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]