Comparison of highly stable forms of iterative conjugate directions methods

Simple and highly stable formulae for conjugate directions methods in case of symmetric matrices and for symmetrized conjugate gradients in case of non-symmetric matrices have been proposed. These methods are compared with highly stable forms of conjugate gradients method and Craig method. It is shown that recurrent algorithm versions are necessary for high round-off stability to be achieved. Conjugate residual method turned out to be the most reliable and fast for symmetric sign-definite and sign-alternating matrices. Symmetrized conjugate gradients method delivered the best results for non-symmetric matrices. These two methods are recommended for developing standard programs. Also a reliable criterion for breaking the count in case of reaching round-off background is constructed.