scholarly journals Generalized Inverses Estimations by Means of Iterative Methods with Memory

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 2
Author(s):  
Santiago Artidiello ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Methods ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Nonsingular Matrix ◽  
Type Method ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
First Time ◽  
With Memory ◽  
Theoretical Results

A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.

SOME NEWTON-TYPE ITERATIVE METHODS WITH AND WITHOUT MEMORY FOR SOLVING NONLINEAR EQUATIONS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350078 ◽  
Author(s):  
XIAOFENG WANG ◽  
TIE ZHANG
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Type Method ◽  
New Methods ◽  
Hermite Interpolating Polynomial ◽  
With Memory ◽  
Theoretical Results ◽  
Very High ◽  
High Computational Efficiency

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, [Formula: see text] and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

scholarly journals Sensitivity Analysis of the Matrix Equation from Interpolation Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Li ◽  
Yuhai Zhang
Keyword(s):  
Sensitivity Analysis ◽  
Approximate Solution ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
Backward Error ◽  
Perturbation Bound ◽  
Interpolation Problems ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Theoretical Results

This paper studies the sensitivity analysis of a nonlinear matrix equation connected to interpolation problems. The backward error estimates of an approximate solution to the equation are derived. A residual bound of an approximate solution to the equation is obtained. A perturbation bound for the unique solution to the equation is evaluated. This perturbation bound is independent of the exact solution of this equation. The theoretical results are illustrated by numerical examples.

scholarly journals Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-27 ◽  
Author(s):  
Predrag S. Stanimirović ◽  
Miroslav Ćirić ◽  
Igor Stojanović ◽  
Dimitrios Gerontitis
Keyword(s):  
Generalized Inverses ◽  
Drazin Inverse ◽  
Time Varying ◽  
Matrix Equations ◽  
Complex Matrices ◽  
Matrix Inverse ◽  
Computational Procedures ◽  
Time Invariant ◽  
Theoretical Results ◽  
Real Matrices

Conditions for the existence and representations of 2-, 1-, and 1,2-inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

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 A Modified Ren’s Method with Memory Using a Simple Self-Accelerating Parameter

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 540 ◽  
Author(s):  
Xiaofeng Wang ◽  
Qiannan Fan
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
New Method ◽  
Convergence Order ◽  
Type Method ◽  
Computational Costs ◽  
With Memory ◽  
Theoretical Results

In this paper, a self-accelerating type method is proposed for solving nonlinear equations, which is a modified Ren’s method. A simple way is applied to construct a variable self-accelerating parameter of the new method, which does not increase any computational costs. The highest convergence order of new method is 2 + 6 ≈ 4.4495 . Numerical experiments are made to show the performance of the new method, which supports the theoretical results.

scholarly journals Solutions and Improved Perturbation Analysis for the Matrix EquationX-A*X-pA=Q  (p>0)

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jing Li
Keyword(s):  
Matrix Equation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Backward Error ◽  
Perturbation Bound ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Theoretical Results ◽  
Definite Solution

The nonlinear matrix equationX-A*X-pA=Qwithp>0is investigated. We consider two cases of this equation: the casep≥1and the case0<p<1.In the casep≥1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case0<p<1, a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

scholarly journals Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 943
Author(s):  
Xiaofeng Wang ◽  
Yingfanghua Jin ◽  
Yali Zhao
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Nonlinear Ordinary Differential Equations ◽  
New Methods ◽  
Derivative Free ◽  
Initial Scheme ◽  
With Memory ◽  
Theoretical Results

Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.

Derivative free iterative methods with memory having higher R-order of convergence

2020 ◽  
Vol 21 (6) ◽  
pp. 641-648
Author(s):  
Pankaj Jain ◽  
Prem Bahadur Chand
Keyword(s):  
Iterative Methods ◽  
Order Of Convergence ◽  
Function Evaluation ◽  
Optimal Methods ◽  
New Methods ◽  
Derivative Free ◽  
Iteration Index ◽  
Order Four ◽  
With Memory ◽  
Theoretical Results

AbstractWe derive two iterative methods with memory for approximating a simple root of any nonlinear equation. For this purpose, we take two optimal methods without memory of order four and eight and convert them into the methods with memory without increasing any further function evaluation. These methods involve a self-accelerator (parameter) that depends upon the iteration index to increase the order of the optimal methods. Consequently, the efficiency of the new methods is considerably high as compared to the methods without memory. Some numerical examples are provided in support of the theoretical results.

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