Some results on convergence of Newton's method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.
A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis co...A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x^＊, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.展开更多
The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vec...The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ-theory are extended.展开更多
The relationship between the convergence ball of the Euler iteration in Banach Spaces and itsexclusive fixed points on Riemann spheres is investigated. By using an exclusive fixed point of the Euleriteration, the conv...The relationship between the convergence ball of the Euler iteration in Banach Spaces and itsexclusive fixed points on Riemann spheres is investigated. By using an exclusive fixed point of the Euleriteration, the convergence ball is determined accurately for a class of operators whose derivatives satisfy somegeneralized Lipschitz condition on Banach spaces.展开更多
Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub＇s method is obtained. An error analysis is given ...Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub＇s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.展开更多
Under the weak Lipschitz condition about the solution of the equation, convergence theorems for a family of iterations with one parameter are obtained. An estimation of the radius of the attraction ball is shown. At l...Under the weak Lipschitz condition about the solution of the equation, convergence theorems for a family of iterations with one parameter are obtained. An estimation of the radius of the attraction ball is shown. At last two examples are given.展开更多
Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we ...Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.展开更多
研究了用Newton-Steffensen法求解非线性算子方程.当非线性算子F的一阶导数满足L-平均Lipschitz条件时,建立了Newton-Steffensen法的三阶收敛判据,同时也给出了收敛球半径的估计.作为应用,当F的一阶导数满足经典的Lipschitz条件时或F满...研究了用Newton-Steffensen法求解非线性算子方程.当非线性算子F的一阶导数满足L-平均Lipschitz条件时,建立了Newton-Steffensen法的三阶收敛判据,同时也给出了收敛球半径的估计.作为应用,当F的一阶导数满足经典的Lipschitz条件时或F满足γ-条件时,建立了Newton-Steffensen法的三阶收敛判据及给出了收敛球半径的估计.从而推广了[Journal of Nonlinear and Convex Analysis,2018,19:433-460]中的相应结果.展开更多
文摘Some results on convergence of Newton's method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.
文摘A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x^＊, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.
基金suported in part by the National Natural Science Foundation of China(Grant No.10271025)Program for New Century Excellent Talents in University.
文摘The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ-theory are extended.
基金This work was supported by the Special Funds for Major State Basic Research Projects(Grant No.G19990328) the National Natural Science Foundation of China(Grant No.19971013) also supported partly by Zhejiang Provincial Natural Science Foundation o
文摘The relationship between the convergence ball of the Euler iteration in Banach Spaces and itsexclusive fixed points on Riemann spheres is investigated. By using an exclusive fixed point of the Euleriteration, the convergence ball is determined accurately for a class of operators whose derivatives satisfy somegeneralized Lipschitz condition on Banach spaces.
基金Supported by the National Natural Science Foundation of China (10871178)the Natural Science Foundation(Y606154)the Foundation of the Eduction Department of Zhejiang Province of China (Y200804008)
文摘Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub＇s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.
基金Supported by Shanghai Municipal Foundation of Selected Academic Research and the National Natural Science Foundation of China（10571059,10571060）.
文摘Under the weak Lipschitz condition about the solution of the equation, convergence theorems for a family of iterations with one parameter are obtained. An estimation of the radius of the attraction ball is shown. At last two examples are given.
基金supported by National Natural Science Foundation of China(11771393,11371320,11632015)Zhejiang Natural Science Foundation(LZ14A010002,LQ18A010008)Scientific Research Fund of Zhejiang Provincial Education Department(FX2016073)
文摘Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.
文摘研究了用Newton-Steffensen法求解非线性算子方程.当非线性算子F的一阶导数满足L-平均Lipschitz条件时,建立了Newton-Steffensen法的三阶收敛判据,同时也给出了收敛球半径的估计.作为应用,当F的一阶导数满足经典的Lipschitz条件时或F满足γ-条件时,建立了Newton-Steffensen法的三阶收敛判据及给出了收敛球半径的估计.从而推广了[Journal of Nonlinear and Convex Analysis,2018,19:433-460]中的相应结果.
