  We consider using the preconditioned-Krylov subspace method to solve the
system of linear equations with a three-by-three block structure. By making use
of the three-by-three block structure, eight inexact block factorization
preconditioners, which can be put into a same theoretical analysis frame, are
proposed based on a kind of inexact factorization. By generalizing Bendixson
Theorem and developing a unified technique of spectral equivalence, the bounds
of the real and imaginary parts of eigenvalues of the preconditioned matrices
are obtained. The comparison to eleven existed exact and inexact
preconditioners shows that three of the proposed preconditioners can lead to
high-speed and effective preconditioned-GMRES in most tests.
