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diff --git a/using_sage/script.rst b/using_sage/script.rst new file mode 100644 index 0000000..e32ac0d --- /dev/null +++ b/using_sage/script.rst @@ -0,0 +1,224 @@ +.. Objectives +.. ---------- + +.. By the end of this tutorial you will -- + +.. 1. Get an idea of the range of things for which Sage can be used. +.. #. Know some of the functions for Calculus +.. #. Get some insight into Graphs in Sage. + + +.. Prerequisites +.. ------------- + +.. Getting Started -- Sage + +.. Author : Puneeth + Internal Reviewer : Anoop Jacob Thomas<anoop@fossee.in> + External Reviewer : + Language Reviewer : Bhanukiran + Checklist OK? : <06-11-2010, Anand, OK> [2010-10-05] + +Script +------ + +{{{ show the welcome slide }}} + +Hello Friends. Welcome to this tutorial on using Sage. + +{{{ show the slide with outline }}} + +In this tutorial we shall quickly look at a few examples of using Sage +for Linear Algebra, Calculus, Graph Theory and Number theory. + +{{{ show the slide with Calculus outline }}} + +Let us begin with Calculus. We shall be looking at limits, +differentiation, integration, and Taylor polynomial. + +{{{ show sage notebook }}} + +We have our Sage notebook running. In case, you don't have it running, +start is using the command, ``sage --notebook``. + +To find the limit of the function x*sin(1/x), at x=0, we say +:: + + lim(x*sin(1/x), x=0) + +We get the limit to be 0, as expected. + +It is also possible to the limit at a point from one direction. For +example, let us find the limit of 1/x at x=0, when approaching from +the positive side. +:: + + lim(1/x, x=0, dir='above') + +To find the limit from the negative side, we say, +:: + + lim(1/x, x=0, dir='below') + +Let us now see how to differentiate, using Sage. We shall find the +differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``. We +shall first define the expression, and then use the ``diff`` function +to obtain the differential of the expression. +:: + + var('x') + f = exp(sin(x^2))/x + + diff(f, x) + +We can also obtain the partial differentiation of an expression w.r.t +one of the variables. Let us differentiate the expression +``exp(sin(y - x^2))/x`` w.r.t x and y. +:: + + var('x y') + f = exp(sin(y - x^2))/x + + diff(f, x) + + diff(f, y) + +Now, let us look at integration. We shall use the expression obtained +from the differentiation that we did before, ``diff(f, y)`` --- +``e^(sin(-x^2 + y))*cos(-x^2 + y)/x``. The ``integrate`` command is +used to obtain the integral of an expression or function. +:: + + integrate(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y) + +We get back the correct expression. The minus sign being inside or +outside the ``sin`` function doesn't change much. + +Now, let us find the value of the integral between the limits 0 and +pi/2. +:: + + integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2) + +Let us now see how to obtain the Taylor expansion of an expression +using sage. Let us obtain the Taylor expansion of ``(x + 1)^n`` up to +degree 4 about 0. +:: + + var('x n') + taylor((x+1)^n, x, 0, 4) + +This brings us to the end of the features of Sage for Calculus, that +we will be looking at. For more, look at the Calculus quick-ref from +the Sage Wiki. + +Next let us move on to Matrix Algebra. + +{{{ show the equation on the slides }}} + +Let us begin with solving the equation ``Ax = v``, where A is the +matrix ``matrix([[1,2],[3,4]])`` and v is the vector +``vector([1,2])``. + +To solve the equation, ``Ax = v`` we simply say +:: + + x = solve_right(A, v) + +To solve the equation, ``xA = v`` we simply say +:: + + x = solve_left(A, v) + +The left and right here, denote the position of ``A``, relative to x. + +#[Puneeth]: any suggestions on what more to add? + +Now, let us look at Graph Theory in Sage. + +We shall look at some ways to create graphs and some of the graph +families available in Sage. + +The simplest way to define an arbitrary graph is to use a dictionary +of lists. We create a simple graph by +:: + + G = Graph({0:[1,2,3], 2:[4]}) + +We say +:: + + G.show() + +to view the visualization of the graph. + +Similarly, we can obtain a directed graph using the ``DiGraph`` +function. +:: + + G = DiGraph({0:[1,2,3], 2:[4]}) + + +Sage also provides a lot of graph families which can be viewed by +typing ``graph.<tab>``. Let us obtain a complete graph with 5 vertices +and then show the graph. +:: + + G = graphs.CompleteGraph(5) + + G.show() + + +Sage provides other functions for Number theory and +Combinatorics. Let's have a glimpse of a few of them. + + +:: + + prime_range(100, 200) + +gives primes in the range 100 to 200. + +:: + + is_prime(1999) + +checks if 1999 is a prime number or not. + +:: + + factor(2001) + +gives the factorized form of 2001. + +:: + + C = Permutations([1, 2, 3, 4]) + C.list() + +gives the permutations of ``[1, 2, 3, 4]`` + +:: + + C = Combinations([1, 2, 3, 4]) + C.list() + +gives all the combinations of ``[1, 2, 3, 4]`` + +That brings us to the end of this session showing various features +available in Sage. + +.. #[[Anoop: I feel we should add more slides, a possibility is to add + the code which they are required to type in, I also feel we should + add some review problems for them to try out.]] + +{{{ Show summary slide }}} + +We have looked at some of the functions available for Linear Algebra, +Calculus, Graph Theory and Number theory. + +This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India + +Hope you have enjoyed and found it useful. +Thank you! + |