A global analysis of Newton iterations for determining turning points

Vladimír Janovský; Viktor Seige

Applications of Mathematics (1993)

  • Volume: 38, Issue: 4-5, page 323-360
  • ISSN: 0862-7940

Abstract

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The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a b i f u r c a t i o n s i n g u l a r i t y (i.e., as a d e g e n e r a t e turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship between the scenario and the actual performance of Newton method is studied. Both theoretical and experimental arguments are presented in order to quaetion the claim that a particular bifurcation singularity o r g a n i y e s the Newton method assuming small parameter perturbations.

How to cite

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Janovský, Vladimír, and Seige, Viktor. "A global analysis of Newton iterations for determining turning points." Applications of Mathematics 38.4-5 (1993): 323-360. <http://eudml.org/doc/15758>.

@article{Janovský1993,
abstract = {The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a $bifurcation singularity$ (i.e., as a $degenerate$ turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship between the scenario and the actual performance of Newton method is studied. Both theoretical and experimental arguments are presented in order to quaetion the claim that a particular bifurcation singularity $organiyes$ the Newton method assuming small parameter perturbations.},
author = {Janovský, Vladimír, Seige, Viktor},
journal = {Applications of Mathematics},
keywords = {detection of turning points; Newton method; Newton flow; basins of attraction; qualitative analysis; normal forms of the flow; global convergence; singularity theory; bifurcation singularity; imperfect bifurcation; global convergence; Newton method; singularity theory; turning points; bifurcation singularity; imperfect bifurcation},
language = {eng},
number = {4-5},
pages = {323-360},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A global analysis of Newton iterations for determining turning points},
url = {http://eudml.org/doc/15758},
volume = {38},
year = {1993},
}

TY - JOUR
AU - Janovský, Vladimír
AU - Seige, Viktor
TI - A global analysis of Newton iterations for determining turning points
JO - Applications of Mathematics
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 38
IS - 4-5
SP - 323
EP - 360
AB - The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a $bifurcation singularity$ (i.e., as a $degenerate$ turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship between the scenario and the actual performance of Newton method is studied. Both theoretical and experimental arguments are presented in order to quaetion the claim that a particular bifurcation singularity $organiyes$ the Newton method assuming small parameter perturbations.
LA - eng
KW - detection of turning points; Newton method; Newton flow; basins of attraction; qualitative analysis; normal forms of the flow; global convergence; singularity theory; bifurcation singularity; imperfect bifurcation; global convergence; Newton method; singularity theory; turning points; bifurcation singularity; imperfect bifurcation
UR - http://eudml.org/doc/15758
ER -

References

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