Continuation and path following. (English) Zbl 0792.65034

Allgower, Eugene L.; Georg, Kurt

Iserles, A. (ed.), Acta Numerica 1993. Cambridge: Cambridge University Press. 1-64 (1993).

This is a nice up-date survey article to explore the main ideas of path following by predictor-corrector and piecewise-linear methods. The review is developed with applications to homotopy methods, nonlinear and linear eigenvalue problems, systems of polynomial equations, interior point methods for linear programming, parametric programming, complex bifurcation, and even complexity issues. Available software is listed.
In Section 5 (Piecewise linear methods) the concept of regularity is introduced. Perhaps it should be made clear that there is in any case no degeneracy in piecewise linear methods no matter regular or not [cf. T. Gao and Z. Wang, in K. Tan ed., Fixed Point Theory and Application, Worls Scientific, Singapore (1992)]. The significance of introducing regularity is no more for avoiding degeneracy but only for simplicity of description.
For the entire collection see [Zbl 0777.00039].

Reviewer: Wang Zeke (Guangzhou)

MSC:

65H20	Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65F15	Numerical computation of eigenvalues and eigenvectors of matrices
65H10	Numerical computation of solutions to systems of equations
65H17	Numerical solution of nonlinear eigenvalue and eigenvector problems

Keywords:

predictor-corrector methods; research survey; path following; piecewise- linear methods; homotopy methods; eigenvalue problems; systems of polynomial equations; interior point methods; linear programming; parametric programming; complex bifurcation; regularity