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It is known that the matrix square root has a significant role in linear algebra computations arisen in engineering and sciences. Any matrix with no eigenvalues in R- has a unique square root for which every eigenvalue lies in the open right half-plane. In this research article, a relationship between a scalar root finding method and matrix computations is exploited to derive new iterations to the matrix square root. First, two algorithms that are cubically convergent with conditional stability will be proposed. Then, for solving the stability issue, we will introduce coupled stable schemes that can compute the square root of a matrix with very acceptable accuracy. Furthermore, the convergence and stability properties of the proposed recursions will be analyzed in details. Numerical experiments are included to illustrate the properties of the modified methods. Keywords: Matrix square root, Positive definite matrix, Iterative method, Convergence, Stability

We propose a quadratically convergent algorithm for inverse symmetric eigenvalue problems based on matrix equations. The basic idea is seen in a recent study by Ogita and Aishima, while they derive an efficient iterative refinement algorithm for symmetric eigenvalue problems using special matrix equations. In other words, this study is interpreted as a unified view on quadratically convergent algorithms for eigenvalue problems and inverse eigenvalue problems based on matrix equations. To the best of our knowledge, such a unified development of algorithms is provided for the first time. Since the proposed algorithm for the inverse eigenvalue problems can be regarded as the Newton's method for the matrix equations, the quadratic convergence is naturally proved. Our algorithm is interpreted as an improved version of the Cayley transform method for the inverse eigenvalue problems. Although the Cayley transform method is one of the effective iterative methods, the Cayley transform takes O ( n 3 ) arithmetic operations to produce an orthogonal matrix using a skew-symmetric matrix in each iteration. Our algorithm can refine orthogonality without the Cayley transform, which reduces the operations in each iteration. It is worth noting that our approach overcomes the limitation of the Cayley transform method to the inverse standard eigenvalue problems, resulting in an extension to inverse generalized eigenvalue problems.

A fast iterative algorithm for numerically computing an interval matrix containing the principal p th root of an n × n matrix A is proposed. This algorithm is based on a numerical spectral decomposition of A , and is applicable when a computed eigenvector matrix of A is not ill-conditioned. Particular emphasis is put on the computational efficiency of the algorithm which has only O ( n 3 + p n ) operations per iteration. The algorithm moreover verifies the uniqueness of the contained p th root. Numerical results show the efficiency of the algorithm.

This paper considers the problem of measurement matrix optimization for compressed sensing (CS) in which the dictionary is assumed to be given, such that it leads to an effective sensing matrix. Due to important properties of equiangular tight frames (ETFs) to achieve Welch bound equality, the measurement matrix optimization based on ETF has received considerable attention and many algorithms have been proposed for this aim. These methods produce sensing matrix with low mutual coherence based on initializing the measurement matrix with random Gaussian ensembles. This paper, use incoherent unit norm tight frame (UNTF) as an important frame with the aim of low mutual coherence and proposes a new method to construction a measurement matrix of any dimension while measurement matrix initialized by partial Fourier matrix. Simulation results show that the obtained measurement matrix effectively reduces the mutual coherence of sensing matrix and has a fast convergence to Welch bound compared with other methods.

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovic Kovac (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ A AΔ A and Δ B BΔ B are small, for some nonsingular diagonal matrices Δ A and Δ B .

We consider the matrix completion problem that aims to construct a low rank matrix X that approximates a given large matrix Y from partially known sample data in Y . In this paper we introduce an efficient greedy algorithm for such matrix completions. The greedy algorithm generalizes the orthogonal rank-one matrix pursuit method (OR1MP) by creating s ⩾ 1 candidates per iteration by low-rank matrix approximation. Due to selecting s ⩾ 1 candidates in each iteration step, our approach uses fewer iterations than OR1MP to achieve the same results. Our algorithm is a randomized low-rank approximation method which makes it computationally inexpensive. The algorithm comes in two forms, the standard one which uses the Lanzcos algorithm to find partial SVDs, and another that uses a randomized approach for this part of its work. The storage complexity of this algorithm can be reduced by using an weight updating rule as an economic version algorithm. We prove that all our algorithms are linearly convergent. Numerical experiments on image reconstruction and recommendation problems are included that illustrate the accuracy and efficiency of our algorithms.

For given symmetric orthogonal matrices R, S, i.e. RT = R, R2 = I, ST = S, S2 = I, a matrix A∈Cn×s is termed (R, S)-conjugate matrix if RAS=A‾. In this paper, an iterative method is constructed to find the (R, S)-conjugate solutions of the generalised coupled Sylvester matrix equations. The consistency of the considered matrix equations over (R, S)-conjugate matrices is discussed. When the matrix equations have a unique (R, S)-conjugate solution pair, the proposed method is convergent for any initial (R, S)-conjugate matrix pair under a loose restriction on the convergent factor. Moreover, the optimal convergent factor of the presented method is derived. Finally, some numerical examples are given to illustrate the results and effectiveness.

