CHAPTER 09.11: ADEQUACY OF
SOLUTIONS: Number of significant digits correct in my solution vector Example
1 In
this segment we'll take an example to see that how many significant digits we
can trust in a solution. So let’s suppose somebody gives you a set of
equations like this 1 2 2 3.999 x y is equal to 2 4. Somebody says: hey I'm
going to find the solution here for this set of equations can you tell me how
much I can trust the solution? So what I'm going to do is I'm going to say
hey this is of the form AX equal to c can I find the norm of A? So the row
sum norm of A will be 5.999. Because the absolute value of 1 plus the
absolute value of 2 is 3. The absolute value of 2 plus the absolute value of
3.999 is 5.999. So we get norm of A to be 5.999. Now I can also calculate the
inverse of the matrix, so I can calculate the inverse of this matrix and that
turns out to be minus 3.999 2000 2000 and minus 1000 that’s what A inverse turns out to be. So
in this case the norm of A
inverse, the row sum norm turns out to be 5999. So that is the row sum norm of
A and that’s the row sum of A
inverse. What I want to do is I want to calculate my condition number, so the
condition number of A will be norm of A times norm of A inverse. And I get 5.999 times 599 and this number
here turns out to be 35990. So that is the condition number of A. But then
what I'm going to do is take the condition number of A and I'm going to
multiply it by the machine epsilon. So, let's talk about the
i-triple e 754 standards for single position.
The machine epsilon for that is 0.119 times ten to power of minus 6. So this
is the machine epsilon for a single position number 0.119 to the power of
minus 6. And this one gives me 0.4290 times ten to the power of minus 2. And
now I want to be able to see is that
hey what is that largest value of M so I got 0.4290 times ten to the power of
minus 2 and I need to say hey what is the largest value of M for which this
number turns out to be less than or equal to that number. And
if you do the calculations you will get m is equal to 2 or m is less than or
equal to 2 that’s what you're going to get as the answer. Which means that
you cannot trust more than two significant digits in your answer. So this is
the number of significant digits which you can trust in your solution based
on the condition number, knowing what the machine epsilon is, and being
comparing it to this number right here. If we are doing the problem double
precision this number would be different it would be much smaller than this
number right here and so eventually you will find out that you will be able
to trust more significant digits if you were doing double precision but doing
quad precision you would be able to trust even more digits. So it is the
combination of the machine epsilon and the condition number which is going to
tell you whether the system of equation is well-conditioned or
ill-conditioned. And that’s the end of this segment. |