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This paper mainly deals with applying a general class of preconditioners to accelerate the convergence speed of some iterative schemes for solving multi-linear systems whose coefficient tensors are strong M -tensors. Theoretical results are established to analyse the performance of a general class of preconditioners extracted from the majorization matrix associated with the coefficient tensor. Brief discussions are included in using a Krylov subspace method based on the Hessenberg process to solve the mentioned problem. For the sake of generality, we consider multi-linear systems with multiple right-hand sides. Numerical experiments are reported to illustrate the validity of the presented theoretical results. [ABSTRACT FROM AUTHOR]

We consider the absolute value equations (AVEs) with a certain tensor product structure. Two aspects of this kind of AVEs are discussed in detail: the solvability and approximate solution. More precisely, first, some sufficient conditions are provided which guarantee the unique solvability of this kind of AVEs. Furthermore, a new iterative method is constructed for solving AVEs and its convergence properties are investigated. The validity of established theoretical results and performance of the proposed iterative scheme are examined numerically. [ABSTRACT FROM AUTHOR]

We revisit the implementation of the Krylov subspace method based on the Hessenberg process for general linear operator equations. It is established that at each step, the computed approximate solution can be regarded by the corresponding approach as the minimizer of a certain norm of residual corresponding to the obtained approximate solution of the system. Test problems are numerically examined for solving tensor equations with a cosine transform product arising from image restoration to compare the performance of the Krylov subspace methods in conjunction with the Tikhonov regularization technique based on Hessenberg and Arnoldi processes. [ABSTRACT FROM AUTHOR]

We consider the solution of linear discrete ill-posed systems of equations with a certain tensor product structure. Two aspects of this kind of problems are investigated: They are transformed to large linear systems of equations and the conditioning of the matrix of the latter system is analyzed. Also, the distance of this matrix to symmetry and skew-symmetry is investigated. The aim of our analysis is to shed light on properties of linear discrete ill-posed problems and to study the feasibility of using Krylov subspace iterative methods in conjunction with Tikhonov regularization to solve Sylvester tensor equations with severely ill-conditioned coefficient matrices. The performance of several proposed algorithms is studied numerically. Applications include color image restoration and the solution of a 3D radiative transfer equation that is discretized by a Chebyshev collocation spectral method. [ABSTRACT FROM AUTHOR]

Abstract We study the performance of a class of block triangular preconditioners for saddle point systems with nonsymmetric positive definite (1 , 1) -block. The presented results incorporate the established results in Zhang and Zhao (2018) where a (parameter-dependent) preconditioner was suggested to solve the mentioned saddle point systems. The performance of the preconditioner in conjunction with Krylov subspace methods is examined for solving a test problem arising from stabilized finite element discretizations of the Oseen problem. The reported numerical experiments show that there exist parameter free preconditioners which outperform the preconditioner proposed in the earlier referred work. [ABSTRACT FROM AUTHOR]

Recently, Salkuyeh and Fahim [A new iterative refinement of the solution of ill-conditioned linear system of equations, Int. Comput. Math. 88(5) (2011), pp. 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergence properties. Afterward, it is shown that the idea of an orthogonal projection technique together with a basic stationary iterative method can be utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. The convergence of the offered hybrid approach is also established. Numerical examples are examined to demonstrate the feasibility of proposed algorithms and their superiority to some of existing approaches for solving ill-conditioned linear system of equations. [ABSTRACT FROM PUBLISHER]

We review the use of block diagonal and block lower/upper triangular splittings for constructing iterative methods and preconditioners for solving stabilized saddle point problems. We introduce new variants of these splittings and obtain new results on the convergence of the associated stationary iterations and new bounds on the eigenvalues of the corresponding preconditioned matrices. We further consider inexact versions as preconditioners for flexible Krylov subspace methods, and show experimentally that our techniques can be highly effective for solving linear systems of saddle point type arising from stabilized finite element discretizations of two model problems, one from incompressible fluid mechanics and the other from magnetostatics. [ABSTRACT FROM AUTHOR]

Recently, Tang et al. [Numer Algorithms. 2014;66(2):379-397] have offered a cyclic iterative method for determining the unique solution of the coupled matrix equations AiXBi = Fi, i = 1, 2,..., N. Analogues to the gradient-based algorithm, the proposed algorithm relies on a fixed parameter whereas it has wider convergence region. Nevertheless, the application of the algorithm to find the centrosymmetric solution of the mentioned problem has been left as a project to be investigated and the optimal value for the fixed parameter has not been derived. In this paper, we focus on a more general class of the coupled linear matrix equations that incorporate the mentioned ones in the earlier refereed work. More precisely, we first develop the authors' propounded algorithm to resolve our considered coupled linear matrix equations over centro-symmetric matrices. Afterwards, we disregard the restriction of the existence of the unique (centro-symmetric) solution and also modify the authors' algorithm by applying an oblique projection technique which allows to produce a sequence of approximate solutions which gratify an optimality property. Numerical results are reported to confirm the validity of the established results and to demonstrate the superior performance of the modified version of the cyclic iterative algorithm. [ABSTRACT FROM AUTHOR]

A class of quaternion matrices called generalized-(anti-)bi-Hermitian matrices is defined which incorporates the-(anti-)bi-Hermitian matrices mentioned by Yuan et al. [Linear Multilinear Algebra. 63;2015:1849–1863] as special cases. In the earlier referred work, Yuan et al. have derived explicit formulas for the least-squares-(anti-)bi-Hermitian solutions of the coupled matrix equations. In this paper, an efficient iterative algorithm is proposed to numerically find the generalized least-squares-(anti-)bi-Hermitian solutions of the coupled matrix equations The validity and efficiency of the presented algorithm is examined by some test experiments. [ABSTRACT FROM AUTHOR]

