Symbolics with Sage
-------------------

This tutorial on using Sage for symbolic calculation is brought to you
by Fossee group.

.. #[Madhu: Sounds more or less like an ad!]

{{{ Part of Notebook with title }}}

.. #[Madhu: Please make your instructions, instructional. While
     recording if I have to read this, think what you are actually
     meaning it will take a lot of time]

We would be using simple mathematical functions on the sage notebook
for this tutorial.

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

During the course of the tutorial we will learn

{{{ Part of Notebook with outline }}}

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.

.. #[Nishanth]: The formatting is all messed up
                First fix the formatting and compile the rst
                The I shall review
.. #[Madhu: Please make the above items full english sentences, not
     the slides like points. The person recording should be able to
     read your script as is. It can read something like "we will learn
     how to define symbolic expressions in Sage, using built-in ..."]

Using sage we can perform mathematical operations on symbols.

.. #[Madhu: Same mistake with period symbols! Please get the
     punctuation right. Also you may have to rephrase the above
     sentence as "We can use Sage to perform sybmolic mathematical
     operations" or such]

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.

.. #[Madhu: But is not required]
::
    var('y')
   
Now if you type::

    sin(y)

     sage simply returns the expression .

.. #[Madhu: Why is this line indented? Also full stop. When will you
     learn? Yes we can correct you. But corrections are for you to
     learn. If you don't learn from your mistakes, I don't know what
     to say]

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 .

.. #[Madhu: "Thus now"? It sounds like Dus and Nou, i.e 10 and 9 in
     Hindi! Full stop again. "a lot" doesn't mean anything until you
     quantify it or give examples.]

Try out

.. #[Madhu: "So let us try" sounds better]
 ::
   
   var('x,alpha,y,beta') x^2/alpha^2+y^2/beta^2
 
Similarly , we can define many algebraic and trigonometric expressions
using sage .

.. #[Madhu: comma again. Show some more examples?]


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.

.. #[Madhu: This doesn't sound like scripts. How will I read this
     while recording. Also if I were recording I would have read your
     third constant as Oh-Oh i.e. double O. It took me at least 30
     seconds to figure out it is infinity]

For instance::

   n(e)
   
   2.71828182845905

gives numerical value of e.

If you look into the documentation of n by doing

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

::
   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

.. #[Madhu: What does etc .. mean in a script?]

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

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

   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.

.. #[Madhu: Here "a lot" makes sense]
::
     
   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)

.. #[Madhu: What will the person recording show in the documentation
     without a script for it? Please don't assume recorder can cook up
     things while recording. It is impractical]

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
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

.. #[Madhu: Instead of "We will be using ..." how about "Let us define
     a function ..."]
::
      

      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)

.. #[Madhu: Again this doesn't sound like a script]

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 .

.. #[Madhu: I am not able to understand this line. "Use them as
.. series". Use what as series?]

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

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

.. #[Madhu: Shouldn't this be on 2 separate lines?]

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)

.. #[Madhu: What is this? your double colon says it must be code block
     but where is the indentation and other things. How will the
     recorder know about it?]

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

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

.. #[Madhu: I am losing the context. Please add something to bring
     this thing to the context]

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

.. #[Madhu: When you switch to irrelevant topics make sure you use
    some connectors in English like "Moving on let us see how to
    perform simple calculus operations using Sage" or something like
    that]
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 .

.. #[Madhu: Full stop, Full stop, Full stop]

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)

.. #[Madhu: Please try to be more explicit saying third argument]

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

.. #[Madhu: Two separate lines.]

To find the factors of an expression use the "factor" function

.. #[Madhu: See the diff]

::
    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()
    
.. #[Madhu: Separate lines?]

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)

.. #[Madhu: Separate lines?]

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)

.. #[Madhu: Separate lines?]

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

.. #[Madhu: So what?]

We can also define symbolic matrices ::



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

.. #[Madhu: Why don't you break the lines?]

Now lets do some of the matrix operations on this matrix

.. #[Madhu: Why don't you break the lines? Also how do you connect
     this up? Use some transformation keywords in English]
::
    A.det() A.inverse()

.. #[Madhu: Why don't you break the lines?]

You can do ::
    
    A.<Tab>

To see what all operations are available

.. #[Madhu: Sounds very abrupt]

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

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 .

.. #[Madhu: See what Nishanth is doing. He has written this as
     points. So easy to read out while recording. You may want to
     reorganize like that]