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1.
Ioannis K. Argyros 《Journal of Applied Mathematics and Computing》1999,6(2):267-275
In this study we examine the applicability of Newton’s method and the modified Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space. We assume that the Newton-Kantorovich hypothesis for Newton’s method is violated, but the corresponding condition for the modified Newton method holds. Under these conditions there is no guarantee that Newton’s method starting from the same initial guess as the modified Newton’s method converges. Hence, it seems that we must always use the modified Newton method under these conditions. However, we provide a numerical example to demonstrate that in practice this may not be a good decision. 相似文献
2.
The numerical solution of nonlinear equation systems is often achieved by so-called quasi-Newton methods. They preserve the
rapid local convergence of Newton’s method at a significantly reduced cost per step by successively approximating the system
Jacobian though low-rank updates. We analyze two variants of the recently proposed adjoint Broyden update, which for the first
time combines the classical least change property with heredity on affine systems. However, the new update does require, the
evaluation of so-called adjoint vectors, namely products of the transposed Jacobian with certain dual direction vectors. The
resulting quasi-Newton method is linear contravariant in the sense of Deuflhard (Newton methods for nonlinear equations. Springer,
Heidelberg, 2006) and it is shown here to be locally and q-superlinearly convergent. Our numerical results on a range of test problems demonstrate that the new method usually outperforms
Newton’s and Broyden’s method in terms of runtime and iterations count, respectively.
Partially supported by the DFG Research Center Matheon “Mathematics for Key Technologies”, Berlin and the DFG grant WA 1607/2-1. 相似文献
3.
Fadi Awawdeh 《Numerical Algorithms》2010,54(3):395-409
Solving systems of nonlinear equations is a relatively complicated problem for which a number of different approaches have
been proposed. In this paper, we employ the Homotopy Analysis Method (HAM) to derive a family of iterative methods for solving
systems of nonlinear algebraic equations. Our approach yields second and third order iterative methods which are more efficient
than their classical counterparts such as Newton’s, Chebychev’s and Halley’s methods. 相似文献
4.
Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause
a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose
two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong
to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep
iterative sequence within convergence region are empirically analyzed and some conclusions are obtained. 相似文献
5.
Inexact implicit methods for monotone general variational inequalities 总被引:32,自引:0,他引:32
Bingsheng He 《Mathematical Programming》1999,86(1):199-217
Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently,
we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear
smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton–like methods for
smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration.
The method is shown to preserve the same convergence properties as the original implicit method.
Received July 31, 1995 / Revised version received January 15, 1999? Published online May 28, 1999 相似文献
6.
In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with
sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements
of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear
equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new
inverse. Numerical comparisons are made to show the performance of the presented methods. 相似文献
7.
This paper presents a fifth-order iterative method as a new modification of Newton’s method for finding multiple roots of
nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative
method is superior to the existing methods. 相似文献
8.
R. S. Temsah 《Numerical Algorithms》2008,47(1):63-80
Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems
of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations.
To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of
ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments
show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other
methods available in the literature.
相似文献
9.
Aliki D. Muradova 《Advances in Computational Mathematics》2008,29(2):179-206
In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular
von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters
are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational
principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions.
Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is
treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced.
Numerical results of solving the problems for polygonal and ferroconcrete plates are presented.
Communicated by A. Zhou. 相似文献
10.
In this paper, we present a class of A(α)-stable hybrid linear multistep methods for numerical solving stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method considered uses a second derivative like the Enright’s second derivative linear multistep methods for stiff IVPs in ODEs. 相似文献
11.
H. Xu 《Mathematical Programming》1999,84(2):401-420
n to Rm. Under the assumption of semi-smoothness of the mapping, we prove that the approximation can be obtained through the Clarke
generalized Jacobian, Ioffe-Ralph generalized Jacobian, B-subdifferential and their approximations. As an application, we
propose a generalized Newton’s method based on the point-based set-valued approximation for solving nonsmooth equations. We
show that the proposed method converges locally superlinearly without the assumption of semi-smoothness. Finally we include
some well-known generalized Newton’s methods in our method and consolidate the convergence results of these methods.
