Symbolics with Sage
-------------------
Hello friends and welcome to this tutorial on symbolics with sage.
.. #[Madhu: Sounds more or less like an ad!]
{{{ Part of Notebook with title }}}
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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.
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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
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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.
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::
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
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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.
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::
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)
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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 .
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.. series". Use what as series?]
We first define a function f(n) in the way discussed above.::
var('n') function('f', n)
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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)
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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.
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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 .
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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)
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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
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To find the factors of an expression use the "factor" function
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::
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()
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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)
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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)
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as we can see the solution is almost equal to zero .
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We can also define symbolic matrices ::
var('a,b,c,d') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) A
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Now lets do some of the matrix operations on this matrix
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this up? Use some transformation keywords in English]
::
A.det() A.inverse()
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You can do ::
A.<Tab>
To see what all operations are available
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{{{ 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]