CHAPTER 09.12: ADEQUACY OF
SOLUTIONS: Number of significant digits correct in my solution vector Example
2 In
this segment we'll take an example of to see that if somebody ask us to solve
a set of simultaneous equations how can we figure out how many significant
digits we can trust in our solution. So let’s suppose somebody gives you a
set of equations which looks like this: 1 2 2 3 unknown vector x y is equal
to 4 7. So somebody is giving you that as a problem statement. Somebody is
giving you this set of equations saying that "hey can you tell me how
many significant digits can I trust in my
solution?" So in this case what I want to do is I want to say norm of A
times norm of x is equal to c. SO that is the form of this set of equations. This
is the caution matrix, this is the unknown vector, and this is the right hand
side vector. And I want to find norm of A and norm of A in this case will be
5. Because the addition of these two absolute value addition of these two
elements is absolute value of one plus absolute value of two is three.
Absolute value of two plus absolute value of three is 5 so the row sum norm
is 5. And then we're going to find the inverse, so what we'll do is I'll give
you the inverse of a matrix. There are several different ways of finding the
inverse of a matrix. So the inverse of this matrix is minus 3, 2, 2, minus 1.
That’s what we get. So in this case the norm of A inverse will be also 5 based on the infinity norm
of the row sum norm. Because absolute value of minus 3 plus absolute value of
2 is 5. Absolute value of 2 plus absolute value of minus 1 is 3 so the
maximum of those is five so that’s why the inverse of norm of the inverse of a is 5. So what that gives us is that the condition number
of A will be norm of a times norm of a
inverse. And
if that is the case, then norm of A is 5. Norm of a inverse is also 5. That gives us 25. So now what
I want to do is I want to compare this with I want to find this number:
Condition number of a times machine epsilon. So the machine epsilon can be
correspondent to whatever position of the position we're using. Whether we're
using a single position, double position or quad position. So let’s suppose we
are using single position and in this case it'll be 25 times 0.119 times 10
to the power of minus 6. And that is the machine epsilon for the single position
real number. And in this case it turns out to be equal to 0.2980 times ten to
the power of minus. So then what we'll do is we have to see whether this
number right here how does that compare with the how does this number which
we get as the product of condition number and machine epsilon. How does that
compare with this 0.5 times to the power of minus m.
So we're looking for the largest value m, integer value of m, positive
integer value m for which this inequality is true. And this turns out to be m
is equal to 5. Of course this means that m is less than or equal to 5, that’s
what you get when you solve this inequality. But
we're looking for the largest value of m for which this inequality is valid.
That when is the condition number times the machine epsilon which is the
major of the relative change in the solution vector to the relative change in
the right hand side vector or the caution. This is what we as condition
number times the machine epsilon. And we'll compare it with this number and
we get m is less than equal to 5. So5 significant digits can be trusted in
our solution when we are working on a single positon machine for this case.
We're going to use double position or quad position surely this number will increase
accordingly. And that’s the end of this segment. |