CHAPTER 09.07: ADEQUACY OF
SOLUTIONS: Relating changes in coefficient matrix to changes in solution vector In
this segment we will talk about, how if you make certain changes in the
coefficient matrix how they are related to changes in the solution vector and
we are going to do this to the norms of the matrixes involved. We are not going to show the proof of the
theorem but we are going to talk about what it means. So let suppose somebody
gives you an equation like this. A times X is equal to C. What you do in
order to be able to figure out whether this set of equations is well
conditioned or ill conditioned? Which
means that if I make a small change on the right hand side or in my
coefficient matrix do I get smaller or large changes in my solution vector. What
we are going to do is in this case, we are going to change our coefficient matrix
a little bit. And we will call the new matrix A prime and then we will get a
different solution for that. Because if we are going to change the
coefficient matrix and keep the other side the same then we are surely going
to get a new solution vector. Now if we define our delta A matrix as the coefficient
matrix has changed minus the original coefficient matrix. We define our delta
X as the changed in the solution vector as the solution vector as we get here
and the solution vector we get from the original set of equations. If we
define that then there is a theorem which relates the norms of some of these
matrixes. It is given as follows: that
the lI? XIl divided by lI? X+? Il is less than equal to lI AIl times lI A^-1Il
times ll? All over lIAIl so we are basically saying
here is that the relative change in the solution vector and this is the
relative change in the coefficient matrix in terms of the norm. So
the relative change in the norm of the solution vector is related to the
relative change in the coefficient matrix by this quantity right here. So
what that basically means is that if
you make a relative change in your coefficient matrix the relative change in
the solution vector can get amplified
by the as much as this number right here. It doesn’t mean it gets amplified by
that number but because there is a less than or equal to sign here that it
can get amplified by as much as by this number here, which is simply the I
AIl times lI A^-1Il so the amplification factor that you are seeing here is
finding the of coefficient matrix, taking its inverse and then find in the of
that multiplying the two and you get I AIl times lI A^-1Il. So this I AIl
times lI A^-1Il is actually defined as the conditioned of the A matrix.
Conditioned number of A. Which makes sense because when we are talking about
well conditioning and ill conditioning system of equations and this is going
to be the amplification factor for the relative change of the norm of the
solution vector to the relative change in the coefficient matrix then if this
is the amplification factor that is what is going to determine how much
change that is going to be and that is why we call this quantity here to be
the conditioned number of the matrix. And now how this is related to
quantifying the relative change in the solution vector to the relative change
in the coefficient matrix, we will talk about in a different segment. And
that the end of this segment. |