Symbolics with Sage

-------------------

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

by Fossee group.

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We would be using simple mathematical functions on the sage notebook

for this tutorial .

During the course of the tutorial we will learn

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

function.  Integration , differentiation using sage .  Defining

matrices.  Defining Symbolic functions .  Simplifying and solving

symbolic expressions and functions

Using sage we can perform mathematical operations on symbols .

On the sage notebook type::

    sin(y)

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

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

    var('y')

Now if you type::

    sin(y)

     sage simply returns the expression .

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

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

expressions .

Try out ::

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

Similarly , we can define many algebraic and trigonometric expressions

using sage .

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

mathematics .

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

    constants.

For instance::

   n(e)

   2.71828182845905

gives numerical value of e.

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

   n(<Tab>

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

helpful if you look at the documentation of all functions introduced

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

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

   n(pi, digits = 10)

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

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

lets try some 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 in the following way.::

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

       extend this line }}} 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 function that are not continuous but defined

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

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

screen::

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

      documentation extend this line }}}

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

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

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

Apart from operations on expressions and functions one can also use

them for 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)

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

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

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

interpret pi.

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

divergent' :: 

	var('n') 

	function('f', n) 

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

We can perform simple calculus operation using sage

For example lets try an expression first ::

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

The diff function differentiates an expression or a function . Its

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 differentiation we need to add an extra argument

for order ::

   diff(<tab> 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

To find factors of an expression use the function factor

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

    factor(y)

One can also simplify complicated expression using sage ::

    f.simplify_full()

This simplifies the expression fully . You 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)

Lets 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 the solution is almost equal to zero .

We can also define symbolic matrices ::

   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()

You can do ::

    A.<Tab>

To see what all operations are available

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So in this tutorial we learnt how to

We learnt about defining symbolic expression and functions .  

And some built-in constants and functions .  

Getting value of built-in constants using n function.  

Using Tab to see the documentation.  

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

diff() and integrate() for calculus operations .  

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

factor() , simplify_full, simplify_exp , simplify_trig .

Substituting values in expression using substitute function.

And finally creating symbolic matrices and performing operation on them .