[petsc-users] Is there efficeint method for matrix with one extremely small eigen value?

[Gong Ding]

Hi,
Can some one gives me advise on how to solve the ill conditioned problem
efficiently with iterative method (since the problem size is big). 

I calculated the smallest eigen values as well as the largest eigen values.
There exist one extremely small eigen value, which made the system ill conditioned.
I guess method such as Tikhonov regularization may work? 
Or there are some cheaper method works, if I can endure some inaccuracy in the solution.
  

Smallest 0 eigen value: -2.112144e-15 with error 9.452618e-14
Smallest 1 eigen value: -2.480170e-04 with error 6.150216e-04
Smallest 2 eigen value: -2.787193e-04 with error 2.808614e-04
Smallest 3 eigen value: -2.825241e-04 with error 6.620491e-04
Smallest 4 eigen value: -2.825241e-04 with error 6.620491e-04
Smallest 5 eigen value: -2.833565e-04 with error 2.990142e-04
Smallest 6 eigen value: -3.020135e-04 with error 6.313397e-04
Smallest 7 eigen value: -3.020149e-04 with error 4.939515e-04
Smallest 8 eigen value: -3.083228e-04 with error 1.114806e-03
Largest  0 eigen value: -4.076308e+03 with error 2.403326e-08
Largest  1 eigen value: -3.894209e+03 with error 6.314489e-08
Largest  2 eigen value: -3.893185e+03 with error 3.924167e-08
Largest  3 eigen value: -3.855228e+03 with error 3.504644e-09
Largest  4 eigen value: -3.739288e+03 with error 1.689236e-08


Thanks.