We construct an alternating splitting iteration scheme for solving and preconditioning a block two-by-two linear system arising from numerical discretizations of the bidomain equations. The convergence theory of this class of splitting iteration methods is established and some useful properties of the preconditioned matrix are analyzed. The potential of this approach is illustrated by numerical experiments.

In this paper, we apply Kronecker product and vectorization operator to extend the conjugate residual squared (CRS) method for solving a class of coupled Sylvester-transpose matrix equations. Some numerical examples are given to compare the accuracy and efficiency of the new matrix iterative method with other methods presented in the literature. Numerical results validate that the proposed method can be much more efficient than some existing methods.

This paper studies linear conjugacy of PL-RDK systems, which are kinetic systems with power law rate functions whose kinetic orders are identical for branching reactions, i.e. reactions with the same reactant complex. Mass action kinetics (MAK) systems are the best known examples of such systems with reactant-determined kinetic orders (RDK). We specify their kinetics with their rate vector and T matrix. The T matrix is formed from the kinetic order matrix by replacing the reactions with their reactant complexes as row indices (thus compressing identical rows of branching reactions of a reactant complex to one) and taking the transpose of the resulting matrix. The T matrix is hence the kinetic analogue of the network’s matrix of complexes Y with the latter’s columns of non-reactant complexes truncated away. For MAK systems, the T matrix and the truncated Y matrix are identical. We show that, on non-branching networks, a necessary condition for linear conjugacy of MAK systems and, more generally, of PL-FSK (power law factor span surjective kinetics) systems, i.e. those whose T matrix columns are pairwise different, is $$T = T'$$ , i.e. equality of their T matrices. This motivated our inclusion of the condition $$T = T'$$ in exploring extension of results from MAK to PL-RDK systems. We extend the Johnston–Siegel Criterion for linear conjugacy from MAK to PL-RDK systems satisfying the additional assumption of $$T = T'$$ and adapt the MILP algorithms of Johnston et al. and Szederkenyi to search for linear conjugates of such systems. We conclude by illustrating the results with several examples and an outlook on further research.

In this paper, we reformulate a class of nonlinear complementarity problems as the implicit fixed-point equations. We demonstrate accelerated modulus-based matrix splitting iteration method. We show their convergence by assuming that the system matrix is positive definite or the splitting of the system matrix are $$H_+$$ -compatible splitting and discuss the choice of the optimal parameter. Furthermore, we give two-step accelerated modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. Numerical experiments are presented to show the effectiveness of the method.

We consider a Strang-type splitting method for an abstract semilinear evolution equation $${\partial _t}u = Au + F\left( u \right).$$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators A and F. Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting method which is split into three parts and prove that our proposed method is convergent at a second order rate.

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a semigroup property depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.

[EN] A general family of iterative methods including a free parameter is derived and proved to be convergent for computing matrix sign function under some restrictions on the parameter. Several special cases including global convergence behavior are dealt with. It is analytically shown that they are asymptotically stable. A variety of numerical experiments for matrices with different sizes is considered to show the effectiveness of the proposed members of the family. (C) 2016 Elsevier B.V. All rights reserved., This research was supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROME-TEO/2016/089.

In this paper, a new type nonlinear conjugate gradient method based on the ScaleMatrix is derived. The new method has the decent and globally convergentproperties under some assumptions. Numerical results indicate the efficiency ofthis method to solve the given test problems.

In this article we introduce the notion of P-proper splitting for square matrices. For an inconsistent linear system of equations $$Ax =b$$ , we associate an iterative method based on a P-proper splitting of A, which if convergent, converges to the best least squares solution of this system. We extend a result of Stein, using which we prove that if A is positive semidefinite, then the said iterative method converges. Also, we generalize Sylvester’s law of inertia and as an application of this generalization we establish some properties of P-proper splittings. Finally, we prove a comparison theorem for iterative methods associated with P-proper splittings of a positive semidefinite matrix.

In this study, we consider the iteration solutions of the generalized Sylvester-conjugate matrix equation: A X B + C X ¯ D = E by a modified conjugate gradient method. When the system is consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial value given Hamiltonian matrix. Furthermore, we can get the minimum-norm solution X ∗ by choosing a special kind of initial matrix. Finally, some numerical examples are given to demonstrate the algorithm considered is quite effective in actual computation.

