Newton's Method for the Matrix Square Root

1986 ◽  
Vol 46 (174) ◽  
pp. 537 ◽  
Author(s):  
Nicholas J. Higham
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Square Root ◽  
Matrix Square Root ◽  
The Matrix

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 Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Matthew Rhudy ◽  
Yu Gu ◽  
Jason Gross ◽  
Marcello R. Napolitano
Keyword(s):  
Sensor Fusion ◽  
Unscented Kalman Filter ◽  
Attitude Estimation ◽  
Square Root ◽  
Fusion Algorithm ◽  
Matrix Square Root ◽  
Global Positioning ◽  
The Matrix ◽  
Aerial Vehicle ◽  
Diagonalization Method

Using an Unscented Kalman Filter (UKF) as the nonlinear estimator within a Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithm for attitude estimation, various methods of calculating the matrix square root were discussed and compared. Specifically, the diagonalization method, Schur method, Cholesky method, and five different iterative methods were compared. Additionally, a different method of handling the matrix square root requirement, the square-root UKF (SR-UKF), was evaluated. The different matrix square root calculations were compared based on computational requirements and the sensor fusion attitude estimation performance, which was evaluated using flight data from an Unmanned Aerial Vehicle (UAV). The roll and pitch angle estimates were compared with independently measured values from a high quality mechanical vertical gyroscope. This manuscript represents the first comprehensive analysis of the matrix square root calculations in the context of UKF. From this analysis, it was determined that the best overall matrix square root calculation for UKF applications in terms of performance and execution time is the Cholesky method.

scholarly journals A note on machine method for root extraction

2022 ◽  
Vol 40 ◽  
pp. 1-6
Author(s):  
Saroj Kumar Padhan ◽  
S. Gadtia
Keyword(s):  
Real Number ◽  
Newton’S Method ◽  
Newton's Method ◽  
Extraction Methods ◽  
Critical Study ◽  
Square Root ◽  
Machine Method ◽  
Iteration Formula ◽  
Root Extraction

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

Newton’s method for solving the tensor square root problem

2019 ◽  
Vol 98 ◽  
pp. 57-62 ◽  
Author(s):  
Xue-Feng Duan ◽  
Cun-Yun Wang ◽  
Chun-Mei Li
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Square Root ◽  
Tensor Square

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.

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