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RE: Very ill-conditioned symmetric positive definite systems [ Reply ]
By: Xavier Lacoste on 2013-07-19 14:38
[forum:125950]
Hello Guo,

We dont' have yet computation of the condition numbers of the matrix, I'll have a look at it.

XL.

RE: Very ill-conditioned symmetric positive definite systems [ Reply ]
By: Guo Luo on 2013-07-17 23:52
[forum:124892]
Hi Xavier,

Thanks a lot! Do you think it is possible to compute the error bound also (the last line of section 2.3 in MUMPS manual)? That is the quantity that I am really interested in. Thanks again!

Regards,
Guo

RE: Very ill-conditioned symmetric positive definite systems [ Reply ]
By: Xavier Lacoste on 2013-07-17 07:29
[forum:124650]
Hi Guo,

I looked inside MUMPS manual http://graal.ens-lyon.fr/MUMPS/doc/userguide_4.10.0.pdf (2.3)

They are using for each equation, |b - Ax |_i / (|b| + |A||x|)_i except if b-Ax is non zero and the denominator is too small.
In that case, it computes : ( | b- Ax |_i/((|A||x|)_i + |A_i|_inf|x|_inf)
And then they return the maximum value of this on all rows.
We can had it I think, I'll try it.

Xavier.


RE: Very ill-conditioned symmetric positive definite systems [ Reply ]
By: Guo Luo on 2013-07-16 22:46
[forum:124636]

error_analysis_mumps4.10.0.pdf (112) downloads
Hi XL,

Thanks a lot for your quick response. I have been using MUMPS before which can perform classical error analysis based on the residuals. The diagnostics that MUMPS computes can be found in the attached file. The reference [12] cited in the file is the following:

[12] M. Arioli, J. Demmel, and I. S. Duff. Solving sparse linear systems with sparse backward error. SIAM Journal on Matrix Analysis and Applications, 10:165–190, 1989.

It would be great if similar diagnostics can be added to pastix.

I am not sure if I am asking for too much. If you prefer I can read the paper myself and then tell you the explicit formulas used by MUMPS to compute all these diagnostics.

Thanks again for your time. And have a nice day.

Guo

RE: Very ill-conditioned symmetric positive definite systems [ Reply ]
By: Xavier Lacoste on 2013-07-16 06:30
[forum:123987]
Hello,

(1) When DPARM_EPSILON_MAGN_CTRL is set to a positive number, the threshold for pivoting is set to norm1(A)*sqrt(epsilon), else it is set to -epsilon.
In your case you get 0.
This mean that no static pivoting is performed.
In all other cases, when a value smaller than epsilon is detected on the diagonal, it is replaced by epsilon.

(2) Computing ||Ax-b||/||b|| is the only test on accuracy we are performing. If you want us to add a new one, we could do it.

Thanks for you feed back,

XL.

Very ill-conditioned symmetric positive definite systems [ Reply ]
By: Guo Luo on 2013-07-13 01:50
[forum:123768]
Dear Pastix developers,

I am using pastix 5.2.1 to solve large ill-conditioned SPD systems resulting from a Galerkin discretization of a 2D Poisson equation. I didn't have a precise estimate of the condition number of my system, but believe it could be as large as 10^9 or even higher.

The system was solved using the parameters:

IPARM_FACTORIZATION = API_FACT_LLT;
DPARM_EPSILON_REFINEMENT = 1e-13;
DPARM_EPSILON_MAGN_CTRL = 0;

and all others being default. The solution completes successfully, but I am a bit concerned with the actual accuracy of the solution.

I am wondering

(1) what pastix precisely does when DPARM_EPSILON_MAGN_CTRL is set to 0?
(2) whether pastix provides any backward error estimates, besides the residual DPARM_RELATIVE_ERROR?

Thank you very much.

Guo