scholarly journals A Geometric Newton Method for Oja's Vector Field

2009 ◽  
Vol 21 (5) ◽  
pp. 1415-1433 ◽  
Author(s):  
P.-A. Absil ◽  
M. Ishteva ◽  
L. De Lathauwer ◽  
S. Van Huffel
Keyword(s):  
Vector Field ◽  
Newton Method ◽  
Newton’S Method ◽  
Matrix Equation ◽  
Orthogonal Group ◽  
Newton's Method ◽  
Newton Algorithm ◽  
The Matrix ◽  
Geometric Techniques ◽  
The Way

Newton's method for solving the matrix equation [Formula: see text] runs up against the fact that its zeros are not isolated. This is due to a symmetry of F by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a “geometric” Newton algorithm that finds the zeros of F. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method.

scholarly journals Newton’s Method for the Matrix Nonsingular Square Root

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Chun-Mei Li ◽  
Shu-Qian Shen
Keyword(s):  
Stability Analysis ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Results ◽  
Square Root ◽  
Convergence Theorems ◽  
The Matrix ◽  
New Algorithms

Two new algorithms are proposed to compute the nonsingular square root of a matrixA. Convergence theorems and stability analysis for these new algorithms are given. Numerical results show that these new algorithms are feasible and effective.

scholarly journals Newton's method for the eigenvalue problem of a symmetric matrix

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 75-82
Author(s):  
V. V. Verbitskyi ◽  
A. G. Huk
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Symmetric Matrix ◽  
Simple Eigenvalue ◽  
Normalization Condition ◽  
Nonlinear System Of Equations ◽  
Computational Costs ◽  
The Matrix ◽  
Eigenvalue And Eigenvector

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

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