# Calculate number of significant figures EFFECTIVELY obtained

1. Oct 20, 2012

### libelec

1. The problem statement, all variables and given/known data
Solve the following equation system using the Gauss-Seidel method, and the convergence criteria || x(k) - x(k-1) || <= 0.5 * 10-2. For the arithmetics, use a number of significant digits that will allow you to get a result with 3 significant digits. Must not iterate more than 5 times.

3.210*a + 0.943*b + 1.020*c = 2.300
0.745*a - 1.290*c = 0.740
0.875*a - 2.540*b + 0.247*c = 3.390

Calculate the number of significant digits effectively obtained for each vector element, and conclude if the convergence criteria was appropriate to get 3 significant digits.

3. The attempt at a solution

After having obtained the solution through the Gauss-Seidel method, I found this question odd. I never heard of a way to calculate the number of significant digits one effectively found, rather use as many significant digits as the error boundary allows you.

I thought two possible interpretations:

a) It's asking me if the approximation of the absolute error (|| x(k) - x(k-1) ||) is at or smaller than 0.5 * 102 for each component. Which I find weird, because if one got the solution with that convergence criteria, you are certain that, at least, each component has 3 significant numbers.

b) Like with iterative refinement, calculate δ$\bar{x}$ (Ax - A$\bar{x}$) and get the number of significant digits through the log(K(A)), where K(A) is the condition of the matrix A. The results I found make no sense, because I get something lower than 1, and I've never applied this to anything other than iterative refinement of direct methods.

Any ideas?