, \textit{stable focus}, \textit{saddle node},
+etc\dots. For example, if the two eigenvalues are real but differing
+in signs, the fixed point is a \textit{saddle node}. If the real
+parts of both eigenvalues are negative and the eigenvalues are
+complex, the fixed point is a \textit{stable focus}. The complex part
+of the eigenvalue determines whether there is any rotation in the
+matrix transformation, so another way to look at this is to break out
+the scaling and rotation components of the transformation
+$\textbf{M}$. If there is a rotation component, then the fixed point
+will be a \textit{center} or a \textit{focus}. If the scaling
+components are both one, the rotation will be a \textit{center}, if
+they are both less than one (contraction), it will be a \textit{stable
+ focus}. Likewise, if there is no rotation component, the fixed
+point will be a \textit{node}, and the scaling components will
+determine the type of node. If both are less than one, we have a
+\textit{stable node}, if one is greater than one and the other less
+than one, we have a \textit{saddle node}.
+
+\lstinputlisting[label=code:glass_dots1_skel,caption={IGNORED}]{skel/glass_dots1_skel.py}
+
+
+
+\begin{center}%
+\begin{figure}
+\begin{centering}\includegraphics[width=4in]{fig/glass_dots1}\par\end{centering}
+
+
+\caption{\label{fig:glass_dots1}Glass pattern showing a stable focus}
+\end{figure}
+\par\end{center}
Added: trunk/py4science/workbook/intro_linalg.tex
===================================================================
--- trunk/py4science/workbook/intro_linalg.tex (rev 0)
+++ trunk/py4science/workbook/intro_linalg.tex 2007-10-26 19:44:07 UTC (rev 4022)
@@ -0,0 +1,49 @@
+Like matlab, numpy and scipy have support for fast linear algebra
+built upon the highly optimized LAPACK, BLAS and ATLAS fortran linear
+algebra libraries. Unlike Matlab, in which everything is a matrix or
+vector, and the '*' operator always means matrix multiple, the default
+object in numpy is an \texttt{array}, and the '*' operator on arrays means
+element-wise multiplication.
+
+Instead, numpy provides a \texttt{matrix} class if you want to do
+standard matrix-matrix multiplication with the '*' operator, or the
+\texttt{dot} function if you want to do matrix multiplies with plain
+arrays. The basic linear algebra functionality is found in
+\texttt{numpy.linalg}
+
+\begin{lstlisting}
+In [1]: import numpy as npy
+In [2]: import numpy.linalg as linalg
+
+# X and Y are arrays
+In [3]: X = npy.random.rand(3,3)
+In [4]: Y = npy.random.rand(3,3)
+
+# * operator is element wise multiplication, not matrix matrix
+In [5]: print X*Y
+[[ 0.00973215 0.18086148 0.05539387]
+ [ 0.00817516 0.63354021 0.2017993 ]
+ [ 0.34287698 0.25788149 0.15508982]]
+
+# the dot function will use optimized LAPACK to do matrix-matix
+# multiply
+In [6]: print npy.dot(X, Y)
+[[ 0.10670678 0.68340331 0.39236388]
+ [ 0.27840642 1.14561885 0.62192324]
+ [ 0.48192134 1.32314856 0.51188578]]
+
+# the matrix class will create matrix objects that support matrix
+# multiplication with *
+In [7]: Xm = npy.matrix(X)
+In [8]: Ym = npy.matrix(Y)
+In [9]: print Xm*Ym
+[[ 0.10670678 0.68340331 0.39236388]
+ [ 0.27840642 1.14561885 0.62192324]
+ [ 0.48192134 1.32314856 0.51188578]]
+
+# the linalg module provides functions to compute eigenvalues,
+# determinants, etc. See help(linalg) for more info
+In [10]: print linalg.eigvals(X)
+[ 1.46131600+0.j 0.46329211+0.16501143j 0.46329211-0.16501143j]
+
+\end{lstlisting}
Added: trunk/py4science/workbook/intro_sigproc.tex
===================================================================
--- trunk/py4science/workbook/intro_sigproc.tex (rev 0)
+++ trunk/py4science/workbook/intro_sigproc.tex 2007-10-26 19:44:07 UTC (rev 4022)
@@ -0,0 +1,15 @@
+\texttt{numpy} and \texttt{scipy} provide many of the essential tools
+for digital signal processing. \texttt{scipy.signal} provides basic
+tools for digital filter design and filtering (eg Butterworth
+filters), a linear systems toolkit, standard waveforms such as square
+waves, and saw tooth functions, and some basic wavelet functionality.
+\texttt{scipy.fftpack} provides a suite of tools for Fourier domain
+analysis, including 1D, 2D, and ND discrete fourier transform and
+inverse functions, in addition to other tools such as analytic signal
+representations via the Hilbert trasformation (\texttt{numpy.fft} also
+provides basic FFT functions). \texttt{pylab} provides Matlab
+compatible functions for computing and plotting standard time series
+analyses, such as historgrams (\texttt{hist}), auto and cross
+correlations (\texttt{acorr} and \texttt{xcorr}), power spectra and
+coherence spectra (\texttt{psd}, \texttt{csd}, \texttt{cohere} and
+\texttt{specgram}.
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