On the Newton Method for the Matrix Pth Root

2006 ◽  
Vol 28 (2) ◽  
pp. 503-523 ◽  
Author(s):  
Bruno Iannazzo
Keyword(s):  
Newton Method ◽  
The Matrix ◽  
Matrix Pth Root

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 On High-Order Iterative Schemes for the Matrix pth Root Avoiding the Use of Inverses

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 144
Author(s):  
Sergio Amat ◽  
Sonia Busquier ◽  
Miguel Ángel Hernández-Verón ◽  
Ángel Alberto Magreñán
Keyword(s):  
Iterative Methods ◽  
Computational Cost ◽  
Good Alternative ◽  
High Order ◽  
Matrix Inverse ◽  
Numerical Examples ◽  
Iterative Schemes ◽  
The Matrix ◽  
Matrix Pth Root

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix inverse. The second family computes, using the first method, an approximation of the matrix pth root. We analyze the computational cost and the convergence of this family of methods. Finally, we introduce several numerical examples in order to check the performance of this combination of schemes. We conclude that the method without inverse emerges as a good alternative since a similar numerical behavior with smaller computational cost is obtained.

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