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| -rwxr-xr-x | day1/cheatsheet1.tex | 74 | ||||
| -rwxr-xr-x | day1/cheatsheet6.tex | 114 |
2 files changed, 144 insertions, 44 deletions
diff --git a/day1/cheatsheet1.tex b/day1/cheatsheet1.tex index 4439a49..11fd385 100755 --- a/day1/cheatsheet1.tex +++ b/day1/cheatsheet1.tex @@ -37,19 +37,52 @@ Do you really want to exit ([y]/n)? y \end{lstlisting} %$ \section{Plotting} +\subsection{linspace} +\typ{In []: x = linspace(start, stop, num)}\\ +\typ{linspace} returns array of length \typ{num}, for which \typ{x[0] = start} and \typ{x[num-1] = stop} \\ +\emph{Please note indices of array starts from zero(0)} -\subsection{Label} +\subsection{plot} +\typ{In []: plot(X, Y)}\\ +For given arrays of equal length(above case X and Y), \typ{plot} plots the correspoding *x* and *y* pairs taken from X and Y. + +\subsection{Colors of plots} +\typ{In []: plot(y, sin(y), 'g')}\\ +Plots graph with green color. Other options available are: +\begin{lstlisting} + 'r' ---> Red + 'b' ---> Blue + 'r' ---> Red + 'c' ---> Cyan + 'm' ---> Magenta + 'y' ---> Yellow + 'k' ---> Black + 'w' ---> White +\end{lstlisting} +One can also set the line width of plot using optional argument \typ{linewidth}. For example:\\ +\typ{In []: plot(x, cos(x), 'r', linewidth=2)}\\ +Plots the line with linewidth = 2 +\subsection{label and title} +\typ{In []: xlabel('Length') #sets *x* axis label to Length}\\ +\typ{In []: ylabel('Time') #sets *y* axis label to Time.}\\ +\typ{In []: title('Sinusoids') #sets title of plot}\\ +\\ +\textbf{Additionally}\\ Pylab accepts TeX equation expressions in any text expression. To get something like:\\ $\sigma_i=15$ \\ on title of figure use: \begin{lstlisting} - title('$\sigma_i=15$') +In []: title('$\sigma_i=15$') \end{lstlisting} Same way one can have TeX expression on xlabel, ylabel etc. \subsection{legends} +\typ{In []: legend('sin(x)',loc=center)} \\ +Placec a legend on the current plot at location *loc*.\\ Apart from \kwrd{center}, some other \kwrd{loc} which can be specified are: \begin{lstlisting} +'best' +'right' 'upper right' 'upper left' 'lower left' @@ -64,26 +97,20 @@ One can also mention explicit co-ordinates for placement of legend. \begin{lstlisting} In []: legend(['sin(2y)'], loc=(.8,.1)) \end{lstlisting} -\typ{loc = 0, 1} (left top position of graph)\\ +\typ{loc = 0, 1} (top left position of graph)\\ \typ{loc = 0.5, 0.5} (center of graph). -\subsection{Saving figures} -One can save figure in any of these formats: png, pdf, ps, eps and svg. +\subsection{Annotate} +\typ{In []: annotate('local max', xy=(1.5, 1))}\\ +Annotates current plot with text, 'local max', at position specified to \typ{xy}. -\subsection{Colors of plots} -\typ{In []: plot(y, sin(y), 'g')}\\ -Plots graph with green color. Other options available are: -\begin{lstlisting} - 'r' ---> Red - 'b' ---> Blue - 'r' ---> Red - 'c' ---> Cyan - 'm' ---> Magenta - 'y' ---> Yellow - 'k' ---> Black - 'w' ---> White -\end{lstlisting} +\subsection{Saving figures} +\typ{In []: savefig('sinusoids.png')}\\ +Saves the current figure with file name 'sinusoids.png' in current working directory. One can save figure in any of these formats: png, pdf, ps, eps and svg. +\subsection{Miscellaneous} +\typ{In []: clf() #Clears the current plot area}\\ +\typ{In []: close() #Closes the figure} \section{Saving and running scripts} \begin{itemize} \item \typ{\%hist}\\ @@ -105,14 +132,13 @@ In []: xlabel('x') In []: title('Sinusoidal Waves') In []: legend(['sin(x)', 'cos(x)']) In []: annotate('origin', xy=(0, 0)) -In []: xmin, xman = xlim() # Without arguments gets -In []: ymin, ymax = ylim() # values - -In []: xlim(0, 2 * pi) # With values, sets the -In []: ylim(ymin - 0.2, ymax + 0.2) # specified values +In []: xmin, xman = xlim() # returns current X axis limits. +In []: ymin, ymax = ylim() +In []: xlim(0, 2 * pi) # sets the X axis limits to passed values +In []: ylim(ymin - 0.2, ymax + 0.2) In []: savefig('sin.png') # Save figure -In []: close() # Closes the figure +In []: close() \end{lstlisting} \section{References} diff --git