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For solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems. [ABSTRACT FROM AUTHOR]

We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by one. This condition means that some auxiliary system of linear equations has no binary solution. Some examples are provided that show that the lower bound can be achieved. This lower bound on the matrix rank allows the problem of finding a binary solution to a system of linear equations with a sufficiently large number of linearly independent equations to be reduced to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each differing from the binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a large system of linear algebraic equations. Estimates of the time required for calculating the matrix rank in the SymPy computer algebra system are also provided. It is shown that the rank of a matrix over the field of residues modulo prime number is calculated faster than it generally takes to calculate the rank of a matrix of the same order over the field of rational numbers. [ABSTRACT FROM AUTHOR]

We investigate the randomized Kaczmarz method that adaptively updates the stepsize using readily available information for solving inconsistent linear systems. A novel geometric interpretation is provided which shows that the proposed method can be viewed as an orthogonal projection method in some sense. We prove that this method converges linearly in expectation to the unique minimum Euclidean norm least-squares solution of the linear system, and provide a tight upper bound for the convergence of the proposed method. Numerical experiments are also given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of the present research article is to solve the system of linear equations using common fixed point theorems in the context of bicomplex valued metric spaces. To obtain our main objective, we introduce generalized contractive conditions in bicomplex valued metric spaces and establish common fixed point theorems for self mappings. We also give a significant example to demonstrate the legitimacy of the given theorems. [ABSTRACT FROM AUTHOR]

In order to improve the convergence property and computational behavior of the randomized extended Kaczmarz method, we propose a multi-step randomized extended Kaczmarz method, in which we repeatedly update the iterate several times at each iteration step, obtaining a nonstationary inner-outer iteration scheme for solving large-scale, sparse, and inconsistent system of linear equations. For this multi-step randomized extended Kaczmarz method, we prove its convergence, derive an upper bound for its convergence rate, and demonstrate that this upper bound can be smaller than that of the randomized extended Kaczmarz method for several typical choices of the numbers of inner iteration steps. Numerical experiments also show that the multi-step randomized extended Kaczmarz method can perform better than the randomized extended Kaczmarz method if we choose the numbers of inner iteration steps appropriately. [ABSTRACT FROM AUTHOR]

This manuscript contains several new notions including intuitionistic fuzzy  N b  metric space, intuitionistic fuzzy quasi- S b -metric space, intuitionistic fuzzy pseudo- S b -metric space, intuitionistic fuzzy quasi- N -metric space and intuitionistic fuzzy pseudo  N b  fuzzy metric space. We prove decomposition theorem and fixed-point results in the setting of intuitionistic fuzzy pseudo  N b  fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications. [ABSTRACT FROM AUTHOR]

The results of a parallel implementation of a randomized vector algorithm for solving systems of linear equations are presented in the paper. The solution is represented in the form of a Neumann series. The stochastic method computes this series by sampling only random columns, avoiding multiplication of matrix by matrix and matrix by vector. We consider the case when the matrix is too large to fit in random-access memory (RAM). We use two approaches to solve this problem. In the first approach, the matrix is divided into parts that are distributed among MPI processes and stored in the available RAM of the cluster nodes. In the second approach, the entire matrix is stored on each node's hard drive, loaded into RAM, and processed in parts. Independent Monte Carlo experiments for random column indices are distributed among MPI processes or OpenMP threads for both approaches to matrix storage. The efficiency of parallel implementations is analyzed. Results are given for a system governed by dense matrices of size 10 4 and 10 5 . [ABSTRACT FROM AUTHOR]

In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the cover order and the cover length, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with non-negativity constraints on variables for both homogeneous and nonhomogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including linear programming (LP) problems and non-negative least squares (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem. [ABSTRACT FROM AUTHOR]

Let M n , k , t be the inclusion matrix of k-subsets and t-subsets of a set of n elements. In this note, we give a new proof for the property that M n , k , t is either of full row rank or of full column rank. Our proof is based on a nontrivial use of system of linear equations. [ABSTRACT FROM AUTHOR]

