Incomplete LU and condition number

Dear all,

I have an equation where the coefficient matrix is sparse and ill-posed matrix (with very large condition number and hence very close to be singular). I solve the system using incomplete LU factorization, and then GMRES, iteratively. I would like to get the condition number of the coefficient matrix, before and after preconditioning. Please note that I can not simply multiply the incomplete factors, and then invert, etc. to find the condition number after preconditioning since the system is very large. Is there any efficient way to know the condition number of the system after preconditioning?

Thanks for your help,

D.