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diff --git a/lecture-notes/advanced-python/sage.rst b/lecture-notes/advanced-python/sage.rst new file mode 100644 index 0000000..46fcef4 --- /dev/null +++ b/lecture-notes/advanced-python/sage.rst @@ -0,0 +1,947 @@ +Sage notebook +============= + +In this section, we will learn what Sage is, what is Sage notebook, how to +start and use the sage notebook. In the notebook we will be specifically +learning how to execute our code, how to write annotations and other +content, typesetting the content and how to use the offline help available. + +To start with, What is Sage? Sage is a free, open-source mathematical +software. Sage can do a lot of math stuff for you including, but not +limited to, algebra, calculus, geometry, cryptography, graph theory among +other things. It can also be used as aid in teaching and research in any of +the areas that Sage supports. So let us start Sage now + +We are assuming that you have Sage installed on your computer now. If not +please visit the page +http://sagemath.org/doc/tutorial/introduction.html#installation for the +tutorial on how to install Sage. + + +Let us now learn how to start Sage. On the terminal type + +:: + + sage + +This should start a new Sage shell with the prompt sage: which looks like +this + +:: + + sage: + +So now we can type all the commands that Sage supports here. But Sage comes +bundled with a much more elegant tool called Sage Notebook. It provides a +web based user interface to use Sage. So once we have a Sage notebook +server up and running, all we need is a browser to access the Sage +functionality. + +We could also use a server run by somebody, else like the official instance +of the Sage server at http://sagenb.org. So all we need is just a browser, +a modern browser, to use Sage and nothing else! The Sage notebook also +provides a convenient way of sharing and publishing our work, which is very +handy for research and teaching. + +However we can also run our own instances of Sage notebook servers on all +the computers we have a local installation of Sage. To start the notebook +server just type + +:: + + notebook() + +on the Sage prompt. This will start the Sage Notebook server. If we are +starting the notebook server for the first time, we are prompted to enter +the password for the admin. Type the password and make a note of it. After +this Sage automatically starts a browser page, with the notebook opened. + +If it doesn't automatically start a browser page check if the Notebook +server started and there were no problems. If so open your browser and in +the address bar type the URL shown in the instructions upon running the +notebook command on the sage prompt. + +If you are not logged in yet, it shows the Notebook home page and textboxes +to type the username and the password. You can use the username 'admin' and +the password you gave while starting the notebook server for the first +time. There are also links to recover forgotten password and to create new +accounts. + +Once we are logged in with the admin account we can see the notebook admin +page. A notebook can contain a collection of Sage Notebook worksheets. + +Worksheet is basically a working area. This is where we enter all the Sage +commands on the notebook. + +The admin page lists all the worksheets created. On the topmost part of +this page we have the links to various pages. + +The home link takes us to the admin home page. The published link takes us +to the page which lists all the published worksheets. The log link has the +complete log of all the actions we did on the notebook. We have the +settings link where we can configure our notebook, the notebook server, +create and mangage accounts. We have a link to help upon clicking opens a +new window with the complete help of Sage. The entire documentation of Sage +is supplied with Sage for offline reference and the help link is the way to +get into it. Then we can report bugs about Sage by clicking on Report a +Problem link and there is a link to sign out of the notebook. + +We can create a new worksheet by clicking New Worksheet link + +Sage prompts you for a name for the worksheet. Let us name the worksheet as +nbtutorial. Now we have our first worksheet which is empty. + +A worksheet will contain a collection of cells. Every Sage command must be +entered in this cell. Cell