I am trying to solve Ax = b system where A is a square, highly ill-conditioned, unsymmetric, partly dense, partly sparse, complex matrix. The linear system arises out of a physical problem where the exact solution to the problem is known and thus the accuracy of the solution x can be verified. I have been able to solve this system using ZGELSS with a fixed RCOND = -1.0D0 so that LAPACK uses machine precision as the SVD threshold. As I cannot predict what the condition number of A is going to be in advance, choosing RCOND = -1.0D0 seemed a good idea and the results with this setting in ZGELSS are satisfactory. The only problem with ZGELSS is that it is very slow. I was aware that ZGELSY will be faster as it uses QR decomposition but do not know how to choose RCOND for ZGELSY. I did try ZGELSY for my linear system and found that the results depend greatly on the input value of RCOND. So the questions are:
a) How do I choose RCOND if I do not know the condition number of A in advance for ZGELSY or
b) Alternatively, is there a faster solver than ZGELSS that relies on SVD which could be used instead? I want to stick to SVD as the condition number of A is likely to remain between 10^16 to 10^22 all the time. Some search on the internet indicates that Jacobi SVD (work of Drmac et al) will be faster and more accurate (?) but I could not find the solver based on Jacobi SVD in LAPACK.
Thanks in advance for help/suggestions.