Comparing robust forms of iterative methods of conjugate directions

Simple and robust formulas of the conjugate direction method for symmetric matrices and of the symmetrized conjugate gradient method for nonsymmetric matrices have been constructed. These methods were compared with robust forms of the conjugate gradient method and the Craig method using test problems. It is shown that stability for the round-off error can be attained when recurrent variants of the methods are used. The most reliable and efficient method for symmetric signdefinite and indefinite matrices appears to be the method of conjugate residuals. For nonsymmetric matrices, the best results have been obtained by the method of symmetrized conjugate gradients. These two methods are recommended for writing standard programs. A reliable criterion has also been constructed for the termination of the calculation on reaching background values due to the round-off errors.