1 Symbolics with Sage

     2 -------------------

     3 

     4 Hello friends and welcome to the tutorial on symbolics with sage.

     5 

     6 {{{ Show welcome slide }}}

     7 

     8 

     9 .. #[Madhu: What is this line doing here. I don't see much use of it]

    10 

    11 During the course of the tutorial we will learn

    12 

    13 {{{ Show outline slide  }}}

    14 

    15 * Defining symbolic expressions in sage.  

    16 * Using built-in costants and functions. 

    17 * Performing Integration, differentiation using sage. 

    18 * Defining matrices. 

    19 * Defining Symbolic functions.  

    20 * Simplifying and solving symbolic expressions and functions.

    21 

    22 We can use Sage for symbolic maths. 

    23 

    24 On the sage notebook type::

    25    

    26     sin(y)

    27 

    28 It raises a name error saying that y is not defined. But in sage we

    29 can declare y as a symbol using var function.

    30 

    31 

    32 ::

    33     var('y')

    34    

    35 Now if you type::

    36 

    37     sin(y)

    38 

    39 sage simply returns the expression.

    40 

    41 

    42 Thus sage treats sin(y) as a symbolic expression . We can use

    43 this to do  symbolic maths using sage's built-in constants and

    44 expressions..

    45 

    46 

    47 So let us try ::

    48    

    49    var('x,alpha,y,beta') 

    50    x^2/alpha^2+y^2/beta^2

    51  

    52 taking another example

    53    

    54    var('theta')

    55    sin^2(theta)+cos^2(theta)

    56 

    57 

    58 Similarly, we can define many algebraic and trigonometric expressions

    59 using sage .

    60 

    61 

    62 Sage also provides a few built-in constants which are commonly used in

    63 mathematics .

    64 

    65 example : pi,e,infinity , Function n gives the numerical values of all these

    66     constants.

    67 

    68 {{{ Type n(pi)

    69    	n(e)

    70 	n(oo) 

    71     On the sage notebook }}}  

    72 

    73 

    74 

    75 If you look into the documentation of function "n" by doing

    76 

    77 .. #[Madhu: "documentation of the function "n"?]

    78 

    79 ::

    80    n(<Tab>

    81 

    82 You will see what all arguments it takes and what it returns. It will be very

    83 helpful if you look at the documentation of all functions introduced through

    84 this script.

    85 

    86 

    87 

    88 Also we can define the no. of digits we wish to use in the numerical

    89 value . For this we have to pass an argument digits.  Type

    90 

    91 .. #[Madhu: "no of digits"? Also "We wish to obtain" than "we wish to

    92      use"?]

    93 ::

    94 

    95    n(pi, digits = 10)

    96 

    97 Apart from the constants sage also has a lot of builtin functions like

    98 sin,cos,log,factorial,gamma,exp,arcsin etc ...

    99 lets try some of them out on the sage notebook.

   100 

   101 

   102 ::

   103      

   104    sin(pi/2)

   105    

   106    arctan(oo)

   107      

   108    log(e,e)

   109 

   110 

   111 Given that we have defined variables like x,y etc .. , We can define

   112 an arbitrary function with desired name in the following way.::

   113 

   114        var('x') 

   115        function('f',x)

   116 

   117 

   118 Here f is the name of the function and x is the independent variable .

   119 Now we can define f(x) to be ::

   120 

   121      f(x) = x/2 + sin(x)

   122 

   123 Evaluating this function f for the value x=pi returns pi/2.::

   124 	   

   125 	   f(pi)

   126 

   127 We can also define functions that are not continuous but defined

   128 piecewise.  Let us define a function which is a parabola between 0

   129 to 1 and a constant from 1 to 2 .  Type the following as given on the

   130 screen

   131 

   132 ::

   133       

   134 

   135       var('x') 

   136       h(x)=x^2 g(x)=1 

   137       f=Piecewise(<Tab>

   138 

   139 {{{ Show the documentation of Piecewise }}} 

   140     

   141 ::

   142       f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) f

   143 

   144 

   145 

   146 

   147 We can also define functions which are series 

   148 

   149 

   150 We first define a function f(n) in the way discussed above.::

   151 

   152    var('n') 

   153    function('f', n)

   154 

   155 

   156 To sum the function for a range of discrete values of n, we use the

   157 sage function sum.

   158 

   159 For a convergent series , f(n)=1/n^2 we can say ::

   160    

   161    var('n') 

   162    function('f', n)

   163 

   164    f(n) = 1/n^2

   165 

   166    sum(f(n), n, 1, oo)

   167 

   168  

   169 Lets us now try another series ::

   170 

   171 

   172     f(n) = (-1)^(n-1)*1/(2*n - 1)

   173     sum(f(n), n, 1, oo)

   174 

   175 

   176 This series converges to pi/4. 

   177 

   178 

   179 Moving on let us see how to perform simple calculus operations using Sage

   180 

   181 For example lets try an expression first ::

   182 

   183     diff(x**2+sin(x),x) 

   184     2x+cos(x)

   185 

   186 The diff function differentiates an expression or a function. Its

   187 first argument is expression or function and second argument is the

   188 independent variable.

   189 

   190 We have already tried an expression now lets try a function ::

   191 

   192    f=exp(x^2)+arcsin(x) 

   193    diff(f(x),x)

   194 

   195 To get a higher order differential we need to add an extra third argument

   196 for order ::

   197  

   198    diff(<tab> diff(f(x),x,3)

   199 

   200 in this case it is 3.

   201 

   202 

   203 Just like differentiation of expression you can also integrate them ::

   204 

   205      x = var('x') 

   206      s = integral(1/(1 + (tan(x))**2),x) 

   207      s

   208 

   209 

   210 

   211 Many a times we need to find factors of an expression ,we can use the "factor" function

   212 

   213 ::

   214     factor(<tab> 

   215     y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) 

   216     f = factor(y)

   217 

   218 One can  simplify complicated expression ::

   219     

   220     f.simplify_full()

   221 

   222 This simplifies the expression fully . We can also do simplification

   223 of just the algebraic part and the trigonometric part ::

   224 

   225     f.simplify_exp() 

   226     f.simplify_trig()

   227     

   228 

   229 

   230 One can also find roots of an equation by using find_root function::

   231 

   232     phi = var('phi') 

   233     find_root(cos(phi)==sin(phi),0,pi/2)

   234 

   235 Lets substitute this solution into the equation and see we were

   236 correct ::

   237 

   238      var('phi') 

   239      f(phi)=cos(phi)-sin(phi)

   240      root=find_root(f(phi)==0,0,pi/2) 

   241      f.substitute(phi=root)

   242 

   243 as we can see when we substitute the value the answer is almost = 0 showing 

   244 the solution we got was correct.

   245 

   246 

   247 

   248 

   249 Lets us now try some matrix algebra symbolically ::

   250 

   251 

   252 

   253    var('a,b,c,d') 

   254    A=matrix([[a,1,0],[0,b,0],[0,c,d]]) 

   255    A

   256 

   257 Now lets do some of the matrix operations on this matrix

   258 

   259 

   260 ::

   261     A.det() 

   262     A.inverse()

   263 

   264 

   265 

   266 {{{ Part of the notebook with summary }}}

   267 

   268 So in this tutorial we learnt how to

   269 

   270 

   271 * We learnt about defining symbolic expression and functions.  

   272 * Using built-in constants and functions.  

   273 * Using <Tab>  to see the documentation of a function.  

   274 * Simple calculus operations .  

   275 * Substituting values in expression using substitute function.

   276 * Creating symbolic matrices and performing operation on them .

   277