CHAPTER 09.01: ADEQUACY OF
SOLUTIONS: Ill-conditioned and well-conditioned system of equations In
this segment we'll talk about how to differentiate between ill-conditioned and
well-conditioned systems of equations. So let’s suppose somebody gives you a
system of equations which is in the matrix form and turns out to be ax equals
c. What 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 solving or setting up simultaneous linear equations we write
them in the form of a times x equals c where a is the coefficient matrix x is
the solution vector and c is the right hand side vector. 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 A 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 change in a small amount. Because
we are going to 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 the calculation of
the c matrix I 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. 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 that if from a simple example
if a system of equations is well-conditioned and ill-conditioned. Let’s
suppose somebody says is this particular system of equations 1 2 2 3.99 xy 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 the set of equations it does have a simple solution for
example. What is the solution to this one? It's two comma one. So x y is two
comma one. So if we take x equal two and y equal one. In fact, if you plug x
equal to two and y equal to one in here you will get 4 and 7.999. So that's
itself a 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 going to 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 islet’s suppose I make a small change in
my caution matrix. So What I'm doing is as follows. I'm taking 1.001 so I'm
making a small change in the caution matrix. Changing by the thousands I get
3.998 so what I'm doing is that I’m making a change of about a magnitude of
point zero zero one for all of these elements which are here in the caution
matrix. And I'm not changing the right hand side. And
I want to see that if 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 caution 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 lets
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 caution 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 minus 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 questions these two unknowns by calculator, Matlab, whatever
we find out that the solution which I get is different. This was 2 now it is
minus 3.999. This was 1 and now it’s 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-conditioned
and 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-conditioned or ill-conditioned. The solution to this
set of equations if you would either solve it or plug in these values of x equals
2 and y equal to 1 you are going to get four and 7 or if you solve these two
questions two unknowns you will get x y to be two and one. 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 caution
matrix. So
let me make a small change in my caution 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 caution matrix right here by changing the cautions 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 caution matrix does not result in a large
change in my solution vector. A small change in the caution 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 2 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 saw 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 in 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. Just for illustration
purposes 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.
That is the end of this segment |