A study of accelerated Newton methods for multiple polynomial roots

Abstract  We analyze and compare several accelerated Newton methods with built in multiplicity estimates. We also introduce the concept
of indicator functions and discuss the Crouse-Putt method. It is shown that many of the accelerated Newton methods not only
derive from Schröder’s classic approach but are equivalent. The related computational experiments show that the built in multiplicity
estimates can significantly decrease the number of Newton iterations, while the error of these estimates may significantly
increase.

• Content Type Journal Article
• Category Original Paper
• DOI 10.1007/s11075-009-9332-x
• Authors
  
  
  • Aurél Galántai, Budapest Tech John von Neumann Faculty of Informatics Bécsi út 96/b 1034 Budapest Hungary
  • Csaba J. Hegedűs, Eötvös Loránd University Pázmány Péter sétány 1/c 1117 Budapest Hungary
  
  

• Journal Numerical Algorithms
• Online ISSN 1572-9265
• Print ISSN 1017-1398

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Original source : http://www.springerlink.com/content/a34147h1g45630...

Analisys of Hamiltonian Boundary Value Methods (HBVMs) for the numerical solution of polynomial Hamiltonian dynamical systems. (arXiv:0909.5659v1 [math.NA])

One main issue, when numerically integrating autonomous Hamiltonian systems,
is the long-term conservation of some of its invariants, among which the
Hamiltonian function itself. For example, it is well known that standard (even
symplectic) methods can only exactly preserve quadratic Hamiltonians. In this
paper, a new family of methods, called Hamiltonian Boundary Value Methods
(HBVMs), is introduced and analyzed. HBVMs are able to exactly preserve, in the
discrete solution, Hamiltonian functions of polynomial type of arbitrarily high
degree. These methods turn out to be symmetric, perfectly $A$-stable, and can
have arbitrarily high order. A few numerical tests confirm the theoretical
results.

Published by

Original source : http://arxiv.org/abs/0909.5659...