CHAPTER 09.04: ADEQUACY OF
SOLUTIONS: How is the norm related to the conditioning of a system of
equations Part 1 of 2 In
this segment we will talk about how the norm is related to the conditioning
of a system of equations. So in the previous segment we talked about if we
have a certain system of equations and we make a small change to the coefficient
matrix of the right hand side and does it make a small change in the solution
vector or a large change in the solution vector. If it makes a small change
in the solution vector then we consider the system that we wish to be well
conditioned, if it makes a large change in the solution vector then we consider
it to be ill-conditioned. But
the exam that you are taking we are just by doing it simply looking at the
solution vectors. So what we will see is if we will be able to quantify this
well conditioning and ill conditioning and see that it is related to the norm
of a matrix. So we're going to go through an example to illustrate this
concept to you. So let’s say that somebody gave you an equation like 1, 2, 2,
3.999, xy. So somebody says hey I got these two equations and two unknowns,
then we know that this the solution is xy is equal to 2, 1. In this case if
we want to note this by A times x is equal to C, so if that is how the notation
is going to be in the matrix form for this set of equations. Then,
if somebody asks "hey what is the norm of X?", now X will simply be equal to 2.
Because that is the max value here and out of these two is two, so we're talking
about the infinite norm or the row sum norm. If someone says hey what is the
norm of C, the infinite or row sum norm of C it would be again it would be
7.999. Now what we want to be able to do is say hey let’s go and change the
right inside vector and see what happen to the norms of the resulting
solution vector or the changes in the solution vector. So let’s go and do
that. Let’s make a small change in C.
So we're making a small change in the right inside vector. So let’s rewrite
down the equations 1, 2, 2, and 3.999. Again we got a caution matrix is
staying the same and we are making a small change in the right inside vector.
And we will make this to be instead of
a 4 we're going to call it 4.001 and instead of 7.999 we're going to make this
7.998. In this case when we solve equations we get XY is equal to minus 3.999
4.000. Now if we denote the set of equations as Ax prime equal to C prime, because of what
we have done is that we're not change the caution matrix we have kept it the
same. We have change the right hand side which is turning out to this one
here and then so we are going to get a different solution of course to which
we're calling it X prime. So if we look at the difference so if it is called
delta C, as the difference between the right hand side so we call it C prime
minus C, that will turn out to be 0.001 and minus 0.001. Simply
because this being the new C which we which we have and the old C we have.
And then we look at what is the change in the solution vector, it would be
the new solution vector minus the old solution vector. That one turns out to
be equal to minus 5.999 and 3.000. So, these are the differences which we are
getting so the only thing that A I made a small change in my C vector
resulting in a large change in my solution vector is the difference being in
the new solution and the old solution or the next solution and the previous
solution here with the different C which I'm using. Then let’s go and see
what we get for the norms, norm of delta C infinity is 0.001. So that is the
infinite norm of that matrix, but the infinite norm of the X matrix, this one
right here is 5.999. So
if we write down all these norms, delta x, norm turned out to be 5.999, our norm of x turned to be 2, then norm of
delta c which is the change in the right side was 0.001. The norm of C, which
is your original right hand side turns out to 7.999. So right here we are able to see that A is
the change in the solution and this one possesses the change in the right
hand side vector. So, let’s go and see what we get for delta X infinity
divided by delta by delta x infinity norm. We get 5.999 divided by 2 which is
2.9995. What is delta C infinity divided by the norm of C that is equal to
0.001 divided by 7.999 and that is 1.250 times 10 to the power of minus four. So now
we are able to see that A for this particular set of equations that we had we
made a small change on the right hand side vector we resulted in a large
change in our solution vector from a relative point of view. We
can't just look at this number or this number and be able to say that hey
what are the relative changes or whether the change was large or small but
now we are finding the relative change in the solution vector is this much.
The relative change in the right side vector the norm is this much. So, the relative change in the norm of the
solution vector is this much. The relative Change in the norm of the right
hand side vector is this much we can
see that the a small relative change in the small relative change in the norm
of the right hand side vector is resulting a large change in the solution and the
relative change in the norm of the solution vector. If we calculate that
number then we get delta x divided by delta x norm divided by norm delta C divided
by the norm of C it turned out to be 2.9995 divided by 1. 250 times ten to the
power of negative 4. That number turns out to be 23993.So you will see that A
is the condition of this set of equations which we had when we started with
may not be good, be ill conditioned. Because this. The relative change in the
solution vector divided by the relative change in the right side vector is
turning out the huge number like this one. How much ill conditioned it is,
we'll look at that in a later segment. Just keep in mind this mind this
number as you see the next segment. And that is the end of this segment. |