More, J.J., and Sorensen, D.C., 1984, Newton's Method, in Golub, G.H., Ed., Studies in Numerical Analysis: Mathematical Association of America, 29-82.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for some i , then there exists an open set such that for every starting point in the Newton iterates will remain in and converge to i . Within , Newton s method produces a quadratically convergent sequence of model iterates , m,i , i (3. 20) where (More and Sorensen, 1984). Therefore, not only is Newton s method locally convergent, but it is also very efficient within the neighbourhood . The problem with implementing Newton s method as a global method is that the region may be small. Although we try to make reasonable initial estimates of the model ....

....changes the local quadratic model of equation 3.16 in such a way that the new model has a unique minimum. The location of this minimum defines the modified Newton step. This step can then be paired with a line search technique to produce a convergent sequence of iterates. Theorem (4. 8) in (More and Sorensen, 1984) gives the strongest convergence result for line search 42 CHAPTER 3. NON LINEAR PARAMETER ESTIMATION PROCEDURE 0.01 1.03092 1.00499 1.01695 99.9 80.1 0.01778 1.00006 1.00028 1.00388 23.5 37.3 0.05623 0.99290 1.00230 1.00375 31.2 147.0 0.100 0.99101 1.00218 1.00352 ....

More, J.J., and Sorensen, D.C., 1984, Newton's Method, in Golub, G.H., Ed., Studies in Numerical Analysis: Mathematical Association of America, 29-82.

....[8, pp. 26 30] The step length parameters are = 10 Gamma4 and j = 0:9. The line search is based on using a safeguarded polynomial interpolation to find an approximate minimizer of the univariate function OE k (ff) f(x k ffp k ) Gamma f(x k ) Gamma ffg T k p k (see Mor e and Sorensen [21]) The step ff k is the first member of a minimizing sequence fff i k g that satisfies the Wolfe conditions. The sequence is usually started with ff 0 k = 1 (see below) If ff k satisfies the strong Wolfe conditions, it follows that s T k y k Gamma(1 Gamma j)g T k s k 0 and the BFGS ....

J. J. Mor' e and D. C. Sorensen, Newton's method, in Studies in Mathematics, Volume 24. Studies in Numerical Analysis, G. H. Golub, ed., vol. 24, The Mathematical Association of America, 1984, pp. 29--82.

.... books by (Dennis and Schnabel, 1983) Fletcher, 1987) and (Gill, Murray and Wright, 1981) We will concentrate on line search methods because most of our knowledge on trust region methods for unconstrained optimization was obtained before 1982, and is described in the excellent survey papers by (Mor e and Sorensen, 1984) and (Mor e, 1983) However in section 8 we will briefly compare the convergence properties of line search and trust region methods. 3. The Basic Convergence Principles One of the main attractions of the theory of unconstrained optimization is that a few general principles can be used to study ....

J. Mor'e and D.C. Sorensen (1984), "Newton's method", in Studies in Numerical Analysis (G.H.

....region are compatible. Thus we expect the step v generated by this normal subproblem to be an efficient restoration step that is robust in the presence of ill conditioning. Note that (2.1. 3) is a standard trust region problem and can be solved to any accuracy using an iterative approach (see [51]) It has the property that the Hessian of the objective is positive semi definite and singular. The problem (2.1.3) can also be solved using a CG iteration, using Steihaug s stopping tests [72] This has the advantage that the Jacobian matrix A is not explicitly needed, but only matrix vector ....

J. J. Mor'e and D. C. Sorensen. Newton's method. In Studies in Numerical Analysis, volume 24 of MAA Studies in Mathematics, pages 29--82. The Mathematical Association of America, 1984.

....Newton s direction and that, in a neighbourhood of an isolated local minimumpoint of Problem (IC) it is a descent direction. By using these results, a globally and superlinearly convergent algorithm for the minimization of U(x; can be defined by using any stabilization technique (see, e.g. [19, 59]) Part (ii) shows that Algorithm NT converges locally in one iteration if the problem considered is a quadratic programming problem and, hence, in this case, it takes advantage of the simple structure of the problem. Algorithm NT provides a link with Quasi Newton approaches for the solution of ....

J. J. Mor' e and D. C. Sorensen, Newton's method, in Studies in Numerical Analysis, G. H. Golub, ed., The Mathematical Association of America, Washington, DC, 1984, pp. 29--82.

....and (1.2) Fletcher [5] suggested that it is possible to compute a sequence of nested intervals that contain points that satisfy (1.1) and (1.2) but he did not prove any result along these lines. This suggestion led to the algorithms developed by Al Baali and Fletcher [2] and Mor e and Sorensen [11]. In this paper we provide a convergence analysis, implementation details, and numerical results for the algorithm of Mor e and Sorensen [11] The search algorithm for T ( is defined in Section 2. We show that the search algorithm produces a sequence of iterates that converge to a point in T ( ....

....(1.2) but he did not prove any result along these lines. This suggestion led to the algorithms developed by Al Baali and Fletcher [2] and Mor e and Sorensen [11] In this paper we provide a convergence analysis, implementation details, and numerical results for the algorithm of Mor e and Sorensen [11]. The search algorithm for T ( is defined in Section 2. We show that the search algorithm produces a sequence of iterates that converge to a point in T ( and that, except for pathological cases, the search algorithm produces a finite sequence ff 0 ; ff m of trial values in [ff min ; ff ....

[Article contains additional citation context not shown here]

J. J. Mor' e and D. C. Sorensen, Newton's method, in Studies in Numerical Analysis, G. H. Golub, ed., The Mathematical Association of America, 1984, pp. 29--82.

No context found.

J.J. MOR ' E and D.C. SORENSEN. Newton's method. In Gene H. Golub, editor, Studies in Numerical Analysis, volume 24 of MAA Studies in Mathematics. The Mathematical Association of America, 1984.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC