Results 1  10
of
11
Local convergence of inexact methods under the Hölder condition
, 2008
"... We study the convergence properties for some inexact Newtonlike methods including the inexact Newton methods for solving nonlinear operator equations on Banach spaces. A new type of residual control is presented. Under the assumption that the derivative of the operator satisfies the HoÌlder condit ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We study the convergence properties for some inexact Newtonlike methods including the inexact Newton methods for solving nonlinear operator equations on Banach spaces. A new type of residual control is presented. Under the assumption that the derivative of the operator satisfies the HoÌlder condition, the radius of convergence ball of the inexact Newtonlike methods with the new type of residual control is estimated, and a linear and/or superlinear convergence property is proved, which extends the corresponding result of [B. Morini, Convergence behaviour of inexact Newton methods, Math. Comput. 68 (1999) 1605â1613]. As an application, we show that the inexact Newtonlike method presented in [R.H. Chan, H.L. Chung, S.F. Xu, The inexact Newtonlike method for inverse eigenvalue problem, BIT Numer. Math. 43 (2003) 7â20] for solving inverse eigenvalue problems can be regarded equivalently as one of the inexact Newtonlike methods considered in this paper. A numerical example is provided to illustrate the convergence performance of the algorithm.
An Inexact Cayley Transform Method For Inverse Eigenvalue Problems
"... The Cayley transform method is a Newtonlike method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
(Show Context)
The Cayley transform method is a Newtonlike method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. In this paper, we develop an inexact version of the Cayley transform method. Our method can reduce the oversolving problem and improves the efficiency with respect to the exact version. We show that the convergence rate of our method is superlinear and that a good tradeoff between the required inner and outer iterations can be obtained.
On Some Inverse Singular Value Problems with ToeplitzRelated Structure
"... In this paper, we consider some inverse singular value problems for Toeplitzrelated matrices. We construct a ToeplitzplusHankel matrix from prescribed singular values including a zero singular value. Then we find a solution to the inverse singular value problem for Toeplitz matrices which have do ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
In this paper, we consider some inverse singular value problems for Toeplitzrelated matrices. We construct a ToeplitzplusHankel matrix from prescribed singular values including a zero singular value. Then we find a solution to the inverse singular value problem for Toeplitz matrices which have double singular values including a double zero singular value.
A Note on the Ulmlike Method for Inverse Eigenvalue Problems
"... A Ulmlike method is proposed in [13] for solving inverse eigenvalue problems with distinct given eigenvalues. The Ulmlike method avoids solving the Jacobian equations used in Newtonlike methods and is shown to be quadratically convergent in the root sense. However, the numerical experiments in [3 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
A Ulmlike method is proposed in [13] for solving inverse eigenvalue problems with distinct given eigenvalues. The Ulmlike method avoids solving the Jacobian equations used in Newtonlike methods and is shown to be quadratically convergent in the root sense. However, the numerical experiments in [3] only show that the Ulmlike method is comparable to the inexact Newtonlike method. In this short note, we give a numerical example to show that the Ulmlike method is better than the inexact Newtonlike method in terms of convergence neighborhoods. Keywords. Inverse eigenvalue problem, Ulmlike method, inexact Newtonlike method. AMS subject classifications. 65F18, 65F10, 65F15.
Convergence criterion of inexact methods for operators with Hölder continuous derivatives
"... Abstract. Convergence criterion of the inexact methods is established for operators with hölder continuous first derivatives. An application to a special nonlinear Hammerstein integral equation of the second kind is provided. 1. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Convergence criterion of the inexact methods is established for operators with hölder continuous first derivatives. An application to a special nonlinear Hammerstein integral equation of the second kind is provided. 1.
An Analysis on Local Convergence of Inexact NewtonGauss Method for Solving Singular Systems of Equations
, 2014
"... We study the local convergence properties of inexact NewtonGauss method for singular systems of equations. Unified estimates of radius of convergence balls for one kind of singular systems of equations with constant rank derivatives are obtained. Application to the Smale point estimate theory is p ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We study the local convergence properties of inexact NewtonGauss method for singular systems of equations. Unified estimates of radius of convergence balls for one kind of singular systems of equations with constant rank derivatives are obtained. Application to the Smale point estimate theory is provided and some important known results are extended and/or improved.
Inexact Newton Methods for Inverse Eigenvalue Problems Zhengjian Bai∗
"... In this paper, we survey some of the latest development in using inexact Newtonlike methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems. One can solve the approximate Jacobian equation by iterative methods. However, iterative ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper, we survey some of the latest development in using inexact Newtonlike methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems. One can solve the approximate Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newtonlike (outer) iterations. The inexact methods can reduce or minimize the oversolving problem and improve the efficiency. The convergence rates of the inexact methods are superlinear and a good tradeoff between the required inner and outer iterations can be obtained.
Kantorovichtype convergence . . .
 APPLIED NUMERICAL MATHEMATICS 59 (2009) 1599–1611
, 2009
"... ..."
Inexact Numerical Methods for Inverse Eigenvalue Problems
, 2006
"... In this paper, we survey some of the latest development in using inexact Newtonlike methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems. One can solve the approximate Jacobian equation by iterative methods. However, iterative ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper, we survey some of the latest development in using inexact Newtonlike methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems. One can solve the approximate Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newtonlike (outer) iterations. The inexact methods can reduce or minimize the oversolving problem and improve the efficiency. The convergence rates of the inexact methods are superlinear and a good tradeoff between the required inner and outer iterations can be obtained. AMS Subject Classifications. 65F18, 65F10, 65F15. 1