Abstract: Complex linear equations have a wide application in science and engineering, and an important issue is how to solve it with high efficiency. Until now, complex linear equations are usually solved by either iteration methods or the solution of the real equations transformed from the original equations. Conjugate gradient method (CG method) is discussed from two different viewpoints, and it is proved theoretically that these two kinds of CG methods have the same convergence. Because the convergence speed of the modified conjugate gradient method (MCG method) and conjugate gradient method are essentially similar, MCG method is extended to solve complex linear equations. Besides, a preconditioned MCG method is proposed in order to improve the convergence speed. Finally, the consistency of algorithms and theoretical analysis and effectiveness of the proposed precondition algorithm are validated by numerical examples.

Key words: complex linear systems, modified conjugate gradient method (MCG method), precondition method, convergence