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 letís 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. Thatís 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 thatís 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 letís 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, thatís 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 thatís the end of this segment.