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 semi‐conjugate 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. Copyright © 2004 John Wiley & Sons, Ltd.