By E. L. Allgower, C.-S. Chien, K. Georg (auth.), Hans D. Mittelmann, Dirk Roose (eds.)
The research of parameter-dependent nonlinear has got a lot consciousness lately. Numerical continuation concepts let the effective computation of resolution branches in a one-parameter challenge. in lots of circumstances continuation systems are used as a part of a extra entire research of a nonlinear challenge, in keeping with bifurcation thought and singularity idea. those theories give a contribution to the certainty of many nonlinear phenomena in nature and so they shape the root for varied analytical and numerical instruments, which offer qualitative and quantitative effects approximately nonlinear structures. during this factor we have now amassed a couple of papers facing continuation innovations and bifurcation difficulties. Readers accustomed to the notions of continuation and bifurcation will locate contemporary learn effects addressing various facets during this factor. those that intend to profit in regards to the box or a particular subject in it could possibly locate it worthy to first seek advice prior literature at the numerical remedy of those difficulties including a few theoretical heritage. The papers during this factor fall certainly into diverse groups.
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Extra info for Continuation Techniques and Bifurcation Problems
A method closely related to that will be discussed here and the results of extensive computations will be reported that shed more light on a phenomenon first observed in . The difference between the continuation method presented here and that of  is the transition to IL = A-I made below which results in a symmetric Jacobian of the system permitting the application of certain conjugate gradient methods. 1c) where A is a symmetric positive-definite matrix. We further denote the gradient and Hessian of g by b and B respectively, and for reasons that will become clear later we introduce IL = A-I.
This root-bracketing procedure can be repeated to quickly isolate limit points [23,27]. ) More sophisticated step-size control mechanisms, which attempt to estimate the local convergence ball for Newton's method or sense curvature in a more sophisticated way, might help in some instances . Other possibilities include: (1) the use of a higher-order predictor; (2) a Newton damping strategy that interacts directly with the step-size selection procedure such as suggested recently by Bank and Mittelmann .
6). This made it possible to use efficient conjugate gradient methods. A further increase of efficiency may be obtained through a multigrid approach [5,6]. 6), permits the solution of problems with 10 4 variables with modest storage and computing time demands. E. Miersemann, H. Mittelmann / Nonlinear variational inequalities 29 It is beyond the scope of this work to analyze the numerical method or to present exhaustive results obtained with it. A phenomenon first observed in  will be illustrated in more detail.
Continuation Techniques and Bifurcation Problems by E. L. Allgower, C.-S. Chien, K. Georg (auth.), Hans D. Mittelmann, Dirk Roose (eds.)