The Kaczmarz method is a classical while effective iteration method for solving very large-scale consistent systems of linear equations, and the randomized Kaczmarz method is an important and valuable variant of the Kaczmarz method. By theoretically analyzing and numerically experimenting several criteria typically adopted in the non-randomized and the randomized Kaczmarz method for selecting the working row, we derive sharper upper bounds for the convergence rates of some of the correspondingly induced Kaczmarz-type methods including those with respect to the maximal residual, maximal distance, and distance selection rules of the working row, and, for this whole suite of iteration methods consisting of the Kaczmarz methods with respect to the uniform, non-uniform, residual, distance, maximal residual, and maximal distance selection rules of the working row, we reveal their comparable relationships in terms of both mean-squared distance and mean-squared error, and show their computational effectiveness and numerical robustness based upon implementing a large number of test examples. Here the mean-squared distance is defined as the mean-value of the squared Euclidean norm of the current update increment of the iteration, and the mean-squared error is defined as the mean-value of the squared Euclidean norm of the current error that is the difference between the current iterate and the true solution of the target linear system. [ABSTRACT FROM AUTHOR]

The Jacobi iterative algorithm has the characteristic of low computational load, and multiple components of the solution can be solved independently. This paper applies these characteristics to the ternary optical computer, which can be used for parallel optimization because it has a large number of data bits and reconfigurable processor bits. Therefore, a new parallel design scheme is constructed to solve the problem of slow efficiency in solving large linear equations. And the elaborate experiment is used to verify. The experimental method is to simulate the calculation on the ternary optical computer experimental platform. Then, the resource consumption is numerically calculated and summarized to measure the feasibility of the parallel design. Eventually, the results show that the parallel design has obvious advantages in computing speed. The Jacobi iterative algorithm is optimized in parallel on ternary optical processor for the first time. There are two parallel highlights of the scheme. First, the n components are calculated in full parallel. Second, the modified signed-digit (MSD) multiplier based on the minimum module and one-step MSD adder are used to calculate each component to eliminate the impact of large amount of data on calculation time. The research provides a new method for fast solution of large linear equations. [ABSTRACT FROM AUTHOR]

For the single-step matrix splitting iteration paradigm proposed and analyzed recently by Bai (Numer. Linear Algebra Appl. 2023; e2510), we further provide a simpler and more accurate convergence result when the coefficient matrix of the linear system is a nonsingular matrix. Moreover, we demonstrate its asymptotic convergence when the coefficient matrix is an H -matrix. These results then enrich the convergence theory and enlarge the application spectrum of this paradigm. [ABSTRACT FROM AUTHOR]

For solving large-scale inconsistent systems of linear equations by iteration methods, with the application of the greedy randomized Kaczmarz method to a consistent augmented linear system, we find an appropriate balance between the updates of the two iteration vectors in the randomized extended Kaczmarz method, and, based on this balance, we propose a greedy randomized augmented Kaczmarz method. We prove the convergence of the greedy randomized augmented Kaczmarz method and derive an upper bound for its expected convergence rate. Numerical results show that the greedy randomized augmented Kaczmarz method can be much more effective than the randomized extended Kaczmarz method as well as the partially randomized extended Kaczmarz method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a normalization method for solving a system of linear equations over tropical semirings. We use a normalization method to construct an associated normalized matrix, which gives a technique for solving the system. If solutions exist, the method can also determine the degrees of freedom of the system. Moreover, we present a procedure to determine the column rank and the row rank of a matrix. Flowcharts for this normalization method and its applications are included as well. [ABSTRACT FROM AUTHOR]

The Gauss-Seidel and Kaczmarz methods are two classic iteration methods for solving systems of linear equations, which operate in column and row spaces, respectively. Utilizing the connections between these two methods and imitating the exact analysis of the mean-squared error for the randomized Kaczmarz method, we conduct an exact closed-form formula for the mean-squared residual of the iterate generated by the randomized Gauss-Seidel method. Based on this new formula, we further estimate an upper bound for the convergence rate of the randomized Gauss-Seidel method. Theoretical analysis and numerical experiments show that this bound measurably improves the existing ones. Moreover, these theoretical results are also extended to the more general extrapolated randomized Gauss-Seidel method. [ABSTRACT FROM AUTHOR]