a/day1/cheatsheet6.tex b/day1/cheatsheet6.tex index 2e9de97..5a9f652 100755 --- a/day1/cheatsheet6.tex +++ b/day1/cheatsheet6.tex @@ -1,6 +1,20 @@ \documentclass[12pt]{article} \title{Solving Equations \& ODEs} \author{FOSSEE} +\usepackage{listings} +\lstset{language=Python, + basicstyle=\ttfamily, +commentstyle=\itshape\bfseries, +showstringspaces=false, +} +\newcommand{\typ}[1]{\lstinline{#1}} +\usepackage[english]{babel} +\usepackage[latin1]{inputenc} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ae,aecompl} +\usepackage{mathpazo,courier,euler} +\usepackage[scaled=.95]{helvet} \begin{document} \date{} \vspace{-1in} @@ -9,27 +23,87 @@ \large{FOSSEE} \end{center} \section{Solving linear equations} -\begin{verbatim} - In []: A = array([[3,2,-1], - [2,-2,4], - [-1, 0.5, -1]]) - In []: b = array([[1], [-2], [0]]) - In []: x = solve(A, b) - In []: Ax = dot(A,x) - In []: allclose(Ax, b) - Out[]: True -\end{verbatim} +Condier following sets of equations:\\ + \begin{align*} + 3x + 2y - z & = 1 \\ + 2x - 2y + 4z & = -2 \\ + -x + $\frac{1}{2}$y -z & = 0 + \end{align*}\\ +The matrix representation is:\\ +\begin{center} +$A*x = B$ +\end{center} +Where A is coefficient matrix(in this case 3x3)\\ +B is constant matrix(1x3)\\ +x is the required solution.\\ +\begin{lstlisting} +In []: A = array([[3,2,-1], [2,-2,4], [-1, 0.5, -1]]) +In []: B = array([[1], [-2], [0]]) +In []: x = solve(A, B) +\end{lstlisting} +Solve the equation $A x = B$ for $x$.\\ +To check whether solution is correct try this: +\begin{lstlisting} +In []: Ax = dot(A,x) #Matrix multiplication of A and x(LHS) +In []: allclose(Ax, B) +Out[]: True +\end{lstlisting} +\typ{allclose} Returns \typ{True} if two arrays(in above case Ax and B) are element-wise equal within a tolerance. +\newpage \section{Finding roots} -\begin{verbatim} - In []: coeffs = [1, 6, 13] - In []: roots(coeffs) -\end{verbatim} -Finding the roots of a function -\begin{verbatim} -In []: fsolve(sin(x)+cos(x)**2, 0) -\end{verbatim} +\subsection{Polynomials} +\begin{center} + $x^2+6x+13=0$ +\end{center} +to find roots, pylab provides \typ{roots} function. +\begin{lstlisting} +In []: coeffs = [1, 6, 13] #list of all coefficients +In []: roots(coeffs) +\end{lstlisting} +\subsection{functions} +Functions can be defined and used by following syntax: +\begin{lstlisting} +def func_name(arg1, arg2): + #function code + return ret_value +\end{lstlisting} +A simple example can be +\begin{lstlisting} +def expression(x): + y = x*sin(x) + return y +\end{lstlisting} +Above function when called with a argument, will return $xsin(x)$ value for that argument. +\begin{lstlisting} +In [95]: expression(pi/2) +Out[95]: 1.5707963267948966 + +In [96]: expression(pi/3) +Out[96]: 0.90689968211710881 +\end{lstlisting} +\subsection{Roots of non-linear eqations} +For Finding the roots of a non linear equation(defined as $f(x)=0$), around a starting estimate we use \typ{fsolve}:\\ +\typ{In []: from scipy.optimize import fsolve}\\ +\typ{fsolve} is not part of \typ{pylab}, instead it is part of \textbf{optimize} package of \textbf{scipy}, and hence we \textbf{import} it.\\ +%\typ{fsolve} takes first argument as name of function, which evaluates $f(x)$, whose roots one wants to find. And second argument is starting estimate, around which roots are found. +For illustration, we want to find roots of equation: +\begin{center} + $f(x)=sin(x)+cos(x)^2$ +\end{center} +So just like we did above, we define a function: +\begin{lstlisting} +In []: def f(x): + ....: return sin(x)+cos(x)**2 + ....: +\end{lstlisting} +Now to find roots of this non linear equation, around a initial estimate value, say 0, we use \typ{fsolve} in following way: +\begin{lstlisting} +In []: fsolve(f, 0) #arguments are function name and estimate +Out[]: -0.66623943249251527 +\end{lstlisting} + \section{ODE} -\begin{verbatim} +\begin{lstlisting} In []: def epid(y, t): .... k, L = 0.00003, 25000 .... return k*y*(L-y) @@ -40,5 +114,5 @@ In []: fsolve(sin(x)+cos(x)**2, 0) In []: y = odeint(epid, 250, t) In []: plot(t, y) -\end{verbatim} +\end{lstlisting} \end{document} |