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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 equations (gif) and (gif) gives a recurrence

which can be written in matrix form as

where is an tridiagonal matrix

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 -norm by obtaining

that minimizes

Now we exploit the fact that if , then is an orthonormal transformation with respect to the current Krylov subspace:

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 (see Paige and Saunders [168]).

Another possibility is to solve the system , as in the CG method ( is the upper part of ). Other than in CG we cannot rely on the existence of a Cholesky decomposition (since is not positive definite). An alternative is then to decompose by an -decomposition. This again leads to simple recurrences and the resulting method is known as SYMMLQ (see Paige and Saunders [168]).   

Jack Dongarra
Mon Nov 20 08:52:54 EST 1995