CS3414 Afterclass Notes --- 24 June, 2002
Linear Systems (Chapter 6)
- Error analysis for Ax=b
- The residual
- Def. r = b-Ax*, where x*
is an approximate solution.
- Not necessarily a good predictor of error. Example ...
See HW5, Problem 1.
- Conditioning of Ax=b
- Theorem: if
- If you replace the perturbation `delta b' in the above result
by the residual r, you get a similar result that relates
the size of the error to the size of the residual. So
a large condition number means that the residual may not
be a good estimate of the error.
- There are other interpretations for cond(A):
- The cond(A) is a measure of the degree of linear
dependence that the columns (or rows) of A have.
- If cond(A) is about 10k, then you should not
be surprised to lose k decimal digits of accuracy in solving
- Fortunately, there are relatively inexpensive
(O(n2)) ways to get good estimates for the norm
- Stability of Gaussian Elimination
- Can show that G.E. with complete pivoting is stable.
- Can show that G.E. with partial pivoting may not be stable.
- But in the overwhelming majority of cases, G.E. with partial
pivoting is stable enough.