CHAPTER 09.09: ADEQUACY OF
SOLUTIONS: Relating changes in right hand side vector to changes in
solution vector In
this segment we will talk about how we can relate changes in the right hand
side vector to changes in the solution vector. So let’s suppose we are
solving a set of equations like A X is equal to C. And then what we do is, we
make a change to the right hand side. So we get C prime. So then what we are
going to get is A times X prime because surely if we are going to change our
right hand side vector then the solution vector is also going to change. So
what we want to be able to say is, hey how is the change in the right hand
side related to the change in the solution vector? So if we define two more vectors,
delta X as X prime, which is the new solution minus the previous solution
right here. And then we will also define delta C as the right hand side which
we have changed it to minus the original right-hand side. Then we have this theorem
right here which says that delta X, norm of delta X divided by the norm of X
is less than equal to the norm of A times the norm of A
inverse times delta C, norm of delta C divided by norm of C. So
what that basically means is that if we are making change in the right hand
side vector let’s suppose a small change in the right hand side vector or any
change in the right hand side vector. We want to see is hey, how much does it amplify the change in the solution vector. SO this
is the relative change in the norm of the solution vector. This is the
relative change on the norm of the right hand side. And we can see is, is
that hey, it get amplified by as much as this quantity right here. And that
quantity there, which is the norm of A times the norm of A inverse, is nothing but the conditioned number of
the A matrix. So we can see that conditioned A can be calculated by simply
multiplying the norm of the coefficient matrix by the norm of the inverse of
the coefficient matrix and that gives us the conditioned number of the
matrix. And that the relative change that is taking place in the solution
vector can get amplified by that quantity, condition A. We what that rate to
be, if we have an ill conditioned system of equations then the conditioned A
will be large. If the system of equations is going to be well conditioned
then we are looking at a conditioned number of A the coefficient matrix to be
small. So what you have to also realize is that the ill conditioning and well
conditioning of a system of equations only depends on the coefficient matrix
not on the right hand side vector as can be seen by this particular problem
here. We are not proving this theorem at this stage we are just showing what
this theorem is all about. And this is the end of this segment. |