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 itís 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 thatís 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 thatís the end of this segment.