Non-Linear Functions : Part 1 (Introduction)

Introduction to basics of functions and iteration.

Notes at https://gist.github.com/nibrahim/e71a442d36e571e9dfb8

Hello and welcome. This is a
presentation of non linear functions.

And I have developed this as a public
class at the Licium in December of 2014

but had not recorded it back then.

So, I'm recording it as a screen cast so
that I can upload it

on to Youtube and I can make it even
more available.

This is the first time that I am doing
this converting my class

into a screen cast so I would appreciate
your feedback.

This presentation is on non linear
functions.

Non linear functions are models,
mathematical models of

physical phenomena that we
often encounter

However, unlike linear functions they are
a little more harder to analyze and

predict. The behavior is harder to
predict.

In many circumstances when we try to model
physical phenomena we see that

higher order terms have very very small
coefficients.

So we just ignore them but those small
terms can actually affect

the behavior of functions in certain
circumstances.

So this presentation is going to be an
introduction to these types of

functions and how they behave and what
you can do with them.

So to start off with let's just do a quick
recap.

A recap of basic ideas which I know many
of you may already be familiar with

If you already know about all this you can
skip this and go to the next

The basic ideas we are going to discuss
are functions.

What are functions? Functions are
mathematical concepts which we use to

describe how elements of one set change
into elements of another set.

For example I can say F of x equal to
x squared.

Which is a function that is squared and
simpled.

Or you can say F of x equal to sine of x

The trigonometric sine of x, expressed in
radians. Things like that.

These are examples of functions,
simple functions

These two are non-linear
but we can have linear ones as well.

Say x + 5, or something like 2x
or something like that

These are linear functions.
These last two are linear.

The first two are non-linear because
they have higher order terms

Higher order terms meaning terms with
a power of more than 1

Now, usually when we talk about functions
we talk about solving them

So we say: y equal to x squared, and you
can plot it

Or you can find the value of the function
for a given value of x, and something like that.

But in this series we're going to be
mostly talking about iteration functions

This is something which you might not
have done.

So the basic idea is to take a function,
say this one which we have up here

F of x equal to x squared
And apply this function repeatedly

So if I were to take F(x) = x^2,
and I say F of 2 will be 4

And then you apply F on this one,
so you say F of 4 equal to 4 squared, 16

And then you say F of 16, it's the square
of 16, then you say square of that

Then square of that, then square of that.

This is called iterating. You apply a
function repeatedly on something.

And this actually gives you certain kinds
of behaviour

which we'll find interesting in the
upcoming segments.

So let's say if I wanted to apply
F(x) on a function, so F(x) = x^2

So if I apply this three times, it will be
F of F of F of x

I want to apply the squaring function on a
number (say 2), three times

You can write it like this. You will get
the third "iterate" of the number 2 under
the function F

And as a matter of notation, we tend to
write it like this, F three times of 2.

So this is F of F of F of 2

So this is the notation which we're going
to use repeatedly in this series.

So this is iterational functions.

And what we're interested in is finding
behaviours of functions when we do this.

Now let's move to the next phase.

We have a notation and we have the basic
thing that we're going to do.

What we're going to discuss right now is
what I call Orbits.

The term might sound technical, but it's a
simple thing

It's just the successive iterates of a
number under a function.

So if I were to take a number, say in this
case let's just say a simpler function

Let's just take F of x equal to 2x (double
the number). Then you can start with 1

We'll start with Orbit of 1 under F.
The first thing would be 1,

then you apply F once, it would become 2.
You apply F to that, you get 4,

you apply F to that you get 8, 16, 32, 64,
128, etc.

These are the iterates of 1 under the
function F(x) = 2x.

And this series is called the Orbit of 1.

So the Orbit of 1 is successive iterates
of a number under a function.

So we talk about what is the orbit of this
number, what is the orbit of that number,

how does the orbit behave? And things like
that. So it's this orbit thing that we're
interested in.

And we can actually make predictions about
orbits and where this thing will go

if you start iterating and things like
that.

Now given that we know what orbits are now
there is some interesting kinds of points

We'll discuss many of them in the upcoming
segments

but the first and simplest one is what's
called a "Fixed Point"

It's something important, just in case.

A Fixed Point is a point that is fixed.
This means that F(x) will be equal to x

for that point. And this is something
which varies depending on functions.

If I were to take F(x)=2x, this function
which I just talked about earlier

Zero is a fixed point.

Why? Because F(0) will be 0, and then you
apply that as many times as you want

It stays there it doesn't move, it's fixed

On the other hand, we just saw over here
that if you take 1, it's not fixed

It just keeps on increasing

If you take 0, it's what we call
a Fixed Point.

And you can see how this behaves, you can
see how the functions behave now.

Let's take a few simple examples and see
how this thing does.

So consider this, we take an interpreter
so that we can do this easily

So that I have square root, sine, cosine
and things like that.

And take a number, so let's say 5. I'm
going to say square root of 5, we get this

Square root of that, we get this.
And I keep doing this again and again

So this is iterating manually

And we can see that as I iterate, this
thing slowly becomes smaller and smaller.

It tends towards 1, right? And 1 is under
square root of 1 a Fixed Point

Because square root of 1 is 1

Alright, consider the other one. Consider
sine of something expressed in radians

So let's take sine of pi. Sine of pi is
this, it's infinitesimally small.

Let's take sine of pi/2. This is 1.

Sine of pi/1.9, just a little bit less.
If you iterate all of this...

This is sine of some random number, just
do 2 or something

Sine of 3, and then sine of this. So if
you apply sine repeatedly on a number

you see that it slowly goes down lower and
lower and lower

and ultimately it tends to 0.

You can see this happening if you write a
small function

Iterate, this is the function, this is the
start point, this the number of iterations

And you can see S equal to start, and
say for i in range of iterations

And you can say S equal to the function
applied on S

And you just say Print. Simple function.

So just try this, iterate sine starting
from 2, a hundrer times

You can see that it starts off from 0.9
and keeps reducing

Reduces, reduces, reduces it goes down
like this

If I were to take this to 1000, it would
go extremely lower

10,000 it's even lower.

So it's slowly tending towards 0. And this
seems to hold for many of the points

For example for 3, for example for pi.
It's going down to almost 0.

Pi/2, anything. So anything you give,
it slowly tends towards 0

And 0 of course is a Fixed Point.

So 0 is a Fixed Point for this function.
0 is also a Fixed Point for sine of x.

Now let's take another one. Let's try
cos of x which is looking more interesting

Let's just take pi, and we'll take
cos of x

Cos of x and cos of a function, and then
now we try this.

You can see that it goes to 0.739085133215

And this seems to happen for anything
usually: 3, 2, 1, anything.

And this is a Fixed Point of cos of x.

This seemingly random number is a Fixed
Point for cos of x.

Because of my definition you know that
this is a point at which cos(x) equal to x

But it's not straightforward what's so
particular about this number.

So this is the concept of a Fixed Point.

So that ends our first segment, which is
just an introduction to functions

What is iteration, what is an Orbit, and
what is a Fixed Point.

Now we'll discuss more types of points
in the next segment.