The conjugate gradient method can be viewed as a special variant of the Lanczos method for positive definite symmetric systems. The minimal residual method (MINRES) and
symmetric LQ method (SYMMLQ) methods 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 methods is a variant of the conjugate
gradient method that avoids the LU decomposition
and does not suffer from breakdown. MINRES minimizes the residual in the 2-norm.
The convergence behavior of the conjugate gradient and MINRES methods for indefinite
systems was analyzed by Paige et al. (1995).

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.

The element in the position of can be annihilated by a simple Givens rotation and the
resulting upper bidiagonal system (the other subdiagonal elements having been removed
in previous iteration steps) can simply be solved, which leads to the MINRES method
(Paige and Saunders 1975).