[petsc-users] minres and gmres difference

[Umut Tabak]

Dear list,

While testing an ill-conditioned system solution with minres and 
gmres(default 30 restarts, and no other options) in PETSc, I came up 
with the result that the system is solved in 155 iterations by minres to 
a relative default tolerance but it could not be solved by gmres.

The original system is indefinite and had a condition number on the 
order of 10e+6, with a diagonal conditioning left and right, I could 
lower the condition number to the order of 10e+4 (however still 
indefinite with negative eigenvalues) and try to solve this system. On 
the other hand, I am also trying to understand the details of minres and 
gmres, so they both minimize the 2 norm of residual by solving a least 
squares problem where the latter is for unsymmetric systems, and the 
other is using a 3-term recurrence relation similar to cg (which I am 
trying to figure out now.). I would expect both to give the same result 
in PETSc however gmres fails. I wanted to check also gmres because I 
read in

"Iterative Krylov Methods for Large Linear Systems" van der Vorst, 
Chap.6 Gmres and Minres, page 86

where it is mentioned that gmres could be more robust and less 
vulnerable to rounding errors than minres.

What could be the reason for this? Any comments are appreciated

Best,
Umut