CGNE is a general-purpose non-symmetric method. It works well when the singular values are much better behaved than eigenvalues. A unitary matrix is a classic example where CGNE converges in one iteration, but GMRES and CGS need N iterations (see Nachtigal, Reddy, and Trefethen, "How fast are nonsymmetric matrix iterations", 1992). If you intend to solve least squares problems, use KSPLSQR.
This is NOT a different algorithm than used with KSPCG, it merely uses that algorithm with the matrix defined by A^t*A and preconditioner defined by B^t*B where B is the preconditioner for A.
This method requires that one be able to apply the transpose of the preconditioner and operator as well as the operator and preconditioner. If the transpose of the preconditioner is not available then the preconditioner is used in its place so one ends up preconditioning A'A with B B. Seems odd?
This only supports left preconditioning.
This object is subclassed off of KSPCG