In this segment, we'll talk about how to differentiate
between ill-conditioned and
well-conditioned systems of equations. So
lets suppose somebody gives you a system
of equations which is in the matrix form and
turns out to be [A][X]=[C]. Where [A] is
the coefficient matrix and [C] is the right
hand side vector and of course [X] is our
solution vector or what we call as the unknown
vector. So whenever we are
setting up simultaneous linear equations,
we write them in the form of [A] times
[X] equal to [C] where [A] is the coefficient matrix
[X] is the solution vector or the unknown vector and [C] is the
right hand side. What you would like
to see is that if you wanted to find
whether this particular system of equations
is well-conditioned or ill-conditioned
is to say the following: that hey if I make
a small change in the elements of the
[A] matrix then how much change is it making
in the solution vector? Or if I make a
small change in my [C] vector, then how much
change does it make in my solution vector?
Because you would like that hey if I make a
small change in the coefficient matrix, you
would like the solution vector to change in
a small amount, or if you change the right
hand side vector you would like the solution
vector to also change in a small amount. Because
as we go through the process
of setting up simultaneous linear equations
for real life problems, those might be set
up through a program where we're going to
have round off errors
in the calculation of the [A] matrix and in the
calculation of the [C] matrix let's suppose.
We don't
want such a round off errors or the lack of
our use of precision when we use only single
precision
rather than double precision or quad precision
to affect adversely what the solution vector
is.
And if it does we want to have a mechanism of
knowing whether it is doing so. So, let's
look
at some examples right here to see 
from a simple example if a system of equations
is well-conditoned and ill-conditioned. Let's
suppose somebody says is this particular system
of equations [1 2, 2 3.999], [x, y] is equal to [4, 7.999]
well-conditioned or ill-conditioned? We want
to be able to make the difference between
saying that hey if someone gave me a system
of
equations like this one, is it well-conditioned
or ill-conditioned? I can see that if I wanted
to solve this set of equations, it does have
a simple solution, for example. What is the
solution
to this one? It's [1, 2]. So [x, y] is
[2, 1]. So if we take x equal 2 and
y
equal 1. In fact, if you plug x equal to
2 and y equal to 1 in here you will get
4 and
7.999. So that's itself the solution for that.
And we want to see that whether if a small
change
in the caution matrix is it going to result
in a very different value x and y I'm gonna
get.
Or if I make a small change in my right hand
side vector, is it going to make a make change
in my x and y? And what I'm doing now here
is let's suppose I make a small change in
my
coefficient matrix. So what I'm doing is as
follows. I'm taking 1.001 so I'm making a
small change
in the coefficient matrix . Changing by the
thousandths, I get 3.998. So what I'm doing is
that I'm
making a change of about a magnitude of 0.001
for all of these elements which
are here in the coefficient matrix. And I
will not change the right hand side. And I
want to see that
hey does it make a big difference in my
solution? And what I find out is that hey
that if I solve
these two equations two unknowns by hand or
by using MATLAB or a new kind of calculator,
the answer that I should get is as follows
[3.994, 0.001388]. And you can see that just
by watching it
or just by looking at it, you can see that
this value of 2 has changed to almost 4. This
value of 1
has to changed to almost a value of 001. So
very different with very small change being
made
in the coefficient of the [A] matrix. So we
can very well see that it is not a well-conditioned
system
of equations. In order to complete the argument
let's go and change the right hand side a little
bit. So let's suppose I have [1 2, 2 3.999] here.
And what I do is I keep the coefficient matrix
the same
but I change the right hand side a little
bit. So let me make this to be again one thousandths
off
of a difference there and 7.9998 here and
one thousandths of a difference right here.
And let's go
and see what I get for values of x and y.
So I get x and y here and if I calculate the
value I
get -3.999 and 4.00. So again you're
finding out that from the original set of
equations you
had right here where the solution was 2 and
1 I made a very small change. I changed 4
to 4.001.
I changed 7.999 to 7.998. Small change, small
relative change in the right hand side vector.
But
when I saw these two equations two unknowns
by calculator, MATLAB, whatever, you find out
that
the solution which I get is different. This
was 2 now it is -3.999. This was 1 and
now it is 4.
So if somebody had to ask me just by intuition
if this is a well-conditioned system of equations
or an ill-conditioned system of equations
I would tell them this to be an ill-conditioned
system of
equations. Now in the later segments we'll
talk about how do we do this quantitatively
without
having to do this. But as an illustration
to illustrate the point that what does it
mean that a
particular system of equations well-conditoned
or ill-conditioned, this is a very good example
to follow. Let's say if this system of equations
is [1 2, 2 3] [x, y] is it well-conditioned system
of equations
or ill-conditioned. We just want to illustrate
the fact whether it is well-conditioned or
ill-conditioned.
So if you look at the, I didn't put the right
hand side here it should be four and seven.
So if we have
this system of equations is it well-conditoned
or ill-conditioned? The solution to this set
of equations
if you would either solve it or plug in these
values of x equal to 2 and y equal to 1, you are going to get
4 and 7 or if you solve these two equations
two unknowns you will get [x, y] to be 2 and
1.
In order to find out whether this previous
system of equations is well-conditioned or
ill-conditioned
I'm going to conduct two experiments. I'm
going to make a small change in my coefficient
matrix. So let me
make a small change in my coefficient matrix.
I'm going to make one-thousandth of a difference
to one. I'm
going to make 2 to be 2.001. I'm going to
make this to be 2.001 and I'm going to make
this to be 3.001.
I'm going to say [x, y] is equal to 4 and 7.
So I have not made any changes in the right
hand side but I made
a very small change to the coefficient matrix
right here by changing the coefficients by a thousandth.
And so what
I get is [x, y], when I solve these two equations
two unknowns by hand or by a calculator or
by MATLAB, I get
2.003 and 0.997. And you can very well see
that 2.003 is very close to 2 and 0.997 is
very close to 1. So a
small change in the coefficient matrix did
not result in a large change in my solution
vector. A small
change in the coefficient matrix resulted
in a small change in my solution vector. Let's
compare for the sake
of completion of the experiment lets go and
change the right hand side vector a little
bit. So we have 
[1 2, 2 3] [x, y] equal to--we'll change the right
hand side a little bit. So let's suppose we
make this to
be 4.001 and 7.001. So we are changing it
by a thousandth here, the right hand side vector.
And when I solve
this set of equations, I found out that hey
my x and y turned out to be equal to 1.999
and 1.001. So again
this number is very close to two and this
number is very close to one. So a small change
in the right hand side
resulted in a small change in my solution
vector; it did not result in a large change.
So in this case, if I'm
conducting this experiment I'm finding out
this particular system of equations is well-conditioned.
Again, we want
to figure out what we mean by well-conditioned
and ill-conditioned systems of equations quantitatively;
not by
just conducting these simply experiments right
here. This is just for illustration purposes and we'll
do that in the later
segments. But this is a good example of getting
started on at least understanding the concept
of ill-conditioned
and well-conditioned equations. And that is the
end of this segment
