Solution Fields of Nonlinear Equations and Continuation Methods

1980 ◽  
Vol 17 (2) ◽  
pp. 221-237 ◽  
Author(s):  
Werner C. Rheinboldt
Keyword(s):  
Nonlinear Equations ◽  
Continuation Methods

scholarly journals A note on continuation methods for the solution of nonlinear equations

1977 ◽  
Vol 20 (2) ◽  
pp. 157-164 ◽  
Author(s):  
James P. Abbott ◽  
Richard P. Brent
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Continuation Method ◽  
Continuation Methods ◽  
Variable Order ◽  
Rapid Convergence

AbstractIn this note we present a variable order continuation method for the solution of nonlinear equations when only a poor estimate of a solution is known. The method changes continuously from one which improves the global convergence characteristics to one which attains rapid convergence to a solution and proves to be more efficient than methods previously presented in [2].

scholarly journals Dynamical phenomena induced by bottleneck

2010 ◽  
Vol 368 (1928) ◽  
pp. 4543-4562 ◽  
Author(s):  
I. Gasser ◽  
B. Werner
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Equations ◽  
Bifurcation Analysis ◽  
Complex Dynamics ◽  
Standing Waves ◽  
Continuation Methods ◽  
Traffic Dynamics ◽  
Poincaré Maps ◽  
Different Types ◽  
The Way

We study a microscopic follow-the-leader model on a circle of length L with a bottleneck. Allowing large bottleneck strengths we encounter very interesting traffic dynamics. Different types of waves—travelling and standing waves and combinations of both wave types—are observed. The way to find these phenomena requires a good understanding of the complex dynamics of the underlying (nonlinear) equations. Some of the phenomena, like the ponies-on-a-merry-go-round solutions, are mathematically well known from completely different applications. Mathematically speaking we use Poincaré maps, bifurcation analysis and continuation methods beside numerical simulations.

Numerical methods for nonlinear equations

Acta Numerica ◽  
2018 ◽  
Vol 27 ◽  
pp. 207-287 ◽  
Author(s):  
C. T. Kelley
Keyword(s):  
Fixed Point ◽  
Numerical Methods ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Continuation Methods ◽  
Anderson Acceleration ◽  
Point Form

This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form $\mathbf{x}=\mathbf{G}(\mathbf{x})$ and the equations form $\mathbf{F}(\mathbf{x})=0$ and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and discuss less familiar topics such as Anderson acceleration, semi-smooth Newton’s method, and pseudo-arclength and pseudo-transient continuation methods.

Mapped continuation methods for computing all solutions to general systems of nonlinear equations

1990 ◽  
Vol 14 (1) ◽  
pp. 71-85 ◽  
Author(s):  
J.D. Seader ◽  
M. Kuno ◽  
W.-J. Lin ◽  
S.A. Johnson ◽  
K. Unsworth ◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Continuation Methods ◽  
Systems Of Nonlinear Equations ◽  
All Solutions ◽  
General Systems
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