BFGS Methods For Solving Symmetric Nonlinear Equations With Strongly Monotone

Quasi-Newton methods are e?cient methods for solving nonlinear equationsand optimization problems. Under appropriate conditions, these methods possesslocal superlinear convergence property. Moreover, when applied to solving opti-mization problems, most quasi-Newton methods are globally convergent if someline search technique is used. Howover,when applied to solve a system of nonlin-ear equations, a quasi-Newton method may not be globally convergent because thequasi-Newton direction is generally not a descent direction of the norm function. Toenlarge the convergence domain, recently Li-Fukushima (2001) introduced a non-montone line search technique. By the use of this nonmontone line search, it wasshown that Broyden-like methods are globally convergent when they are appliedto solve a system of nonlinear equations. For symmetric nonlinear equations, Li-Fukushima (2001)also developed a Guass-Newton-based BFGS method with non-montone line search and establish its global and superlinear convergence. Quiterecently, Gu-Li-Qi-Zhou (2003) improved the methods proposed by Li-Fukushima(2001) and developed a descent quasi-Newton method. They obtained the globaland superliner convergence of the descent quasi-Newton method. We note that thisdescent quasi-Newton method needs extra computational e?ort to obtain the de-scent direction. In this thesis, we respectively propose a hybria BFGS-type methodand a nonmonotone BFGS-type method for solving the symmetric strong non-monotone nonlinear equations. The methods are well-defined. The directionsproduced by these methods are descents one of norm function. Furthermore, toacquire the direction, there is no need to increase additional calculation. Underappropriate conditions,we prove that these methods are globally and superlinearlyconvergent. Compared with Guass-Newton BFGS method(GNBFGS) proposed byLi-Fukushima (2001) for solving symmetric nonlinear equations, the condition num-ber of Bk in our methods is much less than that of Guass-Newton BFGS method.Finally in the thesis, a numeric test has been done. Our numerical results showthat the proposed BFGS methods has a better numerical result and also prove thatthe condition number in this paper is much smaller than that in GNBFGS.