An iterative method for solving absolute value equations and sufficient conditions for unique solvability

2012 ◽  
Vol 8 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Jiri Rohn ◽  
Vahideh Hooshyarbakhsh ◽  
Raena Farhadsefat
Keyword(s):  
Iterative Method ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
Absolute Value Equations ◽  
Absolute Value

scholarly journals Iterative schemes induced by block splittings for solving absolute value equations

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4171-4188
Author(s):  
Nafiseh Shams ◽  
Alireza Fakharzadeh Jahromi ◽  
Fatemeh Beik
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Convergence Properties ◽  
Absolute Value Equations ◽  
Absolute Value ◽  
Iterative Schemes ◽  
Generalized Newton ◽  
Special Case

In this paper, we develop the idea of constructing iterative methods based on block splittings (BBS) to solve absolute value equations. The class of BBS methods incorporates the well-known Picard iterative method as a special case. Convergence properties of mentioned schemes are proved under some sufficient conditions. Numerical experiments are examined to compare the performance of the iterative schemes of BBS-type with some of existing approaches in the literature such as generalized Newton and Picard(-HSS) iterative methods.

A Preconditioned AOR Iterative Method for the Absolute Value Equations

2017 ◽  
Vol 14 (02) ◽  
pp. 1750016 ◽  
Author(s):  
Cui-Xia Li
Keyword(s):  
Iterative Method ◽  
Comparison Theorems ◽  
Absolute Value Equations ◽  
Absolute Value ◽  
Comparison Results ◽  
Preconditioning Technique ◽  
The Matrix ◽  
The Absolute ◽  
Theoretical Results ◽  
Better Than

In this paper, coupled with preconditioning technique, a preconditioned accelerated over relaxation (PAOR) iterative method for solving the absolute value equations (AVEs) is presented. Some comparison theorems are given when the matrix of the linear term is an irreducible [Formula: see text]-matrix. Comparison results show that the convergence rate of the PAOR iterative method is better than that of the accelerated over relaxation (AOR) iterative method whenever both are convergent. Numerical experiments are provided in order to confirm the theoretical results studied in this paper.

scholarly journals An optimized AOR iterative method for solving absolute value equations

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 459-476
Author(s):  
Alireza Fakharzadeh Jahromi ◽  
Nafiseh Shamsa
Keyword(s):  
Convergence Rate ◽  
Iterative Method ◽  
Optimization Techniques ◽  
Optimal Parameters ◽  
Absolute Value Equations ◽  
Absolute Value ◽  
Sor Methods ◽  
Two Parameters

In recent years, the AOR iterative method has been proposed for solving absolute value equations. This method has two parameters ? and ?. In this paper, we intend to find the optimal parameters of this method to improve convergence rate by suitable optimization techniques. Meanwhile, the convergence of the optimized AOR iterative method is discussed. It is both theoretically and experimentally demonstrated efficiency of the optimized AOR iterative method in contrast with the AOR and SOR methods.

scholarly journals Residual Iterative Method for Solving Absolute Value Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Javed Iqbal ◽  
Eisa Al-Said
Keyword(s):  
Iterative Method ◽  
Method Comparison ◽  
Absolute Value Equations ◽  
Projection Technique ◽  
Absolute Value

We suggest and analyze a residual iterative method for solving absolute value equationsAx-x=bwhereA∈Rn×n,b∈Rnare given andx∈Rnis unknown, using the projection technique. We also discuss the convergence of the proposed method. Several examples are given to illustrate the implementation and efficiency of the method. Comparison with other methods is also given. Results proved in this paper may stimulate further research in this fascinating field.

On an iterative method for solving absolute value equations

2011 ◽  
Vol 6 (5) ◽  
pp. 1027-1033 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Javed Iqbal ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said
Keyword(s):  
Iterative Method ◽  
Absolute Value Equations ◽  
Absolute Value

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

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