CHAPTER 09.10: ADEQUACY OF
SOLUTIONS: Number of significant digits correct in my solution vector Theory We
now know that a condition number is the way to figure out whether a previous
system of equations is well-conditioned or ill-conditioned. Well how do we
relate this to the number of significant digits without correcting my
solution vector? So for example we have looked at this particular theorem
that delta x divided by norm of x is less than or equal to condition number
of a times norm of delta c divided by norm of c for ax equal to c. So if we
have a system of equations like ax
equal to c and we make a change of delta c to the right hand side vector it’s
going to make a change of delta x to the solution vector. And we have proved
the theorem here that the relative change in the solution vector can get
amplified by as much as the condition number when we look at the relative
change in the right hand side vector. So based on this we need to figure out
how many significant digits can I trust in my solution so we can very well
say the hey the possible relative error which you're going to get in the
solution vector will be based on the value of the condition number multiplied
by what is the possible change that is going to take in the right hand side
vector. Since
we are representation numbers in the folding point format we can at least say
that hey there is a possibility that the amount of relative error I'm going
to get just by representing numbers will be of the size of epsilon mac. So it
looks like that condition number of a times machine epsilon. So we are saying
that hey if I am representing numbers there is going to be some change in the
number in which I am representing and the number that gets represented by
that'll at least have a relative error of machine epsilon. So this number
here is condition number a, this number here will at least be around the
machine epsilon number. So we are saying that hey this is equal to norm of
delta x divided by norm of x. So what that means that this gives us a sense of
what the relative error in the solution vector is. And that’s the number
which we want to now compare with the number of 0.5 times ten to the power of
two minus m. So what we want to be able to see is that hey is the condition
number of a times the machine epsilon, is it less than or equal to 0.5 times
ten to the power of two minus m. And
why is it, what does m stand for? What is the largest of m can I get say that
this inequality is valid? And whatever the value that m turns out to be that is
the number of significant digits that you can trust in your solution. So this
goes back to the same formula which we have had for relative pre-specified
tolerance 0.5 times ten to the power of two minus m. The reason why the two
is missing is because we are not talking about percentages here this is just a
fractional number right here: condition number 8 times machine epsilon. And
so go and find out the largest integer that you will need for m, positive
integer of m for which this number here the condition number times the
machine epsilon is less than this number that gives you the number. M is the
least number of significant digits we can trust. And that’s the end of this
segment. |