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.

On a new family of high-order iterative methods for the matrix p th root

2015 ◽  
Vol 22 (4) ◽  
pp. 585-595 ◽  
Author(s):  
S. Amat ◽  
J. A. Ezquerro ◽  
M. A. Hernández-Verón
Keyword(s):  
Iterative Methods ◽  
High Order ◽  
New Family ◽  
The Matrix ◽  
High Order Iterative Methods

Certain efficient iterative methods for bipolar fuzzy system of linear equations

2020 ◽  
Vol 39 (3) ◽  
pp. 3971-3985 ◽  
Author(s):  
Muhammad Saqib ◽  
Muhammad Akram ◽  
Shahida Bashir
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Iterative Methods ◽  
Fuzzy Set ◽  
Fuzzy System ◽  
Linear Equations ◽  
Performance Validity ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Iterative Schemes

A bipolar fuzzy set model is an extension of fuzzy set model. We develop new iterative methods: generalized Jacobi, generalized Gauss-Seidel, refined Jacobi, refined Gauss-seidel, refined generalized Jacobi and refined generalized Gauss-seidel methods, for solving bipolar fuzzy system of linear equations(BFSLEs). We decompose n ×  n BFSLEs into 4n ×  4n symmetric crisp linear system. We present some results that give the convergence of proposed iterative methods. We solve some BFSLEs to check the validity, efficiency and stability of our proposed iterative schemes. Further, we compute Hausdorff distance between the exact solutions and approximate solution of our proposed schemes. The numerical examples show that some proposed methods converge for the BFSLEs, but Jacobi and Gauss-seidel iterative methods diverge for BFSLEs. Finally, comparison tables show the performance, validity and efficiency of our proposed iterative methods for BFSLEs.

scholarly journals High-Order Iterative Methods for the DMP Inverse

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Xiaoji Liu ◽  
Naping Cai
Keyword(s):  
Error Estimate ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
High Order ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Sufficient Conditions For Convergence ◽  
Necessary And Sufficient ◽  
High Order Iterative Methods

We investigate two iterative methods for computing the DMP inverse. The necessary and sufficient conditions for convergence of our schemes are considered and the error estimate is also derived. Numerical examples are given to test the accuracy and effectiveness of our methods.

scholarly journals ZNN models for computing matrix inverse based on hyperpower iterative methods

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2999-3014 ◽  
Author(s):  
Igor Stojanovic ◽  
Predrag Stanimirovic ◽  
Ivan Zivkovic ◽  
Dimitrios Gerontitis ◽  
Xue-Zhong Wang
Keyword(s):  
Neural Network ◽  
Iterative Methods ◽  
Generalized Inverses ◽  
Convergence Properties ◽  
Matrix Inverse ◽  
The Family ◽  
The Matrix ◽  
Zhang Neural Network ◽  
Implementation In Matlab

Our goal is to investigate and exploit an analogy between the scaled hyperpower family (SHPI family) of iterative methods for computing the matrix inverse and the discretization of Zhang Neural Network (ZNN) models. A class of ZNN models corresponding to the family of hyperpower iterative methods for computing generalized inverses is defined on the basis of the discovered analogy. The Simulink implementation in Matlab of the introduced ZNN models is described in the case of scaled hyperpower methods of the order 2 and 3. Convergence properties of the proposed ZNN models are investigated as well as their numerical behavior.

scholarly journals Reducing Chaos and Bifurcations in Newton-Type Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
S. Amat ◽  
S. Busquier ◽  
Á. A. Magreñán
Keyword(s):  
Iterative Methods ◽  
Newton’S Method ◽  
Newton's Method ◽  
High Order ◽  
Damping Factor ◽  
Type Method ◽  
Iterative Schemes

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce thebadzones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.

scholarly journals On highly efficient derivative-free family of numerical methods for solving polynomial equation simultaneously

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Computational Cost ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Order Convergence ◽  
Numerical Examples ◽  
Iterative Schemes ◽  
Highly Efficient ◽  
Derivative Free ◽  
Solving Polynomial Equation

AbstractA highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.

XIII.—Studies in Practical Mathematics. VI. On the Factorization of Polynomials by Iterative Methods

1951 ◽  
Vol 63 (2) ◽  
pp. 174-191 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Theoretical Analysis ◽  
Iterative Methods ◽  
Iterative Process ◽  
Numerical Examples ◽  
The Matrix ◽  
Practical Mathematics ◽  
Factorization Of Polynomials ◽  
Latent Roots

SynopsisThe method of iteration of penultimate remainders, introduced by S. N. Lin for approximating by stages to the exact factors of a polynomial, is subjected to theoretical analysis. The matrix governing the iterative process is obtained, and its latent roots and latent vectors are found. Incidental theorems yielding further factorizations are proved, and processes are developed for accelerating convergence. Numerical examples illustrate varying situations likely to arise in practice.

scholarly journals The alternate direction iterative methods for generalized saddle point systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shuanghua Luo ◽  
Angang Cui ◽  
Cheng-yi Zhang
Keyword(s):  
Saddle Point ◽  
Iterative Methods ◽  
Iterative Schemes ◽  
Numerical Example ◽  
Convergence Results ◽  
Alternate Direction ◽  
Saddle Point Matrix

Abstract The paper studies two splitting forms of generalized saddle point matrix to derive two alternate direction iterative schemes for generalized saddle point systems. Some convergence results are established for these two alternate direction iterative methods. Meanwhile, a numerical example is given to show that the proposed alternate direction iterative methods are much more effective and efficient than the existing one.

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