[petsc-dev] KSP convergence test based on error on residual

[Barry Smith]

   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 <mark.adams at columbia.edu> 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.