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given by d A d t = [ A u − A l , A ] , A ( 0 ) = A 0 , where A is a real n × n matrix (not necessarily symmetric), [ A , B ] = A B − B A is the matrix commutator (also known as the Lie bracket), A u is the strictly upper triangular part of A and A l is the strictly lower triangular part of A. We prove that if the initial matrix A 0 is staircase, so is A ( t ) . Moreover, we prove that this flow preserves the certain positivity properties of A 0 . Also we prove that if the initial matrix A 0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A ( t ) converges to a diagonal matrix while preserving the spectrum of A 0 . Some simulations are provided to confirm the convergence properties.

In this paper, not requiring that the Hermitian part of the complex symmetric linear system must be Hermitian positive definite, a class of splitting methods is established by the modified positive/negative-stable splitting (PNS) of the coefficient matrix and is called the MPNS method. Theoretical analysis shows that the MPNS method is absolutely convergent under proper conditions. Some useful properties of the corresponding MPNS-preconditioned matrix are obtained. Numerical experiments are reported to illustrate the efficiency of both the MPNS method and the MPNS preconditioner.

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 equationsAnalogues 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 centro-symmetric 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 w...

For a given symmetric orthogonal matrix R, i.e., RT = R, R2 = I, a matrix A ? Cnxn is termed Hermitian R-conjugate matrix if A = AH, RAR = ?. In this paper, an iterative method is constructed for finding the Hermitian R-conjugate solutions of general coupled Sylvester matrix equations. Convergence analysis shows that when the considered matrix equations have a unique solution group then the proposed method is always convergent for any initial Hermitian R-conjugate matrix group under a loose restriction on the convergent factor. Furthermore, the optimal convergent factor is derived. Finally, two numerical examples are given to demonstrate the theoretical results and effectiveness.

To solve a class of nonlinear complementarity problems, accelerated modulus-based matrix splitting iteration methods are presented and analyzed. Convergence analysis and the choice of the parameters are given when the system matrix is either positive definite or an H+-matrix. Numerical experiments further demonstrate that the proposed methods are efficient and have better performance than the existing modulus-based iteration method in aspects of the number of iteration steps and CPU time.

This paper considers the problem of matrix completion, which is to recover a d 1 × d 2 matrix from observations in a small proportion of indices. We study the nuclear norm minimization method with the restriction of matching the observed entries. Under certain coherence conditions, we prove that the required sample size is of order r 2 d log d via a graphical approach, where d = d 1 + d 2 and r is the rank of the target matrix.

By reformulating a class of weakly nonlinear complementarity problems as implicit fixed-point equations based on splitting of the system matrix, a modified modulus-based matrix splitting algorithm is presented. The convergence analysis of proposed algorithm is established for the case that the splitting of the system matrix is an H-splitting. Numerical experiments on two model problems are given to illustrate the theoretical results and examine the numerical effectiveness.

In this paper, we will extend the two-sweep iteration methods to solve the linear complementarity problems and establish a class of two-sweep modulus-based matrix splitting iteration methods for the implicit fixed-point equation of the linear complementarity problems. Some convergence properties of two-sweep modulus-based matrix splitting iteration methods are discussed when the system matrices are positive-definite matrices and H + -matrices. Numerical experiments are presented to illustrate the efficiency of the proposed methods.

It is shown that the problem of observer design for estimating a set of linear combinations of state variables of a plant can be formulated in terms of linear matrix inequalities. An algorithm for constructing functional observers is proposed based on a nonsingular transformation of a plant model in the state space by matrix canonization with subsequent solution of the system of linear matrix inequalities.

We consider the problem of approximately computing saddle-point of a zero-sum matrix game when either the columns of the matrix are revealed incrementally in time or the matrix is too large to apply traditional methods. We leverage the established adaptive multiplicative weights algorithm but introduce a novel simple criterion to determine whether the approximately computed minimizer's best strategy needs to be re-computed when a new column of the matrix is introduced. Our main results are two-fold. First, we show that our proposed incremental approach achieves the same accuracy as applying the adaptive multiplicative weights algorithm on the entire matrix, if known a priori. Second, when the columns of the matrix are generated independently and from the same distribution, we show that the expected number of times the approximate strategy is re-computed grows at most logarithmically with the number of columns of the matrix, thereby being computationally efficient.

In this paper, a relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus-based matrix splitting iteration methods. The convergence analysis and the strategy of the choice of the parameters are given. Numerical examples show that the proposed methods are efficient and accelerate the convergence performance with less iteration steps and CPU times.

Based on the accelerated Hermitian and skew-Hermitian splitting iteration scheme (Bai and Golub, 2007), we propose a new two-parameter matrix splitting preconditioner in this paper. Spectral properties of the preconditioned matrix are analyzed in detail. Furthermore, based on this preconditioner, an improved version of matrix splitting preconditioner is presented and analyzed. Finally, performance of the preconditioners is compared by using GMRES( m ) as an iterative solver on linear systems arising from the discretization of Stokes and Navier-Stokes equations.

