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1.
An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in
this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations
with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified.
AMS subject classification (2000) 47J05, 47J06, 47J35, 65R30 相似文献
2.
A family of eighth-order iterative methods for the solution of nonlinear equations is presented. The new family of eighth-order methods is based on King’s fourth-order methods and the family of sixth-order iteration methods developed by Chun et al. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus we provide a new example which agrees with the conjecture of Kung–Traub for n=4. Numerical comparisons are made to show the performance of the presented methods. 相似文献
3.
4.
An iterative method for finding a solution of the equation f(x)=0 is presented. The method is based on some specially derived quadrature rules. It is shown that the method can give better results than the Newton method. 相似文献
5.
Alicia Cordero José L. Hueso Eulalia Martínez Juan R. Torregrosa 《Numerical Algorithms》2010,53(4):485-495
In this paper, we present two new three-step iterative methods for solving nonlinear equations with sixth convergence order.
The new methods are obtained by composing known methods of third order of convergence with Newton’s method and using an adequate
approximation for the derivative, that provides high order of convergence and reduces the required number of functional evaluations
per step. The first method is obtained from Potra-Pták’s method and the second one, from Homeier’s method, both reaching an
efficiency index of 1.5651. Our methods are comparable with the method of Parhi and Gupta (Appl Math Comput 203:50–55, 2008). Methods proposed by Kou and Li (Appl Math Comput 189:1816–1821, 2007), Wang et al. (Appl Math Comput 204:14–19, 2008) and Chun (Appl Math Comput 190:1432–1437, 2007) reach the same efficiency index, although they start from a fourth order method while we use third order methods and simpler
arithmetics. We prove the convergence results and check them with several numerical tests that allow us to compare the convergence
order, the computational cost and the efficiency order of our methods with those of the original methods. 相似文献
6.
Finding the zeros of a nonlinear equation is a classical problem of numerical analysis which has various applications in many
sciences and engineering. In this problem we seek methods that lead to approximate solutions. Sometimes the applications of
the iterative methods depended on derivatives are restricted in Physics, chemistry and engineering. In this paper, we propose
two iterative formulas without derivatives. These methods are based on the central-difference and forward-difference approximations
to derivatives. The convergence analysis shows that the methods are cubically and quadratically convergent respectively. The
best property of these schemes are that they are derivative free. Several numerical examples are given to illustrate the efficiency
and performance of the proposed methods. 相似文献
7.
Varsha Daftardar-Gejji Hossein Jafari 《Journal of Mathematical Analysis and Applications》2006,316(2):753-763
An iterative method for solving nonlinear functional equations, viz. nonlinear Volterra integral equations, algebraic equations and systems of ordinary differential equation, nonlinear algebraic equations and fractional differential equations has been discussed. 相似文献
8.
Implicit iterative method acquires good effect in solving linear ill-posed problems. We have ever applied the idea of implicit iterative method to solve nonlinear ill-posed problems, under the restriction that α is appropriate large, we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence, numerical results show that the implicit iterative method for nonlinear ill-posed problems is efficient. In this paper, we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when αk are determined by Hanke criterion. 相似文献
9.
S.M. Shakhno 《Journal of Computational and Applied Mathematics》2009,231(1):222-235
We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension. 相似文献
10.
Numerical Algorithms - For nonlinear equations, the homotopy methods (continuation methods) are popular in engineering fields since their convergence regions are large and they are quite reliable... 相似文献
11.
Ioannis K. Argyros 《Journal of Applied Mathematics and Computing》2007,25(1-2):345-351
Using more precise majorizing sequences we provide a finer convergence analysis than before [1], [7] of Newton’s method in Riemannian manifolds with the following advantages: weaker hypotheses, finer error bounds on the distances involved and a more precise information on the location of the singularity of the vector field. 相似文献
12.
