Study on semi-conjugate direction methods for non-symmetric systems

Some theoretical problems and implementation problems are studied here for the semi-conjugate direction method established by Yuan, Golub, Plemmons and Cecilio (2002). The existence of semiconjugate directions is proved for almost all matrices except skew-symmetric matrices. A new technique is proposed to overcome the breakdown problem appeared in the semi-conjugate direction method. In the implementation of the semi-conjugate direction method, the generation of the semi-conjugate direction is very important and necessary, but very expensive. The technique of limited-memory is introduced to economize the cost of the generation of the semi-conjugate direction in the Yuan–Golub– Plemmons–Cecilio algorithm. Finally, some numerical experiments are given to confirm our theoretical results. Our results illustrate that the semi-conjugate direction method is very nice alternative for solving non-symmetric systems, and the limited-memory left conjugate direction method is a good improvement of the left conjugate direction method