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 letís 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 letís 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.