Mohan Joshi 《Proceedings Mathematical Sciences》1983,92(1):61-65
An approximate solvability scheme for equations of the typeu+K u(u)=w, in a closed convex subsetA of a Hilbert spaceX is given. Here, for eachu ∈ A, K u: X → X is a bounded linear operator. 相似文献
13.
Ioannis K. Argyros 《Journal of Mathematical Analysis and Applications》2007,332(1):97-108
We revisit a fast iterative method studied by us in [I.K. Argyros, On a two-point Newton-like method of convergent order two, Int. J. Comput. Math. 88 (2) (2005) 219-234] to approximate solutions of nonlinear operator equations. The method uses only divided differences of order one and two function evaluations per step. This time we use a simpler Kantorovich-type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case. Moreover we show that in some cases our method compares favorably, and can be used in cases where other methods using similar information cannot [S. Amat, S. Busquier, V.F. Candela, A class of quasi-Newton generalized Steffensen's methods on Banach spaces, J. Comput. Appl. Math. 149 (2) (2002) 397-406; D. Chen, On the convergence of a class of generalized Steffensen's iterative procedures and error analysis, Int. J. Comput. Math. 31 (1989) 195-203]. Numerical examples are provided to justify the theoretical results. 相似文献
14.
Tensor methods for nonlinear equations base each iteration upon a standard linear model, augmented by a low rank quadratic term that is selected in such a way that the mode is efficient to form, store, and solve. These methods have been shown to be very efficient and robust computationally, especially on problems where the Jacobian matrix at the root has a small rank deficiency. This paper analyzes the local convergence properties of two versions of tensor methods, on problems where the Jacobian matrix at the root has a null space of rank one. Both methods augment the standard linear model by a rank one quadratic term. We show under mild conditions that the sequence of iterates generated by the tensor method based upon an ideal tensor model converges locally and two-step Q-superlinearly to the solution with Q-order 3/2, and that the sequence of iterates generated by the tensor method based upon a practial tensor model converges locally and three-step Q-superlinearly to the solution with Q-order 3/2. In the same situation, it is known that standard methods converge linearly with constant converging to 1/2. Hence, tensor methods have theoretical advantages over standard methods. Our analysis also confirms that tensor methods converge at least quadratically on problems where the Jacobian matrix at the root is nonsingular.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Research supported by AFOSR grant AFOSR-90-0109, ARO grant DAAL 03-91-G-0151, NSF grants CCR-8920519 CCR-9101795. 相似文献
15.
《Communications in Nonlinear Science & Numerical Simulation》2010,15(8):2026-2036
In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity. 相似文献
16.
Jinhai Chen 《Computational Optimization and Applications》2008,40(1):97-118
In this paper, inexact Gauss–Newton methods for nonlinear least squares problems are studied. Under the hypothesis that derivative
satisfies some kinds of weak Lipschitz conditions, the local convergence properties of inexact Gauss–Newton and inexact Gauss–Newton
like methods for nonlinear problems are established with the modified relative residual control. The obtained results can
provide an estimate of convergence ball for inexact Gauss–Newton methods. 相似文献
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18.
In this paper, we present a new trust region algorithm for the system of singular nonlinear equations with the regularized
trust region subproblem. The new algorithm preserves the global convergence of the traditional trust region algorithm, and
has the quadratic convergence under some suitable conditions. Finally, some numerical results are given. 相似文献
19.
Maria Rosaria Capobianco Giuliana Criscuolo Peter Junghanns 《Numerical Algorithms》2010,55(2-3):205-221
Different iterative schemes based on collocation methods have been well studied and widely applied to the numerical solution of nonlinear hypersingular integral equations (Capobianco et al. 2005). In this paper we apply Newton’s method and its modified version to solve the equations obtained by applying a collocation method to a nonlinear hypersingular integral equation of Prandtl’s type. The corresponding convergence results are derived in suitable Sobolev spaces. Some numerical tests are also presented to validate the theoretical results. 相似文献