We are concerned with the numerical methods for nonlinear equation and their semilocal convergence in this paper.The construction techniques of iterative methods are induced by using linear approximation,integral inte...We are concerned with the numerical methods for nonlinear equation and their semilocal convergence in this paper.The construction techniques of iterative methods are induced by using linear approximation,integral interpolation,Adomian series decomposition,Taylor expansion,multi-step iteration,etc.The convergent conditions and proof methods,including majorizing sequences and recurrence relations,in semilocal convergence of iterative methods for nonlinear equations are discussed in the theoretical analysis.The majorizing functions,which are used in majorizing sequences,are also discussed in this paper.展开更多
A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided differenc...A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.展开更多
Kantorovich theorem was extended to variational inequalities by which the convergence of Newton iteration,the existence and uniqueness of the solution of the problem can be tested via computational conditions at the i...Kantorovich theorem was extended to variational inequalities by which the convergence of Newton iteration,the existence and uniqueness of the solution of the problem can be tested via computational conditions at the initial point.展开更多
Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:...Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed graph.We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method.Specifically,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this method.Furthermore,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically regular.An example of nonsmooth functions that have p-point-based approximation is given.Moreover,a numerical experiment is given which illustrates the theoretical result.展开更多
The author provides a finer local as well as semilocM convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the m-dimensional Euclidean space （m ...The author provides a finer local as well as semilocM convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the m-dimensional Euclidean space （m ≥ 1 a natural number）.展开更多
In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set...In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed graph.We establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of f.Indeed,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular.An application of this method to variational inequality is given.In addition,a numerical experiment is given which illustrates the theoretical result.展开更多
We provided in[14]and[15]a semilocal convergence analysis for Newton’s method on a Banach space setting,by splitting the given operator.In this study,we improve the error bounds,order of convergence,and simplify the ...We provided in[14]and[15]a semilocal convergence analysis for Newton’s method on a Banach space setting,by splitting the given operator.In this study,we improve the error bounds,order of convergence,and simplify the sufficient convergence conditions.Our results compare favorably with the Newton-Kantorovich theorem for solving equations.展开更多
文摘We are concerned with the numerical methods for nonlinear equation and their semilocal convergence in this paper.The construction techniques of iterative methods are induced by using linear approximation,integral interpolation,Adomian series decomposition,Taylor expansion,multi-step iteration,etc.The convergent conditions and proof methods,including majorizing sequences and recurrence relations,in semilocal convergence of iterative methods for nonlinear equations are discussed in the theoretical analysis.The majorizing functions,which are used in majorizing sequences,are also discussed in this paper.
基金Supported by the National Natural Science Foundation of China (10871178)the Natural Science Foundation of Zhejiang Province of China (Y606154)Foundation of the Education Department of Zhejiang Province of China (20071362)
文摘A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.
文摘Kantorovich theorem was extended to variational inequalities by which the convergence of Newton iteration,the existence and uniqueness of the solution of the problem can be tested via computational conditions at the initial point.
基金supported by CAS-President International Fellowship Initiative (PIFI), Chinese Academy of Sciences, Beijing, Chinasupported by National Natural Science Foundation of China (Grants Nos. 11688101 and 11331012)
文摘Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed graph.We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method.Specifically,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this method.Furthermore,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically regular.An example of nonsmooth functions that have p-point-based approximation is given.Moreover,a numerical experiment is given which illustrates the theoretical result.
文摘The author provides a finer local as well as semilocM convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the m-dimensional Euclidean space （m ≥ 1 a natural number）.
基金supported by CAS-President International Fellowship Initiative(PIFI)from the Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing,China.
文摘In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed graph.We establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of f.Indeed,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular.An application of this method to variational inequality is given.In addition,a numerical experiment is given which illustrates the theoretical result.
文摘We provided in[14]and[15]a semilocal convergence analysis for Newton’s method on a Banach space setting,by splitting the given operator.In this study,we improve the error bounds,order of convergence,and simplify the sufficient convergence conditions.Our results compare favorably with the Newton-Kantorovich theorem for solving equations.
