1 Symbolics with Sage

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

     3 

     4 This tutorial on using Sage for symbolic calculation is brought to you

     5 by Fossee group.

     6 

     7 {{{ Part of Notebook with title }}}

     8 

     9 We would be using simple mathematical functions on the sage notebook

    10 for this tutorial .

    11 

    12 During the course of the tutorial we will learn

    13 

    14 

    15 {{{ Part of Notebook with outline }}}

    16 

    17 To define symbolic expressions in sage .  Use built-in costants and

    18 function.  Integration , differentiation using sage .  Defining

    19 matrices.  Defining Symbolic functions .  Simplifying and solving

    20 symbolic expressions and functions

    21     

    22 

    23 

    24 Using sage we can perform mathematical operations on symbols .

    25 

    26 On the sage notebook type::

    27    

    28     sin(y)

    29 

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

    31 can declare y as a symbol using var function. ::

    32    

    33     var('y')

    34    

    35 Now if you type::

    36 

    37     sin(y)

    38 

    39      sage simply returns the expression .

    40 

    41 thus now sage treats sin(y) as a symbolic expression . You can use

    42 this to do a lot of symbolic maths using sage's built-in constants and

    43 expressions .

    44 

    45 Try out ::

    46    

    47    var('x,alpha,y,beta') x^2/alpha^2+y^2/beta^2

    48  

    49 Similarly , we can define many algebraic and trigonometric expressions

    50 using sage .

    51 

    52 

    53 

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

    55 mathematics .

    56 

    57 example : pi,e,oo , Function n gives the numerical values of all these

    58     constants.

    59 

    60 For instance::

    61 

    62    n(e)

    63    

    64    2.71828182845905

    65 

    66 gives numerical value of e.

    67 

    68 If you look into the documentation of n by doing ::

    69 

    70    n(<Tab>

    71 

    72 You will see what all arguments it can take etc .. It will be very

    73 helpful if you look at the documentation of all functions introduced

    74       

    75 

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

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

    78   

    79    n(pi, digits = 10)

    80 

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

    82 sin,cos,sinh,cosh,log,factorial,gamma,exp,arcsin,arccos,arctan etc ...

    83 lets try some out on the sage notebook. ::

    84      

    85    sin(pi/2)

    86    

    87    arctan(oo)

    88      

    89    log(e,e)

    90 

    91 

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

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

    94 

    95        var('x') function(<tab> {{{ Just to show the documentation

    96        extend this line }}} function('f',x)

    97 

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

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

   100 

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

   102 

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

   104 	   

   105 	   f(pi)

   106 

   107 We can also define function that are not continuous but defined

   108 piecewise.  We will be using a function which is a parabola between 0

   109 to 1 and a constant from 1 to 2 .  type the following as given on the

   110 screen::

   111       

   112 

   113       var('x') h(x)=x^2 g(x)=1 f=Piecewise(<Tab> {{{ Just to show the

   114       documentation extend this line }}}

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

   116 

   117 Checking f at 0.4, 1.4 and 3 :: f(0.4) f(1.4) f(3)

   118 

   119 for f(3) it raises a value not defined in domain error .

   120 

   121 

   122 Apart from operations on expressions and functions one can also use

   123 them for series .

   124 

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

   126 

   127    var('n') function('f', n)

   128 

   129 

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

   131 sage function sum.

   132 

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

   134    

   135    var('n') function('f', n)

   136 

   137    f(n) = 1/n^2

   138 

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

   140 

   141 For the famous Madhava series :: var('n') function('f', n)

   142 

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

   144 

   145 This series converges to pi/4. It was used by ancient Indians to

   146 interpret pi.

   147 

   148 For a divergent series, sum would raise a an error 'Sum is

   149 divergent' :: 

   150 	

   151 	var('n') 

   152 	function('f', n) 

   153 	f(n) = 1/n sum(f(n), n,1, oo)

   154 

   155 

   156 

   157 

   158 We can perform simple calculus operation using sage

   159 

   160 For example lets try an expression first ::

   161 

   162     diff(x**2+sin(x),x) 2x+cos(x)

   163 

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

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

   166 independent variable .

   167 

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

   169 

   170    f=exp(x^2)+arcsin(x) diff(f(x),x)

   171 

   172 To get a higher order differentiation we need to add an extra argument

   173 for order ::

   174  

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

   176 

   177 

   178 in this case it is 3.

   179 

   180 

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

   182 

   183      x = var('x') s = integral(1/(1 + (tan(x))**2),x) s

   184 

   185 

   186 

   187 To find factors of an expression use the function factor

   188 

   189     factor(<tab> y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) f =

   190     factor(y)

   191 

   192 One can also simplify complicated expression using sage ::

   193     f.simplify_full()

   194 

   195 This simplifies the expression fully . You can also do simplification

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

   197 

   198     f.simplify_exp() f.simplify_trig()

   199     

   200 

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

   202 

   203     phi = var('phi') find_root(cos(phi)==sin(phi),0,pi/2)

   204 

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

   206 correct ::

   207 

   208      var('phi') f(phi)=cos(phi)-sin(phi)

   209      root=find_root(f(phi)==0,0,pi/2) f.substitute(phi=root)

   210 

   211 

   212 as we can see the solution is almost equal to zero .

   213 

   214 

   215 We can also define symbolic matrices ::

   216 

   217 

   218 

   219    var('a,b,c,d') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) A

   220 

   221 

   222 Now lets do some of the matrix operations on this matrix ::

   223 

   224 

   225     A.det() A.inverse()

   226 

   227 You can do ::

   228     

   229     A.<Tab>

   230 

   231 To see what all operations are available

   232 

   233 

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

   235 

   236 So in this tutorial we learnt how to

   237 

   238 

   239 We learnt about defining symbolic expression and functions .  

   240 And some built-in constants and functions .  

   241 Getting value of built-in constants using n function.  

   242 Using Tab to see the documentation.  

   243 Also we learnt how to sum a series using sum function.  

   244 diff() and integrate() for calculus operations .  

   245 Finding roots , factors and simplifying expression using find_root(), 

   246 factor() , simplify_full, simplify_exp , simplify_trig .

   247 Substituting values in expression using substitute function.

   248 And finally creating symbolic matrices and performing operation on them .