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The power method for tensor eigenproblems and limiting directions of Newton iterates
Author(s) -
Dupont Todd F.,
Ridgway Scott L.
Publication year - 2013
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1883
Subject(s) - iterated function , mathematics , tensor (intrinsic definition) , newton's method , mathematical analysis , power iteration , limiting , iterative method , geometry , mathematical optimization , physics , nonlinear system , mechanical engineering , quantum mechanics , engineering
SUMMARY We study the angular orientations of convergent iterates generated by Newton's method in multiple space dimensions. We show that the Newton iteration can be interpreted as a fixed‐point algorithm for solving a tensor eigenproblem. We give an extensive computational analysis of this tensor eigenproblem in two dimensions. In a large fraction of cases, the tensor eigenproblem has a discrete number of solutions to which the Newton directions converge quickly, but there is also a large fraction of cases in which the behavior is more complicated. We contrast the angular orientations of iterates generated by Newton's method with the corresponding directions of the continuous Newton algorithm. Copyright © 2013 John Wiley & Sons, Ltd.