Received October 2, 1995 / Revised version received May 5, 1998 Published online October 9, 1998 相似文献
12.
For solving systems of nonlinear equations, we have recently developed a Newton’s method to manage issues with inaccurate function values or problems with high computational cost. In this work we introduce a modification of the above method, reducing the total computational cost and improving, in general, its overall performance. Moreover, the proposed version retains the quadratic convergence, the good behavior over singular and ill-conditioned cases of Jacobian matrix, and its capability to be ideal for imprecise function problems. Numerical results demonstrate the efficiency of the new proposed method. 相似文献
13.
Augmented Lagrangian Homotopy Method for the Regularization of Total Variation Denoising Problems 总被引:1,自引:0,他引:1
L. A. Melara A. J. Kearsley R. A. Tapia 《Journal of Optimization Theory and Applications》2007,134(1):15-25
This paper presents a homotopy procedure which improves the solvability of mathematical programming problems arising from
total variational methods for image denoising. The homotopy on the regularization parameter involves solving a sequence of
equality-constrained optimization problems where the positive regularization parameter in each optimization problem is initially
large and is reduced to zero. Newton’s method is used to solve the optimization problems and numerical results are presented. 相似文献
14.
Wang Haijun 《Numerical Algorithms》2008,48(4):317-325
In this paper, we present new iteration methods with cubic convergence for solving nonlinear equations. The main advantage
of the new methods are free from second derivatives and it permit that the first derivative is zero in some points. Analysis
of efficiency shows that the new methods can compete with Newton’s method and the classical third-order methods. Numerical
results indicate that the new methods are effective and have definite practical utility.
相似文献
15.
In this work, we develop a family of predictor-corrector methods free from second derivative for solving systems of nonlinear equations. In general, the obtained methods have order of convergence three but, in some particular cases the order is four. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Newton’s classical method and with other recently published methods. 相似文献
16.
We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered
three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation
is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and
one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation
for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order
techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order
methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency
index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques. 相似文献
17.
Kanittha Chompuvised Ampon Dhamacharoen 《Applied mathematics and computation》2011,217(24):10355-10360
This paper focuses on solving the two point boundary value problem, in which boundary conditions are systems of nonlinear equations. The shooting method was used together with a combination of Newton’s method and Broyden’s method, to update the initial values of the differential equations. The experiments showed that the proposed method performed well, in the sense that the overall amount of work was less than that of the Newton Shooting method. 相似文献
18.
In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods. 相似文献
19.
Jacobian smoothing Brown’s method for nonlinear complementarity problems (NCP) is studied in this paper. This method is a generalization of classical Brown’s method. It belongs to the class of Jacobian smoothing methods for solving semismooth equations. Local convergence of the proposed method is proved in the case of a strictly complementary solution of NCP. Furthermore, a locally convergent hybrid method for general NCP is introduced. Some numerical experiments are also presented. 相似文献
20.
Cheng Xiaoliang Xu Yuanji Meng Bingquan 《高校应用数学学报(英文版)》2005,20(3):347-351
§1Introduction ConsidertheHamilton-Jacobi-Bellmanequation max1≤v≤m[A(v)u(x)-f(v)(x)]=0,x∈Ω(1.1)withtheboundarycondition u(x)=0,x∈Ω(1.2)whereΩisabounded,smoothdomaininEuclideanspaceRd,d∈N;f(v)(x)aregiven functionsfromC2(Ω);A(v)aresecond-orderuniformlyellipticoperatorsoftheform A(v)=-d i,j=1a(v)ij2xixj+di=1b(v)ixi+c(v).(1.3)Intheaboveexpression(1.3)therearecoefficientsa(v)ij,b(v)i,c(v)∈C2(Ω)satisfying,forall1≤v≤m,a(v)ij(x)=a(v)ji(x),1≤i,j≤d,c(v)≥c0≥0,x∈Ω,a… 相似文献