J. E. Dennis, Jr., J. M. Mart ' inez and X. Zhang, "Triangular decomposition methods for solving reducible nonlinear systems of equations," SIAM Journal on Optimization, Vol. 5, No. 2, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and [35] If k is large the process must be periodically restarted taking B k J(x k ) Broyden s method is a particular case of the family of Least Change Secant Update (LCSU) methods ( 8] 9] 10] 40] 42] which include many algorithms that are useful for particular structures. See [7], 36] 27] 28] 29] 30] 54] 34] 39] etc. With the same hypotheses that imply local convergence of Newton s method, for a good enough initial Jacobian approximation B 0 , LCSU algorithms are well defined, converge to a solution x and the rate of convergence is q superlinear. See ....

Dennis Jr., J.E., Martinez, J.M. and Zhang, X. (1994). Triangular decomposition methods for solving reducible nonlinear systems of equations, SIAM Journal on Optimization 4, 358-382.

No context found.

J. E. Dennis, Jr., J. M. Mart ' inez and X. Zhang, "Triangular decomposition methods for solving reducible nonlinear systems of equations," SIAM Journal on Optimization, Vol. 5, No. 2, 1994.

.... becomes F 1 (x 1 ) 0; F 2 (x 1 ; x 2 ) 0; Fm (x 1 ; x 2 ; xm ) 0; 2) with x = x 1 ; x 2 ; xm ) x i 2 IR n i ; F i : IR n 1 Theta IR n 2 Theta : Theta IR n i IR n i ; i = 1; 2; m; n 1 n 2 : nm = n: Such systems were considered in [4], where a large class of locally convergent p step (quasi) Newton methods was proposed. Of course, in practice one is interested in the case m 1. Systems with the structure (2) appear naturally in many practical applications. For example, the index i 2 f1; mg could represent time and x ....

....as we claimed in the case of local convergence properties, the reason why Newton globalization schemes are efficient in real life is that theoretically justified Inexact Newton counterparts exist. The relation between the results presented in this paper with the BlockNewton method introduced in [4] is the same that exists between the classical Inexact Newton and the exact Newton method. We are going to prove that the Inexact Newton version of the Block Newton method of [4] is locally convergent, which means that, very likely, local convergence of the BlockNewton method will be reflected in ....

[Article contains additional citation context not shown here]

J. E. Dennis, J. M. Mart'inez and X. Zhang, Triangular decomposition methods for solving reducible nonlinear systems of equations, SIAM Journal on Optimization 4 (1994), 358-382.

....F (x) 0, where F 2 R n , x 2 R n and the Jacobian J(x) is sparse. A triangular decomposition approach is to transform a sparse nonlinear system of equations into a block triangular structure so that the computations can be decomposed for parallel processing. Dennis, Mart inez and Zhang[8, 9] develop several efficient parallel methods for solving a sparse nonlinear system of equations. We monitored one of the methods which was implemented on the GP1000. This method is a variation of the Gauss Seidel Newton method. It evaluates the diagonal blocks concurrently instead of successively ....

Dennis, J. E., Jr., Mart ' inez, J. M., and Zhang, X. Triangular decomposition methods for solving reducible nonlinear systems of equations. SIAM J. Optimization, to appear.

....: Theta IR n i IR n i ; i = 1; 2; m; n 1 n 2 : nm = n: The most usual way of solving (1) is the straight decomposition scheme (SDS) that consists on solving sequentially the different n i Theta n i systems. Alternative Newtonian procedures were suggested in [2, 3, 4]. In fact, solving is an ambiguous word in the case of a nonlinear system where to find exact solutions is not possible. In order to understand the necessity of the analysis presented in this paper, let us consider the most simple situation, where we have two blocks of equations (m = 2) ....

....help us to find a reasonable x 2 in the second step. The simple procedure described above suggests an iterative process, which is essentially the one considered in this paper. The possibility of taking advantage of parallel computer architectures, led to the development of related methods in [2, 4]. The idea of these algorithms is to take the system (1) as a whole, though taking into account its block angular structure, in such a way that each iteration can be considered an approximation to a Newton, quasi Newton [2] or inexact Newton iteration [4] See also [1, 7, 9] In this paper we ....

[Article contains additional citation context not shown here]

Dennis, J.E., Mart'inez, J.M. and Zhang, X., Triangular decomposition methods for solving reducible nonlinear systems of equations. SIAM Journal on Optimization, 4 (1994) 358-382.

....nonsingular we say that F is BD regular at x: Such systems were considered in Mart inez and Qi [9] Mifflin [10] Pang and Qi [11] Qi and Sun [13] and other authors. Smooth systems of equations with the block angular structure (1) have been studied in Erickson [5] Dennis, Mart inez and Zhang, [3], Kreji c and Mart inez [8] In [3] Newton like methods were considered, that require the solutions of m linear systems of dimension n i Theta n i at each iteration. When the number of state variables n i is large, direct solution of linear systems can be prohibitive and, so, an inexact Newton ....

....at x: Such systems were considered in Mart inez and Qi [9] Mifflin [10] Pang and Qi [11] Qi and Sun [13] and other authors. Smooth systems of equations with the block angular structure (1) have been studied in Erickson [5] Dennis, Mart inez and Zhang, 3] Kreji c and Mart inez [8] In [3] Newton like methods were considered, that require the solutions of m linear systems of dimension n i Theta n i at each iteration. When the number of state variables n i is large, direct solution of linear systems can be prohibitive and, so, an inexact Newton approach (using iterative linear ....

[Article contains additional citation context not shown here]

J. E. Dennis, J. M. Mart'inez and X. Zhang, Triangular decomposition methods for solving reducible nonlinear systems of equations, SIAM Journal on Optimization 4 (1994) 358-382.

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