Numerical Solution of Nonlinear Equations pp 182-198 | Cite as

# A derivative-free arc continuation method and a bifurcation technique

Conference paper

First Online:

## Abstract

Algorithms and comparison results for a derivative-free predictor-corrector method for following arcs of H(x,t) = ϑ, where H : R^{n} × [0, 1] → R^{n} is smooth, are given. The method uses a least-change secant update for H', adaptive controlled predictor stepsize, and Powell's indexing procedure to preserve linear independence in the updates. Considerable savings in numbers of theoretical function calls are observed over high order methods requiring explicit H'. The framework of a promising technique for handling general bifurcation problems is presented.

## key words

arc continuation quasi-Newton methods least change secant updates Brouwer degree numerical computation nonlinear algebraic systems Powell's method## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A.K. Cline, C.B. Moler, G.W. Stewart, and J.H. Wilkinson, "An estimate for the condition number of a matrix," SIAM J. Numer. Anal. 16, #2, 1979.Google Scholar
- 2.E. Allgower and K. Georg, "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations," SIAM Review 22, no. 1, January, 1980.Google Scholar
- 3.J.E. Dennis, Jr. and R.B. Schnabel, "Least change secant updates for quasi-Newton methods," SIAM Review 21 no. 4, 1979 (pp. 443–459).MathSciNetCrossRefzbMATHGoogle Scholar
- 4.C.B. Garcia and T.Y. Li, "On a path-following method for systems of equations," MRC Technical Summary Report #1983, Mathematics Research Center, University of Wisconsin — Madison, July, 1979.Google Scholar
- 5.C.B. Garcia and W.I. Zangwill, "Finding all solutions to polynomial systems and other systems of equations," Math. Prog. 16, 1979 (pp. 159–176).MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Kurt Georg, "Numerical integration of the Davidenko equation," these proceedings.Google Scholar
- 7.Kurt Georg, "On tracing an implicitly defined curve by quasi-Newton steps and calculating bifurcation by local perturbations," to appear in the SIAM Journal on Scientific and Statistical Computing.Google Scholar
- 8.Harmut Jürgens, Heinz-Otto Peitgen, and Dietmar Saupe, "Topological perturbations in the numerical nonlinear eigenvalue and bifurcation problems," Proc. Conf. Analysis and Computation of Fixed Points, Academic Press, New York-London, 1980, 139–181.zbMATHGoogle Scholar
- 9.R.B. Kearfott, "An efficient degree-computation method for a generalized method of bisection," Numer. Math. 32, 1979 (pp. 109–127).MathSciNetCrossRefzbMATHGoogle Scholar
- 10.R.B. Kearfott, "An improved program for generalized bisection," to appear.Google Scholar
- 11.R.B. Kearfott, "A summary of recent experiments to compute the topological degree," Applied Nonlinear Analysis, ed. V. Lakshmikantham, A.P., 1979.Google Scholar
- 12.R.B. Kearfott, "Some general derivative-free bifurcation techniques," submitted to the SIAM Journal on Scientific and Statistical Computing.Google Scholar
- 13.H.B. Keller, "Numerical solution of bifurcation and nonlinear eigenvalue problems." Applications of Bifurcation Theory, ed. P.H. Rabinowitz, Academic Press, New York, 1979, (pp. 359–384).Google Scholar
- 14.T.Y. Li and J.A. Yorke, "A simple reliable numerical algorithm for following homotopy paths," Technical Summary Report #1984, Mathematics Research Center, University of Wisconsin — Madison, Wisconsin 53706.Google Scholar
- 15.M.J.D. Powell, "A Fortran subroutine for solving systems of nonlinear algebraic equations," Numerical Methods for Nonlinear Algebraic Equations, ed. P. Rabinowitz, Gordon and Breach, 1970.Google Scholar
- 16.L.T. Watson, "A globally convergent algorithm for computing fixed points of C
^{2}maps," Appl. Math. Comput., to appear.Google Scholar

## Copyright information

© Springer-Verlag 1981