A block Newton’s method for computing invariant pairs of nonlinear matrix pencils

2018 ◽  
Vol 33 (1) ◽  
pp. 15-23
Author(s):  
Kirill V. Demyanko ◽  
Yuri M. Nechepurenko
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Linear Matrix ◽  
Inverse Iteration ◽  
Matrix Pencils ◽  
Nonlinear Matrix ◽  
Newton's Methods

Abstract The block inverse iteration and block Newton’s methods proposed by the authors of this paper for computing invariant pairs of regular linear matrix pencils are generalized to the case of regular nonlinear matrix pencils. Numerical properties of the proposed methods are demonstrated with a typical quadratic eigenproblem.

Inverse Iteration, Ill-Conditioned Equations and Newton’s Method

SIAM Review ◽  
1979 ◽  
Vol 21 (3) ◽  
pp. 339-360 ◽  
Author(s):  
G. Peters ◽  
J. H. Wilkinson
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Inverse Iteration

Frequency modification using Newton's method and inverse iteration eigenvector updating

AIAA Journal ◽  
1992 ◽  
Vol 30 (7) ◽  
pp. 1886-1891 ◽  
Author(s):  
Malcolm J. Smith ◽  
Stanley G. Hutton
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Inverse Iteration

scholarly journals Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I

2021 ◽  
Vol 47 (3) ◽  
pp. 1033-1040
Author(s):  
Chacha S Chacha
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Newton’S Method ◽  
Matrix Equation ◽  
Newton's Method ◽  
Fixed Point Method ◽  
Point Method ◽  
Computational Time ◽  
Nonlinear Matrix Equation ◽  
Nonlinear Matrix

The nonlinear matrix equation   was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation  namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms. Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method

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