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The Conjugate Gradient method can be viewed as a special variant of the Lanczos method (see §gif) for positive definite symmetric systems. The MINRES and 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 and SYMMLQ methods are variants of the CG method that avoid the -factorization and do not suffer from breakdown. MINRES minimizes the residual in the -norm . SYMMLQ solves the projected system, but does not minimize anything (it keeps the residual orthogonal to all previous ones ). The convergence behavior of Conjugate Gradients and MINRES  for indefinite systems was analyzed by Paige, Parlett, and Van der Vorst [167].

Breakdown: The occurrence of a zero coefficient that is to be used as a divisor in an iterative method.

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