Solving almost-degenerate linear problem?

jollage · April 7, 2016, 10:05

Hi,

I have a problem of how to solve safely an almost-degenerate linear problem. The problem comes from the nonlinear analysis near the critical condition, at which the growth rate of the disturbance is vanishing, rendering the corresponding linear problem almost degenerate. I just used Matlab to brutally inverse the matrix (using backslash), but I'm concerned about its accuracy.

Therefore, I'm asking if there is any linear algebra package which is designed specifically for solving such problem? What's the tricks not mentioned in the literature of tackling such problem?

Thank you in advance. Any comments will be appreciated.

mprinkey · April 7, 2016, 12:13

If the system is dense, use singular value decomposition. It will find the solution or a reduced basis for the solution if it degenerate.

If the system is sparse, you have a murkier situation. My first recommendation is to try GMRES with a simple preconditioner (Jacobi or Incomplete LU with zero fill-in). GMRES is a very robust workhorse. But elaborate preconditioning may cause the solution of a degenerate system to drift from iteration to iteration. This is just intuition though. I'm sure there are tomes written about analysis of such systems.

April 7, 2016, 10:05	Solving almost-degenerate linear problem?	#1
jollage Member Join Date: Feb 2011 Posts: 41 Rep Power: 15	Hi, I have a problem of how to solve safely an almost-degenerate linear problem. The problem comes from the nonlinear analysis near the critical condition, at which the growth rate of the disturbance is vanishing, rendering the corresponding linear problem almost degenerate. I just used Matlab to brutally inverse the matrix (using backslash), but I'm concerned about its accuracy. Therefore, I'm asking if there is any linear algebra package which is designed specifically for solving such problem? What's the tricks not mentioned in the literature of tackling such problem? Thank you in advance. Any comments will be appreciated.

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April 7, 2016, 12:13		#2
mprinkey Senior Member Michael Prinkey Join Date: Mar 2009 Location: Pittsburgh PA Posts: 363 Rep Power: 25	If the system is dense, use singular value decomposition. It will find the solution or a reduced basis for the solution if it degenerate. If the system is sparse, you have a murkier situation. My first recommendation is to try GMRES with a simple preconditioner (Jacobi or Incomplete LU with zero fill-in). GMRES is a very robust workhorse. But elaborate preconditioning may cause the solution of a degenerate system to drift from iteration to iteration. This is just intuition though. I'm sure there are tomes written about analysis of such systems.