Notes for LGMRES PETSc code
--------------------------------

Developed and tested with Petsc 2.1.3 on solaris.

Files:

 lgmres.h (header included by user code)
 lgmres.c (main file)
 lgmresp.h (private header file for lgmres)
 makefile (lgmres library makefile)

To use:

- Register the lgmres method in your code, for example:

    ierr = KSPRegister("lgmres", KSPCreate_LGMRES); PetscCall(ierr);

  This is done automatically by Petsc 2.1.5

Options:

- Uses the same options as gmres, plus two additional specifications

-ksp_gmres_restart: for lgmres, this indicates the maximum size of the
  approximation space (krylov space size + error approximations)

-ksp_lgmres_augment : number of error approximations to include in the 
  approximation space (default is 2).  If this is set to zero, then
  this method is equivalent to gmres with a restart parameter defined 
  by -ksp_gmres_restart.


-ksp_lgmres_constant: keep the approximation space a constant size for 
  every restart cycle. This only affects the first couple of restart
  cycles when the total number of desired error approximations may not
  be available.  (For example, no error approximations are available for
  the first restart cycle.  If this flag is set then the krylov space
 size = approx. space size for the first cycle.)


Example:

For the arco3 matrix in the petsc test collection, I use the
following options::

-ksp_gmres_modifiedgramschmidt 
-ksp_type lgmres 
-pc_type ilu
-ksp_gmres_restart 30 
-ksp_lgmres_augment 2
-ksp_lgmres_constant
-ksp_rtol 1e-9
-ksp_max_it 15000


More info:

- The technical report describing the LGMRES method can be downloaded
from my website (CU-CS-945-03):

  http://amath.colorado.edu/student/allisonb/research.html

- Augmenting with 1,2, or 3 approximations is generally optimal.  I
usually use 2.  

- This method is an accelerator for gmres - it is not useful for
problems that stall.  When gmres(m) stalls then lgmres with a size m
approx space will also generally stall.

- If gmres(m) converges in a small number of restart cycles, then lgmres
also tends not to be very helpful.
