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A locally and cubically convergent algorithm for computing ðµ âeigenpairs of symmetric tensors
Author(s) -
Zhao Ruijuan,
Zheng Bing,
Liang Maolin,
Xu Yangyang
Publication year - 2020
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.2284
Subject(s) - mathematics , unitary state , convergence (economics) , tensor (intrinsic definition) , computation , quantum entanglement , quadratic equation , symmetric tensor , newton's method , algorithm , nonlinear system , mathematical analysis , quantum , geometry , exact solutions in general relativity , physics , quantum mechanics , political science , law , economics , economic growth
Summary This paper is concerned with computing âeigenpairs of symmetric tensors. We first show that computing âeigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complexâvalued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.