The complex dynamic of conjugate gradient method

Abstract

Conjugate gradient method is a root-finding algorithm to non-linear equations. In this paper, we suggest extending this method for a polynomial to the complex plane. Through the experimental and theoretical mathematics method, we drew the following conclusions: (1) the conjugate gradient is a dynamical system with two complex parameters; (2) locally conditions for convergence to any roots of complex functions is given; (3) the conjugate gradient method may fail to converge to all roots for cubic with three simple roots; (4) the boundary of conjugate gradient basins are fractals in some cases, and depends on the parameters; (5) the algorithm is then improved by introducing a method to determine the optimal parameters.

Keywords:

• basins of attraction
• Cayley's problem
• conjugate gradient
• dynamical systems
• fractals

2000 AMS Subject Classifications :

• 28A80
• 37F50
• 37F10
• 65K10
• 49M37