In this paper, we reformulate the nonlinear complementarity problem as an implicit fixed-point equation. We establish a modulus-based matrix splitting iteration method based on the implicit fixed-point equation and prove its convergence theorem under suitable conditions. Furthermore, we propose a two-step modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. We can obtain many matrix splitting iteration methods by suitably choosing the matrix splittings and the parameters. The proposed methods can be regarded as extensions of the methods for linear complementarity problem. Numerical experiments are presented to show the effectiveness of the proposed methods.

For a given square matrix A, the numerical range for the exponential function e^(At), t in C, is considered. Some geometrical and topological properties of the numerical range are presented.

In this article, generalizations of certain $M$-matrix properties are proved for the group generalized inverse. The proofs use the notion of proper splittings of one type or the other. In deriving certain results, we make use of a recently introduced notion of a $B_{\#}$-splitting. Applications in obtaining comparison results for the spectral radii of matrices are presented.

n this paper, an iterative method to solve one kind of nonlinear matrix equation is discussed. For each initial matrix with some conditions, the matrix sequences generated by the iterative method are shown to lie in a fixed open ball. The matrix sequences generated by the iterative method are shown to converge to the only solution of the nonlinear matrix equation in the fixed closed ball. In addition, the error estimate of the approximate solution in the fixed closed ball, and a numerical example to illustrate the convergence results are given.

This study models a 3-dimensional planetary gear system using the transfer matrix method. The local transfer matrices between each component of the planetary gear set were derived with consideration of the tooth width, and the transfer matrix of a planetary gear system corresponding to the inertial transfer matrix was determined. The eigenvalue analysis of the transfer matrix suggested an analysis method in the form of a lambda matrix, instead of the direct search method through a characteristic polynomial. The boundary conditions at the first and the last stations of the entire transfer matrix were partitioned into known and unknown values to generate a concentrated transfer matrix and a latent equation, and the eigenvalue problem in the lambda matrix was solved. The characteristics of the responses according to the phase state of the harmonic component of the transmission error were reviewed through the steady-state response and mode shape type.

The objective of this paper is to present an example in which matrix functions are used to solve a modern control exercise. Specifically, the solution for the equation of state, which is a matrix differential equation is calculated. To resolve this, two different methods are presented, first using the properties of the matrix functions and by other side, using the classical method of Laplace transform.

In this paper, we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized Sylvester-transpose matrix equation A X B + C X T D = F . The idea is from (Ding and Chen, 2005; Niu et?al., 2011; Wang et?al., 2012) in which some efficient algorithms were developed for solving the Sylvester matrix equation and the Lyapunov matrix equation. On the basis of the information generated in the previous half-step, we further introduce a relaxation factor to obtain the solution of the generalized Sylvester-transpose matrix equation. We show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions. Finally, some numerical examples are given to illustrate that the introduced iterative algorithm is efficient.

In this paper, an iterative method is constructed to find the least-squares solutions of generalized Sylvester equation , where is real matrices group, and satisfies different linear constraint. By this iterative method, for any initial matrix group within a special constrained matrix set, a least squares solution group with satisfying different linear constraint can be obtained within finite iteration steps in the absence of round off errors, and the unique least norm least-squares solution can be obtained by choosing a special kind of initial matrix group. In addition, a minimization property of this iterative method is characterized. Finally, numerical experiments are reported to show the efficiency of the proposed method.

A practical distributed Gauss–Newton method for the near-optimal computation of interference alignment solutions is proposed, based on the block-wise structure of Jacobian matrix of the residual interference function. Also, a simple practical matrix inverse computation algorithm to solve the rank-deficient problem due to the interference sparse matrix is adopted. This proposal achieves convergent performance comparable to centralised Gauss–Newton method, with the major advantage of being practically implemented in a distributed manner.

Various convergent matrix series formulae for the inverse of positive definite matrices are derived starting from series expansions of a simple matrix exponential function. The structure of these series assures a stable numerical computation since successive terms of the series are obtained through a multiplication with a matrix whose condition becomes better and better as the computation progresses.

This paper proposes and investigates a finite-time convergent neural dynamics (FTCND) for online solution of time-varying linear complex matrix equation in complex domain. Different from the conventional gradient-based neural dynamical method, the proposed method utilizes adequate time-derivative information of time-varying complex matrix coefficients. It is theoretically proved that our FTCND model can converge to the theoretical solution of time-varying linear complex matrix equation within finite time. In addition, the upper bound of the convergence time is derived analytically via Lyapunov theory. For comparative purposes, the conventional gradient-based neural dynamics (GND) is developed and exploited for solving such a time-varying complex problem. Computer-simulation results verify the effectiveness and superiorness of the FTCND model for solving time-varying linear complex matrix equation in complex domain, as compared with the GND model.