is equivalent to the prompt on console. When we +create a new worksheet, to start with we will have one empty cell. Let us +try out some math here + +:: + + 2 + 2 + 57.1 ^ 100 + +The hat operator is used for exponentiation. We can observe that only the +output of the last command is displayed. By default each cell displays the +result of only the last operation. We have to use print statement to +display all the results we want to be displayed. + +Now to perform more operations we want more cells. So how do we create a +new cell? It is very simple. As we hover our mouse above or below the +existing cells we see a blue line, by clicking on this new line we can +create a new cell. + +We have a cell, we have typed some commands in it, but how do we evaluate +that cell? Pressing Shift along with Enter evaluates the cell. +Alternatively we can also click on the evaluate link to evaluate the cell + +:: + + matrix([[1,2], [3,4]])^(-1) + +After we create many cells, we may want to move between the cells. To move +between the cells use Up and Down arrow keys. Also clicking on the cell +will let you edit that particular cell. + +To delete a cell, clear the contents of the cell and hit backspace. + +If you want to add annotations in the worksheet itself on the blue line +that appears on hovering the mouse around the cell, Hold Shift and click on +the line. This creates a *What You See Is What You Get* cell. + +We can make our text here rich text. We can make it bold, Italics, we can +create bulleted and enumerated lists in this area + +:: + + This text contains both the **bold** text and also *italicised* + text. + It also contains bulleted list: + * Item 1 + * Item 2 + It also contains enumerate list: + 1. Item 1 + 2. Item 2 + +In the same cell we can display typeset math using the LaTeX like syntax + +:: + + $\int_0^\infty e^{-x} \, dx$ + +We enclose the math to be typeset within $ and $ or $$ and $$ as in LaTeX. + +We can also obtain help for a particular Sage command or function within +the worksheet itself by using a question mark following the command + +:: + + sin? + +Evaluating this cell gives me the entire help for the sin function inline +on the worksheet itself. Similarly we can also look at the source code of +each command or function using double question mark + +:: + + matrix?? + +Sage notebook also provides the feature for autocompletion. To auto-complete +a command type first few unique characters and hit tab key + +:: + + sudo<tab> + +To see all the commands starting with a specific name type those characters +and hit tab + +:: + + plo<tab> + +To list all the methods that are available for a certain variable or a +data-type we can use the variable name followed by the dot to access the +methods available on it and then hit tab + +:: + + s = 'Hello' + s.rep<tab> + +The output produced by each cell can be one of the three states. It can be +either the full output, or truncated output or hidden output. The output +area will display the error if the Sage code we wrote in the cell did not +successfully execute + +:: + + a, b = 10 + +The default output we obtained now is a truncated output. Clicking at the +left of the output area when the mouse pointer turns to hand gives us the +full output, clicking again makes the output hidden and it cycles. + +Lastly, Sage supports a variety of languages and each cell on the worksheet +can contain code written in a specific language. It is possible to instruct +Sage to interpret the code in the language we have written. This can be +done by putting percentage sign(%) followed by the name of the language. +For example, to interpret the cell as Python code we put + +:: + + %python + +as the first line in the cell. Similarly we have: %sh for shell scripting, +%fortran for Fortran, %gap for GAP and so on. Let us see how this works. +Say we have an integer. The type of the integer in default Sage mode is + +:: + + a = 1 + type(a) + + Output: <type 'sage.rings.integer.Integer'> + +We see that Integers are Sage Integers. Now let us put %python as the first +line of the cell and execute the same code snippet + +:: + + %python + a = 1 + type(a) + + Output: <type 'int'> + +Now we see that the integer is a Python integer. Why? Because now we +instructed Sage to interpret that cell as Python code. + +This brings us to the end of the section on using Sage. + +Symbolics +========= + +In this section, we shall learn to define symbolic expressions in Sage, use +built-in constants and functions, perform integration and differentiation +using Sage, define matrices and symbolic functions and simplify and solve +them. + +In addtion to a lot of other things, Sage can do Symbolic Math and we shall +start with defining symbolic expressions in Sage. + +On the sage notebook type + +:: + + sin(y) + +It raises a name error saying that ``y`` is not defined. We need to declare +``y`` as a symbol. We do it using the ``var`` function. + +:: + + var('y') + +Now if you type + +:: + + sin(y) + +Sage simply returns the expression. + +Sage treats ``sin(y)`` as a symbolic expression. We can use this to do +symbolic math using Sage's built-in constants and expressions. + +Let us try out a few examples. + +:: + + var('x, alpha, y, beta') + (x^2/alpha^2)+(y^2/beta^2) + +We have defined 4 variables, ``x``, ``y``, ``alpha`` and ``beta`` and +have defined a symbolic expression using them. + +Here is an expression in ``theta`` + +:: + + var('theta') + sin(theta)*sin(theta)+cos(theta)*cos(theta) + +Sage also provides built-in constants which are commonly used in +mathematics, for instance pi, e, infinity. The function ``n`` gives the +numerical values of all these constants. + +:: + + n(pi) + n(e) + n(oo) + +If you look into the documentation of function ``n`` by doing + +:: + + n? + +You will see what all arguments it takes and what it returns. It will be +very helpful if you look at the documentation of all functions introduced, +in this section. + +Also we can define the number of digits we wish to have in the constants. +For this we have to pass an argument -- digits. + +:: + + n(pi, digits = 10) + +Apart from the constants Sage also has a lot of built-in functions like +``sin``, ``cos``, ``log``, ``factorial``, ``gamma``, ``exp``, ``arcsin`` +etc ... + +Lets try some of them out on the Sage notebook. + +:: + + sin(pi/2) + + arctan(oo) + + log(e,e) + +Given that we have defined variables like x, y etc., we can define an +arbitrary function with desired name + +:: + + var('x') + function('f',x) + +Here f is the name of the function and x is the independent variable . +Now we can define f(x) to be + +:: + + f(x) = x/2 + sin(x) + +Evaluating this function f for the value x=pi returns pi/2. + +:: + + f(pi) + +We can also define functions that are not continuous but defined piece-wise. +Let us define a function which is a parabola between 0 to 1 and a constant +from 1 to 2 . + +:: + + + var('x') + h(x)=x^2 + g(x)=1 + + f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) + f + +We can also define functions convergent series and other series. + +We first define a function f(n) in the way discussed above. + +:: + + var('n') + function('f', n) + + +To sum the function for a range of discrete values of n, we use the +sage function sum. + +For a convergent series , f(n)=1/n^2 we can say + +:: + + var('n') + function('f', n) + f(n) = 1/n^2 + sum(f(n), n, 1, oo) + + +Lets us now try another series + +:: + + + f(n) = (-1)^(n-1)*1/(2*n - 1) + sum(f(n), n, 1, oo) + +This series converges to pi/4. + +Let's now look at some calculus. + +:: + + diff(x**2 + sin(x), x) + +The diff function differentiates an expression or a function. It's first +argument is expression or function and second argument is the independent +variable. + +We have already tried an expression now lets try a function + +:: + + f = exp(x^2) + arcsin(x) + diff(f(x), x) + +To get a higher order differential we need to add an extra third argument +for order + +:: + + diff(f(x), x, 3) + +in this case it is 3. + +Just like differentiation of expression you can also integrate them + +:: + + x = var('x') + s = integral(1/(1 + (tan(x))**2),x) + s + +Many a times we need to find factors of an expression, we can use the +"factor" function + +:: + + y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) + f = factor(y) + +One can simplify complicated expression + +:: + + f.simplify_full() + +This simplifies the expression fully. We can also do simplification of +just the algebraic part and the trigonometric part + +:: + + f.simplify_exp() + f.simplify_trig() + +One can also find roots of an equation by using ``find_root`` function + +:: + + phi = var('phi') + find_root(cos(phi)==sin(phi),0,pi/2) + +Let's substitute this solution into the equation and see we were +correct + +:: + + var('phi') + f(phi)=cos(phi)-sin(phi) + root=find_root(f(phi)==0,0,pi/2) + f.substitute(phi=root) + +as we can see when we substitute the value the answer is almost = 0 showing +the solution we got was correct. + +Lets us now try some matrix algebra symbolically :: + + var('a,b,c,d') + A=matrix([[a,1,0],[0,b,0],[0,c,d]]) + A + +Now lets do some of the matrix operations on this matrix + +:: + A.det() + A.inverse() + + +That brings us to the end of our discussion on symbolics. + +Plotting using Sage +=================== + +In this section we shall look at + + * 2D plotting in SAGE + * 3D plotting in SAGE + +We shall first create a symbolic variable ``x`` + +:: + + x = var('x') + +We shall plot the function ``sin(x) - cos(x) ^ 2`` in the range (-5, 5). + +:: + + plot(sin(x) - cos(x) ^ 2, (x, -5, 5)) + +As we can see, the plot is shown. + +``plot`` command takes the symbolic function as the first argument and the +range as the second argument. + +We have seen that plot command plots the given function on a linear range. + +What if the x and y values are functions of another variable. For instance, +lets plot the trajectory of a projectile. + +A projectile was thrown at 50 m/s^2 and at an angle of 45 degrees from the +ground. We shall plot the trajectory of the particle for 5 seconds. + +These types of plots can be drawn using the parametric_plot function. We +first define the time variable. + +:: + + t = var('t') + +Then we define the x and y as functions of t. + +:: + + f_x = 50 * cos(pi/4) + f_y = 50 * sin(pi/4) * t - 1/2 * 9.81 * t^2 ) + +We then call the ``parametric_plot`` function as + +:: + + parametric_plot((f_x, f_y), (t, 0, 5)) + +And we can see the trajectory of the projectile. + +The ``parametric_plot`` funciton takes a tuple of two functions as the +first argument and the range over which the independent variable varies as +the second argument. + +Now we shall look at how to plot a set of points. + +We have the ``line`` function to achieve this. + +We shall plot sin(x) at few points and join them. + +First we need the set of points. + +:: + + points = [ (x, sin(x)) for x in srange(-2*float(pi), 2*float(pi), 0.75) ] + +``srange`` takes a start, a stop and a step argument and returns a list of +point. We generate list of tuples in which the first value is ``x`` and +second is ``sin(x)``. + +:: + + line(points) + +plots the points and joins them with a line. + +The ``line`` function behaves like the plot command in matplotlib. The +difference is that ``plot`` command takes two sequences while line command +expects a sequence of co-ordinates. + +As we can see, the axes limits are set by SAGE. Often we would want to set +them ourselves. Moreover, the plot is shown here since the last command +that is executed produces a plot. + +Let us try this example + +:: + + plot(cos(x), (x,0,2*pi)) + # Does the plot show up?? + +As we can see here, the plot is not shown since the last command does not +produce a plot. + +The actual way of showing a plot is to use the ``show`` command. + +:: + + p1 = plot(cos(x), (x,0,2*pi)) + show(p1) + # What happens now?? + +As we can see the plot is shown since we used it with ``show`` command. + +``show`` command is also used set the axes limits. + +:: + + p1 = plot(cos(x), (x,0,2*pi)) + show(p1, xmin=0, xmax=2*pi, ymin=-1.2, ymax=1.2) + +As we can see, we just have to pass the right keyword arguments to the +``show`` command to set the axes limits. + +The ``show`` command can also be used to show multiple plots. + +:: + + p1 = plot(cos(x), (x, 0, 2*pi)) + p2 = plot(sin(x), (x, 0, 2*pi)) + show(p1+p2) + +As we can see, we can add the plots and use them in the ``show`` command. + +Now we shall look at 3D plotting in SAGE. + +We have the ``plot3d`` function that takes a function in terms of two +independent variables and the range over which they vary. + +:: + + x, y = var('x y') + plot3d(x^2 + y^2, (x, 0, 2), (y, 0, 2)) + +We get a 3D plot which can be rotated and zoomed using the mouse. + +``parametric_plot3d`` function plots the surface in which x, y and z are +functions of another variable. + +:: + + u, v = var("u v") + f_x = u + f_y = v + f_z = u^2 + v^2 + parametric_plot3d((f_x, f_y, f_z), (u, 0, 2), (v, 0, 2)) + +Using Sage +========== + +In this section we shall quickly look at a few examples of using Sage for +Linear Algebra, Calculus, Graph Theory