@Article{Gordon:2008:CRR,
  author =       "Dan Gordon and Rachel Gordon",
  title =        "{CGMN} Revisited: Robust and Efficient Solution of Stiff Linear Systems Derived from Elliptic Partial Differential 
                 Equations",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "35",
  number =       "3",
  month =        oct,
  year =         "2008",
  pages =        "18:1--18:27",
  URL =          "http://doi.acm.org/10.1145/1391989.1391991",
  abstract =     "Given a linear system $Ax = b$, one can construct a related ``normal equations'' system $AA^T\!y=b,~x=A^T\!y$.  
                 Bj\"orck and Elfving have shown that the SSOR algorithm, applied to the normal equations, can be accelerated by the 
                 conjugate gradient algorithm (CG). The resulting algorithm, called CGMN, is error-reducing and it always converges 
                 (in theory), even when the equation system is inconsistent and/or nonsquare.  
                 SSOR on the normal equations is equivalent to the Kaczmarz algorithm (KACZ), with a fixed relaxation parameter, run 
                 in a double (forward and backward) sweep on the original equations.  
                 CGMN was tested on nine well-known large and sparse linear systems obtained by central-difference discretization of 
                 elliptic convection-diffusion partial differential equations (PDEs). Eight of the PDEs were strongly 
                 convection-dominated, and these are known to produce very stiff systems with large off-diagonal elements. CGMN was 
                 compared with some of the foremost state-of-the art Krylov subspace methods: restarted GMRES, Bi-CGSTAB and CGS.  
                 These methods were tested both with and without various preconditioners.
                 CGMN converged in all the cases, while none of the above algorithm/preconditioner combinations achieved this level 
                 of robustness.  Furthermore, on varying grid sizes, there was only a gradual increase in the number of iterations 
                 as the grid was refined. 
                 On the eight convection-dominated cases, the initial convergence rate of CGMN was better than all the other 
                 combinations of algorithms and preconditioners, and the residual decreased monotonically.  The CGNR algorithm was 
                 also tested, and it was as robust as CGMN, but slower.",
}