Recently, Ramadan et al. have focused on the following matrix equation:A1V+A2V¯+B1W+B2W¯=E1VF1+E2V¯F2+Cand propounded two gradient-based iterative algorithms for solving the above matrix equation over reflexive and Hermitian reflexive matrices, respectively. In this paper, we develop two new iterative algorithms based on a two-dimensional projection technique for solving the mentioned matrix equation over reflexive and Hermitian reflexive matrices. The performance of our proposed algorithms is collated with the gradient-based iterative algorithms. It is both theoretically and experimentally demonstrated that the approaches handled surpass the offered algorithms in the earlier referred work in solving the mentioned matrix equation over reflexive and Hermitian reflexive matrices. In addition, it is briefly discussed that a one-dimensional projection technique can accelerate the speed of convergence of the gradient-based iterative algorithm for solving general coupled Sylvester matrix equations over reflexive matrices without assuming the restriction of the existence of a unique solution. [ABSTRACT FROM AUTHOR]

Recently, Shenet al.[Preconditioned iterative methods for solving weighted linear least squares problems, Appl. Math. Mech.-Engl. Ed. 33(3) (2012), pp. 375–384] have considered four kinds of preconditioned generalized accelerated overrelaxation (GAOR) methods for solving the linear systems based on a class of weighted least-squares problems and examined their convergence rates. More recently, Yun [Comparison results on the preconditioned GAOR method for generalized least squares problems, Int. J. Comput. Math. 89 (2012), pp. 2094–2105] has focused on the same problem and suggested three different types of preconditioned GAOR methods and studied their convergence properties. In this paper, we first introduce the generalized mixed-type splitting (GMTS) iterative method for solving the linear systems corresponding to the weighted least-squares problems. The GMTS iterative method exploits auxiliary matricesL1andD1which gratify certain conditions. In order to improve the convergence rate of the GMTS method, different types of preconditioners are applied. In addition, the convergence of the (preconditioned) GMTS iterative methods is discussed. It is both theoretically and experimentally demonstrated that by appropriate choices of auxiliary matricesL1andD1, the (preconditioned) GMTS method outperforms the (preconditioned) GAOR method. [ABSTRACT FROM AUTHOR]

Recently, the gradient-based iterative algorithms have been widely exploited for finding the (least-squares) solutions of the different kinds of (coupled) linear matrix equations. Nevertheless, so far, the convergence of the propounded gradient-based algorithms has been studied for the case where the mentioned (coupled) linear matrix equations have a unique (least-squares) solution. In the present paper, we consider the consistent general coupled linear matrix equations which incorporate many of the recently investigated (coupled) linear matrix equations as their special instances. It is demonstrated that using a gradient-based iterative algorithm for solving the mentioned coupled linear matrix equations is equivalent to extending the well-known Richardson method for solving the normal equations corresponding to the original coupled linear matrix equations. In addition, we prove the semi-convergence of the Richardson method when the coefficient matrix of the associated normal equations is singular. Finally, some numerical experiments are presented to illustrate the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equationsTν(X)=Fν,ν=1,2,…,N,where X=(X1,X2,…,Xp) is a group of unknown matrices and for ν=1,2,…,N,Tν(X)=∑i=1p∑μ=1s1AνiμXiBνiμ+∑μ=1s2CνiμXiTDνiμ+∑μ=1s3MνiμX¯iNνiμ+∑μ=1s4HνiμXiHGνiμ,in which Aνiμ, Bνiμ, Cνiμ, Dνiμ, Mνiμ, Nνiμ, Hνiμ, Gνiμ and Fν are given matrices with suitable dimensions defined over complex number field. By using the hierarchical identification principle, an iterative algorithm is proposed for solving the above coupled linear matrix equations over the group of reflexive (anti-reflexive) matrices. Meanwhile, sufficient conditions are established which guarantee the convergence of the presented algorithm. Finally, some numerical examples are given to demonstrate the validity of our theoretical results and the efficiency of the algorithm for solving the mentioned coupled linear matrix equations. [ABSTRACT FROM PUBLISHER]

Abstract: In the present paper, we propose the global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method for solving large and sparse general coupled matrix equations where , , ,, are given matrices and , , are the unknown matrices. To do so, first, a new inner product and its corresponding matrix norm are defined. Then, using a linear operator equation and new matrix product, we demonstrate how to employ Gl-FOM and Gl-GMRES algorithms for solving general coupled matrix equations. Finally, some numerical experiments are given to illustrate the validity and applicability of the results obtained in this work. [Copyright &y& Elsevier]

This paper is concerned with studying the performance of some global iterative schemes based on Hessenberg process to solve the Sylvester tensor equation (STE). Furthermore, we briefly mention the implementation of flexible versions. Under certain conditions, we devote part of the work to derive upper bounds for condition number of matrices with a special Kronecker product structure which can be helpful to obtain error bounds related to perturbation analysis of the mentioned STE. In addition, we propose using the Tikhonov regularization technique in conjunction with the tensor form of the flexible Hessenberg process when the STE is ill-posed. Numerical examples are examined for two test problems arising from discretization of three-dimensional partial differential equations. We also present some experimental results related to restoration of color images. [ABSTRACT FROM AUTHOR]

We analyze two types of block preconditioners for a class of saddle point problems arising from the modeling of liquid crystal directors using finite elements. Spectral properties of the preconditioned matrices are investigated, and numerical experiments are performed to assess the behavior of preconditioned iterations using both exact and inexact versions of the preconditioners. [ABSTRACT FROM AUTHOR]