Abstract
For a class of functions, closely related to the complex polynomials, the so-called Gradient Newton flows are studied. Attention is paid to structural stability and global convergence aspects. In the special case of polynomials of degree 3, a complete description of the local and global features of the involved phase-portraits is given.
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References
Branin, F.H., A widely convergent method for finding multiple solutions of simultaneous non-linear equations, IBM J. Res. Develop. (1972), 504–522.
Bröcker, Th., Differentiable Germs and Catastrophes, London Math. Society Lect. Notes 17, Cambridge Univ. Press 1975.
Diener, I., On the global convergence of path-following methods to determine all solutions to a system of non-linear equations, NAM Bericht 47, Göttingen (1985).
Gibson, C.G., Wirthmüller, K., Plessis, A.A. du, Looijenga, E.J.N., Topological Stability of Smooth Mappings, Lect. Notes Math. 552, Springer Verlag 1976.
Jongen, H.Th., Jonker, P., Twilt, F., Nonlinear optimization in IRn, part II: Transversality, Flows, Parametric Aspects, Methoden und Verfahren der Mathematischen Physik, 32, Peter Lang Verlag, Frankfurt a.M., Bern, New York 1986.
Jongen, H.Th., Jonker, P., Twilt, F., On the classification of plane graphs representing structurally stable rational Newton flows, Memorandum Nr. 750, University of Twente, Enschede, The Netherlands (1988); to appear in J. Comb. Theory, series B.
Jongen, H.Th., Jonker, P., Twilt, F., The continuous desingularized Newton method for meromorphic functions, Acta Applicandae Mathematica 13 (1988), 81–121.
Smale, S., A convergent process of price adjustment and global Newton methods, J. Math. Economics 3 (1976), 107–120.
Lang, S., Algebra, Addison Wesley Publ. 1965.
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© 1989 Springer Verlag
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Twilt, F., Jonker, P., Streng, M. (1989). Gradient newton flows for complex polynomials. In: Dolecki, S. (eds) Optimization. Lecture Notes in Mathematics, vol 1405. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083595
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DOI: https://doi.org/10.1007/BFb0083595
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