The Study Of Preconditioned GSOR-Based Iteration Method For Solving Systems Of Nonlinear Equations

Solving systems of nonlinear equations is an important branch in the study of numerical algebra,and it is also a very active problem in modern mathematics research.In this thesis,several new methods based on generalized successive over-relaxation(GSOR)iteration method are proposed for solving systems of nonlinear equations and weakly nonlinear equations.First of all,in this thesis,problems about systems of nonlinear equations and weakly nonlinear equations are briefly introduced,including research background,related concepts,expression,research status,etc.Hereafter,we propose a new iterative method named the modified Newton-preconditioned generalized successive over-relaxation iteration method(MN-PGSOR)for solving systems of nonlinear equations.We use the preconditioned generalized successive over-relaxation method(PGSOR)as an internal iteration and the Newton method as an outer iteration to get the approximate solution of the nonlinear equations.Next,we study convergence of the MN-PGSOR method under the Holder continuous condition and prove that it is locally convergent.The final numerical results show that the MN-PGSOR method can solve systems of nonlinear systems quite effectively.In addition,considering that a special case of nonlinear equation is weakly nonlinear equation,we propose four new iterative methods named Picard-accelerated generalized successive over-relaxation iteration method(P-AGSOR),Picard-preconditioned generalized successive over-relaxation iteration method(P-PGSOR),nonlinear AGSOR-like(NL-AGSOR)method and nonlinear PGSOR-like(NL-PGSOR)method for solving systems of weakly nonlinear equations.We use the accelerated generalized successive over-relaxation method as the internal iteration and the Picard method as the outer iteration to solve systems of weakly nonlinear equations.Next,under specific continuous conditions,we prove the property of local convergence of the two methods P-AGSOR and P-PGSOR by the improved Ostrowski theorem.The final numerical results show that the P-AGSOR and P-PGSOR,NL-AGSOR and NL-PGSOR methods can solve systems of weakly nonlinear equations very effectively.