Twice is enough method for computation of conjugate directions in ABS

In our earlier works we have given the` ¿twice is enough` ¿ type algorithms to determine conjugate directions of positive definite symmetric matrices. In this work we show that it can be generalized to any symmetric matrices. By testing 32 algorithms from the S2 subclass we show that there are 4 algorithms that yield very precise conjugate directions. We compared 13 well-known algorithms like the Lanczos and Hestenes types as an example and the results show the superiority of the best performing algorithms from the S2. Furthermore we show that there are 4 algorithms that give almost exact ranks in for the test problems. We propose some very good algorithms for the difficult and ill-conditioned test problems derived from the Pascal and Hilbert matrices. As a partial result we computed the rank of the matrices as well. In some cases the S2 subclass computes more accurate ranks compared to the MATLAB built in rank function. According to our difficult and ill-conditioned test problems we found that in most cases algorithms the S2 subclass yield far better results than the classical methods.