[petsc-users] Howto understand the output of -ksp_monitor_singular_value

[Gong Ding]

Hi,
I use -ksp_monitor_singular_value to view the approximate condition number of my matrix.
The output during the GMRES iteration looks like

    0 KSP Residual norm 1.072677079362e+02 % max 1 min 1 max/min 1
    1 KSP Residual norm 2.558517256216e+01 % max 0.949986 min 0.949986 max/min 1
    2 KSP Residual norm 7.952391160495e+00 % max 1.00443 min 0.53867 max/min 1.86465
    3 KSP Residual norm 3.147058569003e+00 % max 1.01529 min 0.418892 max/min 2.42375
    4 KSP Residual norm 1.213876539809e+00 % max 1.04559 min 0.365424 max/min 2.86131
    5 KSP Residual norm 4.398018777420e-01 % max 1.0717 min 0.341344 max/min 3.13966
    6 KSP Residual norm 1.681000721101e-01 % max 1.07182 min 0.333065 max/min 3.21806
    7 KSP Residual norm 6.008534584667e-02 % max 1.07287 min 0.330693 max/min 3.24432
    8 KSP Residual norm 2.317545636598e-02 % max 1.07476 min 0.32966 max/min 3.26019
    9 KSP Residual norm 8.732395511044e-03 % max 1.0798 min 0.32928 max/min 3.27929
   10 KSP Residual norm 3.543098250574e-03 % max 1.08891 min 0.328903 max/min 3.31073
   11 KSP Residual norm 1.322114569421e-03 % max 1.1002 min 0.328779 max/min 3.34632
   12 KSP Residual norm 4.278882091191e-04 % max 1.12562 min 0.328666 max/min 3.4248
   13 KSP Residual norm 1.503136522978e-04 % max 1.13558 min 0.328572 max/min 3.45611
   14 KSP Residual norm 5.410712876997e-05 % max 1.1531 min 0.32832 max/min 3.51211
   15 KSP Residual norm 2.200160155820e-05 % max 1.16627 min 0.328089 max/min 3.55474
   16 KSP Residual norm 9.689610704862e-06 % max 1.17549 min 0.327714 max/min 3.58694
   17 KSP Residual norm 4.077334723619e-06 % max 1.18316 min 0.32763 max/min 3.61126
   18 KSP Residual norm 1.522015252918e-06 % max 1.18949 min 0.327542 max/min 3.63158


In the first iteration, the min and max singular value are both 1, and the ratio of max/min increase quickly
in the next iterations.

>From above values, how to extract the condition number of (preconditioned) matrix?