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 letís 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 letís 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 letís go and do that. †Letís make a small change in C. So we're making a small change in the right inside vector. So letís 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 letís 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, letís 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.