Newton Methods to Solve a System of Nonlinear Algebraic Equations
MetadataShow full item record
Fundamental insight into the solution of systems of nonlinear equations was provided by Powell. It was found that Newton iterations, with exact line searches, did not converge to a stationary point of the natural merit function, i.e., the Euclidean norm of the residuals. Extensive numerical simulation of Powell's equations produced the unexpected result that Newton iterations converged to the solution from all initial points, where the function is defined, or from those points where the Jacobian is nonsingular, if no line search is used. The significance of Powell's example is that an important requirement exists when utilizing Newton's method to solve such a system of nonlinear equations. Specifically, a merit function, which is used in a line search, must have properties consistent with those of a Lyapunov function to provide sufficient conditions for convergence. This implies that level sets of the merit function are properly nested, either globally, or in some finite local region. Therefore, they are topologically equivalent to concentric spherical surfaces, either globally or in a finite local region. Furthermore, an exact line search at a point, far from the solution, may be counterproductive. This observation, and a primary aim of the present analysis, is to demonstrate that it is desirable to construct new Newton iterations, which do not require a merit function with associated line searches. © 2014 Springer Science+Business Media New York.
Showing items related by title, author, creator and subject.
On-Line dynamic security assessment of wind farm connected power systems using a class of intelligent algorithmsTiako, Remy (2012)Recently large-scale wind farms are integrated quite commonly into power systems. The stochastic operation of wind plants due to intermittency and intra-interval effects of the wind is a problematic issue to determine the ...
Sansom, E.; Jansen-Sturgeon, T.; Rutten, M.; Devillepoix, Hadrien; Bland, Phil; Howie, Robert; Cox, M.; Towner, Martin; Cupák, M.; Hartig, Ben (2019)Meteoroid modelling of fireball data typically uses a one dimensional model along a straight line triangulated trajectory. The assumption of a straight line trajectory has been considered an acceptable simplification for ...
Shimwell, T.; Barker, R.; Biddulph, P.; Bly, D.; Boysen, R.; Brown, A.; Brown, M.; Clementson, C.; Crofts, M.; Culverhouse, T.; Czeres, J.; Dace, R.; Davies, M.; D’Alessandro, R.; Doherty, P.; Duggan, K.; Ely, J.; Felvus, M.; Feroz, F.; Flynn, W.; Franzen, Thomas; Geisbusch, J.; Genova-Santos, R.; Grainge, K.; Grainger, W.; Hammett, D.; Hobson, M.; Holler, C.; Hurley-Walker, Natasha; Jilley, R.; Kaneko, T.; Kneissl, R.; Lancaster, K.; Lasenby, A.; Marshall, P.; Newton, F.; Norris, O.; Northrop, I.; Odell, D.; Olamaie, M.; Perrott, Y.; Pober, J.; Pooley, G.; Pospieszalski, M.; Quy, V.; Rodriguez-Gonzalvez, C.; Saunders, R.; Scaife, A.; Schammel, M.; Schofield, J.; Scott, P.; Shaw, C.; Smith, H.; Titterington, D.; Velic, M.; Waldram, E.; West, S.; Wood, B.; Yassin, G.; Zwart, J. (2012)We present an interesting Sunyaev–Zel’dovich (SZ) detection in the first of the Arcminute Microkelvin Imager (AMI) ‘blind’, degree-square fields to have been observed down to our target sensitivity of 100 µJy beam-1. ...