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Article
Publication date: 3 July 2017

Masoud Hajarian

The purpose of this paper is to obtain an iterative algorithm to find the solution of the coupled Sylvester-like matrix equations.

Abstract

Purpose

The purpose of this paper is to obtain an iterative algorithm to find the solution of the coupled Sylvester-like matrix equations.

Design/methodology/approach

In this work, the matrix form of the conjugate direction (CD) algorithm to find the solution X of the coupled Sylvester-like matrix equations:

{A1XB1+M1f1(X)N1=F1,A2XB2+M2f2(X)N2=F2,
with fi(X) = X, fi(X) = X¯, fi(X) = XT and fi(X) = XH for i = 1; 2 has been established.

Findings

It is proven that the algorithm converges to the solution within a finite number of iterations in the absence of round-off errors. Finally, four numerical examples were used to test the proficiency and convergence of the established algorithm.

Originality/value

The numerical examples have led the author to believe that the generalized CD (GCD) algorithm is efficient and it converges more rapidly in comparison with the CGNR and CGNE algorithms.

Details

Engineering Computations, vol. 34 no. 5
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 27 May 2014

Masoud Hajarian

The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for…

Abstract

Purpose

The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for solving the systems of linear matrix equations.

Design/methodology/approach

According to the hierarchical identification principle, the authors construct alternating direction gradient-based iterative (ADGI) methods to solve systems of linear matrix equations.

Findings

The authors propose efficient ADGI methods to solve the systems of linear matrix equations. It is proven that the ADGI methods consistently converge to the solution for any initial matrix. Moreover, the constructed methods are extended for finding the reflexive solution to the systems of linear matrix equations.

Originality/value

This paper proposes efficient iterative methods without computing any matrix inverses, vec operator and Kronecker product for finding the solution of the systems of linear matrix equations.

Details

Engineering Computations, vol. 31 no. 4
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 15 November 2011

Mehdi Dehghan and Masoud Hajarian

The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix…

Abstract

Purpose

The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X.

Design/methodology/approach

By applying the principle of hierarchical identification and the Hermitian/skew‐Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift‐splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X+XA2+A3XH+XHA4=B. Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation.

Findings

The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms.

Originality/value

So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew‐Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation.

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Article
Publication date: 13 July 2012

Mehdi Dehghan and Masoud Hajarian

The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Abstract

Purpose

The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Design/methodology/approach

By extending the idea of conjugate gradient (CG) method, the authors present two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Findings

When the general coupled matrix equations are consistent over the generalized centro‐symmetric and central anti‐symmetric matrices, the generalized centro‐symmetric and central anti‐symmetric solutions can be obtained within nite iterative steps. Also the least Frobenius norm generalized centrosymmetric and central anti‐symmetric solutions can be derived by choosing a special kind of initial matrices. Furthermore, the optimal approximation generalized centrosymmetric and central anti‐symmetric solutions to given generalized centro‐symmetric and central anti‐symmetric matrices can be obtained by finding the least Frobenius norm generalized centro‐symmetric and central anti‐symmetric solutions of new matrix equations. The authors employ some numerical examples to support the theoretical results of this paper. Finally, the application of the presented methods is highlighted for solving the projected generalized continuous‐time algebraic Lyapunov equations (GCALE).

Originality/value

By the algorithms, the solvability of the general coupled matrix equations over generalized centro‐symmetric and central anti‐symmetric matrices can be determined automatically. The convergence results of the iterative algorithms are also proposed. Several examples and an application are given to show the efficiency of the presented methods.

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Article
Publication date: 25 May 2012

Mehdi Dehghan and Masoud Hajarian

Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve…

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Abstract

Purpose

Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non‐linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose two new modified Newton's method for solving non‐linear equations. Convergence results show that the order of convergence of the proposed iterative methods for a simple root is four. The iterative methods are free from second derivative and can be used for solving non‐linear equations without computing the second derivative. Finally, several numerical examples are given to illustrate that proposed iterative algorithms are effective.

Design/methodology/approach

In this paper, first the authors introduce two new approximations for the definite integral arising from Newton's theorem. Then by considering these approximations, two new iterative methods are provided with fourth‐order convergence which can be used for solving non‐linear equations without computing second derivatives.

Findings

In this paper, the authors propose two new iterative methods without second derivatives for solving the non‐linear equation f(x)=0. From numerical results, it is observed that the new methods are comparable with various iterative methods. Also numerical results corroborate the theoretical analysis.

Originality/value

The best property of these schemes is that they are second derivative free. Also from numerical results, it is observed that the new methods are comparable with various iterative methods. The numerical results corroborate the theoretical analysis.

Details

Engineering Computations, vol. 29 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

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