Jed beat me to it, For the symmetric positive definite case one just uses the definition of the largest eigenvalue e^T e r^T A^{-1} A^{-1} r \lambda_{max} (A^(-2)) = 1 ------------ = --------------------------- <= ----------- r^T r r^T r \lambda^2_{min}(A) When I did my dissertation I always computed for absolute error reduction using the minimum eigenvalue for the SPD case the and the singular value for the general case. When I started working with Bill and he only cared about the solution to a nonlinear problem I lost the habit but it really should be incorporated in PETSc. Barry On Jul 17, 2011, at 9:31 PM, Jed Brown wrote:

On Sun, Jul 17, 2011 at 21:23, Mark F. Adams <[email protected]> wrote: Humm, the only linear algebra proof that I know gives bounds on the error of the form

| error |_2 <= Condition-number * | residual |_2,

This looks like relative error.

for SPD matrices of course. This is pessimistic but I'm not sure how you could get a bound on error with only the lowest eigen value ...

Suppose you have

| A x - b | < c

Then there is some y such that

A (x + y) - b = 0

and for which

|A y| < c

Suppose s is the smallest singular value of A, thus 1/s is the largest singular value of A^{-1}. Then

|y| = | A^{-1} A y | <= (1/s) |A y| < c/s.

So you can bound the absolute error in the solution if you know the residual and the smallest singular value.