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QuasiNewton Methods With Derivatives
, 1995
"... When the Jacobian of a nonlinear system of equations is fully available, the main drawback for the application of Newton's method is the linear algebra work associated with its basic iteration. In many cases, quasiNewton methods "with cheap linear algebra" can be applied. The availa ..."
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When the Jacobian of a nonlinear system of equations is fully available, the main drawback for the application of Newton's method is the linear algebra work associated with its basic iteration. In many cases, quasiNewton methods "with cheap linear algebra" can be applied
On the convergence of inexact Newton methods
, 2011
"... The inexact Newton method is widely used to solve systems of nonlinear equations. It is wellknown that forcing terms should be chosen relatively large at the start of the process, and be made smaller during the iteration process. This paper explores the mechanics behind this behavior theoretically ..."
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The inexact Newton method is widely used to solve systems of nonlinear equations. It is wellknown that forcing terms should be chosen relatively large at the start of the process, and be made smaller during the iteration process. This paper explores the mechanics behind this behavior
Newton Leaves and the Continuous Newton Method
, 1993
"... Aspects of the continuous Newton method in IR n are discussed from a viewpoint of global optimization. Using an algebraic instead of an analytic definition of Newton trajectories we generalize the usual concept to certain higher dimensional sets (Newton leaves). The study of these sets reveals ins ..."
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Aspects of the continuous Newton method in IR n are discussed from a viewpoint of global optimization. Using an algebraic instead of an analytic definition of Newton trajectories we generalize the usual concept to certain higher dimensional sets (Newton leaves). The study of these sets reveals
Choosing the Forcing Terms in an Inexact Newton Method
 SIAM J. SCI. COMPUT
, 1994
"... An inexact Newton method is a generalization of Newton's method for solving F(x) = 0, F:/ /, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition ]lF(x) + F'(x)s]l _< /]lF(xk)]l for a "forcing term" / [0,1). I ..."
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Cited by 161 (6 self)
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An inexact Newton method is a generalization of Newton's method for solving F(x) = 0, F:/ /, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition ]lF(x) + F'(x)s]l _< /]lF(xk)]l for a "forcing term" / [0
Global Approximate Newton Methods
, 1981
"... . We derive a class of globally convergent and quadratically converging algorithms for a system of nonlinear equations g(u) = 0, where g is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have succes ..."
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Cited by 44 (3 self)
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. We derive a class of globally convergent and quadratically converging algorithms for a system of nonlinear equations g(u) = 0, where g is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling. AMS subject classifications. 65H10 1. Introduction. In this paper we derive an algorithm for solving the nonlinear system g(u) = 0 (1.1) where g = (g 1 ; g 2 ; : : : ; gn ) T is a sufficiently smooth homeomorphism from IR n to IR n . Recall that a homeomorphism is a bijection (11, onto) with both g and the inverse map, g \Gamma1 , continuous. Physically, a homeomorphism means that the process modelled by g has a unique solution x for any set of input conditions y, i.e., g(x) = y, and that the solution x varies continuously with input y. Sometimes this notion is referred to as a "wellposed" ...
QuasiNewton methods for unconstrained optimization
 J. Inst. Math. Appl
, 1972
"... Arevised algorithm is given for unconstrained optimization using quasiNewton methods. The method is based on recurring the factorization of an approximation to the Hessian matrix. Knowledge of this factorization allows greater flexibility when choosing the direction of search while minimizing the a ..."
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Cited by 64 (1 self)
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Arevised algorithm is given for unconstrained optimization using quasiNewton methods. The method is based on recurring the factorization of an approximation to the Hessian matrix. Knowledge of this factorization allows greater flexibility when choosing the direction of search while minimizing
Nonsmooth Equations and Smoothing Newton Methods
 APPLIED MATHEMATICS REPORT AMR 98/10 , SCHOOL OF MATHEMATICS, THE UNIVERSITY OF NEW SOUTH
, 1998
"... In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. ..."
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Cited by 3 (2 self)
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In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented.
Multihomogeneous Newton methods
 MATHEMATICS OF COMPUTATION
, 1999
"... We study multihomogeneous analytic functions and a multihomogeneous Newton’s method for finding their zeros. We give a convergence result for this iteration and we study two examples: the evaluation map and the generalized eigenvalue problem. ..."
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Cited by 18 (5 self)
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We study multihomogeneous analytic functions and a multihomogeneous Newton’s method for finding their zeros. We give a convergence result for this iteration and we study two examples: the evaluation map and the generalized eigenvalue problem.
Numerical experience with limitedMemory QuasiNewton methods and Truncated Newton methods
 SIAM J. Optimization
, 1992
"... Abstract. Computational experience with several limitedmemory quasiNewton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a wellknown test library [J. J. Mor, B. S. Garbow, and K. E. Hillstrom, ACM Trans. Math. Software, 7 (1 ..."
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Cited by 13 (9 self)
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Abstract. Computational experience with several limitedmemory quasiNewton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a wellknown test library [J. J. Mor, B. S. Garbow, and K. E. Hillstrom, ACM Trans. Math. Software, 7
GLOBALLY CONVERGENT INEXACT NEWTON METHODS*
"... Abstract. Inexact Newton methods for finding a zero of F 1 1 are variations of Newton’s method in which each step only approximately satisfies the linear Newton equation but still reduces the norm of the local linear model of F. Here, inexact Newton methods are formulated that incorporate features d ..."
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Abstract. Inexact Newton methods for finding a zero of F 1 1 are variations of Newton’s method in which each step only approximately satisfies the linear Newton equation but still reduces the norm of the local linear model of F. Here, inexact Newton methods are formulated that incorporate features
Results 1  10
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