The algebraic reconstruction technique (ART) and its generalization the block-iterative ART are often used to solve large scale linear systems Ax=b which arise in many applications, such as image processing. Two ingredients, i.e. stepsizes and blocks, play a key role in block-iterative ART. In this paper we address the problem of how to choose stepsizes and blocks of A so as to achieve optimal convergence rate of the block-iterative ART. We introduce a random cyclic selection method for selecting blocks for the block-iterative ART in order to reduce computational complexity. Our numerical experiments demonstrate the efficiency of our selections of stepsizes and blocks in the block-iterative ART. [ABSTRACT FROM AUTHOR]

We enrich the known results about coupled fixed points and coupled best proximity points. We generalize the notion of ordered pairs of cyclic contraction maps and we obtain sufficient conditions for the existence and uniqueness of best proximity points. We get a priori and a posteriori error estimates for the coupled fixed points and for the coupled best proximity points, provided that the underlying Banach space has modulus of convexity of power type in the case of best proximity points, obtained by sequences of successive iterations. We illustrate the main result with an example. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a method for solving fully fuzzy Sylvester equation (FFSE) A X + X B = C , where A , B and C are fuzzy matrices. By using of α -cutting, FFSE is transformed to a generalized Sylvester matrix equation, and then to a crisp system of linear equations. Existence and uniqueness of a solution to this system are investigated. Two numerical examples are given to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

For solving large sparse, overdetermined, and inconsistent system of linear equations by iteration methods, by further reconstructing the randomized extended Kaczmarz method proposed by Zouzias and Freris in 2013 (SIAM J. Matrix Anal. Appl. 34 (2013), 773–793), we propose a partially randomized extended Kaczmarz method. When the coefficient matrix is assumed to be of full column rank, we prove the convergence and derive an upper bound for the expected convergence rate of the partially randomized extended Kaczmarz method. This bound could be smaller than that of the randomized extended Kaczmarz method under certain conditions. Moreover, with numerical results we show that the partially randomized extended Kaczmarz method can be much more effective than the randomized extended Kaczmarz method. [ABSTRACT FROM AUTHOR]

Numerical methods to solve problems on a mathematical safe with an arbitrary finite number of positions of locks are given. The methods are based on TSS-algorithms for constructing sets of basic solutions to systems of linear Diophantine equations in finite fields and rings. [ABSTRACT FROM AUTHOR]

This study explored the mental constructions displayed by pre-service mathematics teachers (PMTs) when applying Cramer's rule. The aim was to reveal the character of mental constructions made around the nature of the solution set of equations and the role of parameters in the solution of equations with parametric coefficients. These mental constructions occur within the Action-Process-Object-Schema (APOS) theory. In this study, data was generated from 31 first year pre-service teachers by means of an activity sheet and interviews. The interviews were used to clarify pre-service teachers' responses to tasks from the activity sheet. The findings revealed that many PMTs displayed procedural understanding of Cramer's rule. It was also found that they understood what the solution of system of linear equations means. This meant that they were operating at the action stage, in terms of APOS. Additionally, it revealed that the lack of construction of related schemata negatively impacted the PMTs' attempt to construct the necessary mental constructions. Therefore, the researchers provided the genetic decomposition (GD) for the use of Cramer's rule to assist teacher educators to analyse the mental constructions of students. This study explored a new application of APOS theory. Analysing the mental constructions of students by means of research is intended to assist in designing alternative teaching strategies. In this way, this study makes a significant contribution to the solution of system of equations pedagogy. [ABSTRACT FROM AUTHOR]

Abstract The Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, 2008) is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the eigenvalues and eigenvectors. Using the Ritz values, we propose a self-contained variant of the Ritz-IDR(s) method (Simoncini and Szyld, 2010) for solving a system of linear equations. In addition, the Ritz vectors are used to speed-up IDR(s) for the solution of sequence of systems of linear equations. [ABSTRACT FROM AUTHOR]

