The conjugate gradient method can be viewed as a special variant of the Lanczos method for positive definite symmetric
systems. The minimal residual method and
symmetric LQ method (SYMMLQ) are variants that can be applied to symmetric indefinite
systems.

The vector sequences in the conjugate gradient method correspond to a factorization of a tridiagonal matrix similar to the coefficient
matrix. Therefore, a breakdown of the algorithm can occur corresponding to a zero
pivot if the matrix is indefinite. Furthermore, for indefinite matrices the minimization
property of the conjugate gradient method is no longer well-defined. The MINRES and
SYMMLQ methods are variants of the CG method that avoid the LU
decomposition and do not suffer from breakdown. SYMMLQ solves the projected system,
but does not minimize anything (it keeps the residual orthogonal to all previous
ones).

When
is not positive definite, but symmetric, we can still construct an orthogonal basis
for the Krylov subspace by three-term recurrence relations. Eliminating the search
directions in the equations of the conjugate
gradient method gives a recurrence

In this case we have the problem that no longer defines an inner product. However
we can still try to minimize the residual in the 2-norm by obtaining

(3)

that minimizes

(4)

(5)

Now we exploit the fact that if

(6)

then
is an orthonormal transformation with respect to the current Krylov subspace:

(7)

and this final expression can simply be seen as a minimum norm least squares problem.

One approach is to solve the system , as in the conjugate
gradient method ( is the upper part of ). However, other than in the conjugate gradient method,
we cannot rely on the existence of a Cholesky
decomposition (since is not positive definite). An alternative is then to decompose
by an LQ decomposition. This leads to simple
recurrences and the resulting method is known as SYMMLQ (Paige and Saunders 1975).