and Number theory. + +Let us begin with Calculus. We shall be looking at limits, differentiation, +integration, and Taylor polynomial. + +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. + +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. + +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. + +Using Sage to Teach +=================== + +In this section, we shall look at + + * How to use the "@interact" feature of SAGE for better demonstration + * How to use SAGE for collaborative learning + +Let us look at a typical example of demonstrating a damped oscillation. + +:: + + t = var('t') + p1 = plot( e^(-t) * sin(2*t), (t, 0, 15)) + show(p1) + +Now let us reduce the damping factor + +:: + + t = var('t') + p1 = plot(e^(-t/2) * sin(2*t), (t, 0, 15)) + show(p1) + +Now if we want to reduce the damping factor even more, we would be using +e^(-t/3). We can observe that every time we have to change, all we do is +change something very small and re evaluate the cell. + +This process can be simplified, using the ``@interact`` feature of SAGE. + +:: + + @interact + def plot_damped(n=1): + t = var('t') + p1 = plot( e^(-t/n) * sin(2*t), (t, 0, 20)) + show(p1) + +We can see that the function is evaluated and the plot is shown. We can +also see that there is a field to enter the value of ``n`` and it is +currently set to ``1``. Let us change it to 2 and hit enter. + +We see that the new plot with reduced damping factor is shown. Similarly we +can change ``n`` to any desired value and hit enter and the function will +be evaluated. + +This is a very handy tool while demonstrating or teaching. + +Often we would want to vary a parameter over range instead of taking it as +an input from the user. For instance we do not want the user to give ``n`` +as 0 for the damping oscillation we discussed. In such cases we use a range +of values as the default argument. + +:: + + @interact + def plot_damped(n=(1..10)): + t = var('t') + p1 = plot( e^(-t/n) * sin(2*t), (t, 0, 20)) + show(p1) + +We get similar plot but the only difference is the input widget. Here it is +a slider unlike an input field. We can see that as the slider is moved, the +function is evaluated and plotted accordingly. + +Sometimes we want the user to have only a given set of options. We use a +list of items as the default argument in such situations. + +:: + + @interact + def str_shift(s="STRING", shift=(0..8), direction=["Left", "Right"]): + shift_len = shift % len(s) + chars = list(s) + if direction == "Right": + shifted_chars = chars[-shift_len:] + chars[:-shift_len] + else: + shifted_chars = chars[shift_len:] + chars[:shift_len] + print "Actual String:", s + print "Shifted String:", "".join(shifted_chars) + +We can see that buttons are displayed which enables us to select from a +given set of options. + +We have learnt how to use the ``@interact`` feature of SAGE for better +demonstration. We shall look at how to use SAGE worksheets for +collaborative learning. + +The first feature we shall see is the ``publish`` feature. Open a worksheet +and in the top right, we can see a button called ``publish``. Click on that +and we get a confirmation page with an option for re publishing. + +For now lets forget that option and simply publish by clicking ``yes``. The +worksheet is now published. + +Now lets sign out and go to the sage notebook home. We see link to browse +published worksheets. Lets click on it and we can see the worksheet. This +does not require login and anyone can view the worksheet. + +Alternatively, if one wants to edit the sheet, there is a link on top left +corner that enables the user to download a copy of the sheet onto their +home. This way they can edit a copy of the worksheet. + +We have learnt how to publish the worksheets to enable users to edit a +copy. Next, we shall look at how to enable users to edit the actual +worksheet itself. + +Let us open the worksheet and we see a link called ``share`` on the top +right corner of the worksheet. Click the link and we get a box where we can +type the usernames of users whom we want to share the worksheet with. We +can even specify multiple users by seperating their names using commas. +Once we have shared the worksheet, the worksheet appears on the home of +shared users. + +.. + Local Variables: + mode: rst + indent-tabs-mode: nil + sentence-end-double-space: nil + fill-column: 75 + End: |