For a wide range of control engineering applications, the problem of solving a system of linear equations is often encountered and has been well studied. Traditionally, this problem has been mainly solved in a centralized manner. However, for applications related to large-scale complex networked systems, centralized algorithms are often subjected to some practical issues due to limited computational power and communication bandwidth. As a promising and viable alternative, distributed algorithms can effectively address the issues associated with centralized algorithms by solving the problem efficiently in a multi-agent setting that accords with the distributed nature of networked systems. Distributed algorithms decompose the entire problem into many sub-problems that are solved by individual agents in a cooperative manner. In this survey paper, we provide a detailed overview of the state of the art relevant to distributed algorithms for solving a system of linear equations. We will first review basic distributed algorithms including both discrete-time and continuous-time algorithms. Then we will discuss the extended algorithms to achieve communication efficiency. Furthermore, we will also introduce distributed algorithms to obtain the minimum-norm solution for a system of linear equations with multiple solutions, as well as the least-squares solution when there is no solution. Finally, the relationship of distributed algorithms for solving a system of linear equations to the existing distributed optimization algorithms is discussed. [ABSTRACT FROM AUTHOR]

For consistent system of linear equations with the coefficient matrix being flat, we conduct an exact closed-form formula for the mean squared error of the iterate generated by the randomized Kaczmarz method, which completes the existing closed-form formula derived only for the tall coefficient matrix. Based upon these formulas, we further estimate an upper bound for the convergence rate of the randomized Kaczmarz method. Both theoretical analysis and numerical experiments demonstrate that this bound can significantly improve the existing ones. [ABSTRACT FROM AUTHOR]

For solving large sparse systems of linear equations by iteration methods, we further generalize the greedy randomized Kaczmarz method by introducing a relaxation parameter in the involved probability criterion, obtaining a class of relaxed greedy randomized Kaczmarz methods. We prove the convergence of these methods when the linear system is consistent, and show that these methods can be more efficient than the greedy randomized Kaczmarz method if the relaxation parameter is chosen appropriately. [ABSTRACT FROM AUTHOR]

Solving a linear system of n × n equations can be very difficult for the computer, especially if one needs the exact solution, even when the number n - of equations and of unknown variables is relatively small (a few thousands). All existing methods have to overcome at least one of the following problems: 1. Computational complexity, which is expressed with the number of arithmetic operations required in order to determine a solution; 2. The possibility of overflow and underflow problems; 3. Causing variations in the values of some coefficients in the initial system, which may be leading to instability of the solution; 4. Requiring additional conditions for convergence; 5. In cases of a large number of equations and unknown variables it is often required that the systems matrix be: either sparse, or symmetrical, or diagonal, etc. This paper presents a method for solving a system of linear equations of arbitrary order (any number of equations and unknown variables) to which the problems listed above do not reflect. [ABSTRACT FROM AUTHOR]

In the resolution of linear systems it is common to analyze not only a system AX = B, but several, in which the matrix A is conserved and only the vector B is altered. The Gauss Elimination method consists of performing elementary transformations on the equations of a system AX = B until an upper triangular system UX = C, equivalent to the given system. Such a method would oblige us to solve everything from the beginning, for each new vector B(RUGGIERO; LOPES, 1988). Alternatively, the LU factorization of A allows the solution of each system to occur only by changing the vector B. The present work is the result of a scientific initiation study of the Degree in Mathematics and linked to the Tutorial Education Program (PET). In this work, the objective is to present the Gauss elimination method with partial pivoting, the algorithm for solving triangular linear systems and the LU factorization method of the coefficient matrix. [ABSTRACT FROM AUTHOR]

We consider the numerical method for non-self-adjoint positive definite linear differential equations, and its application to the unsteady discrete elliptic problem, which is derived from spatial discretization of the unsteady elliptic problem with Dirichlet boundary condition. Based on the idea of the alternating direction implicit (ADI) iteration technique and the Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation (WR) iteration method for solving the non-self-adjoint positive definite linear differential equations, called the WR–HSS method. We analyze the convergence property of the WR–HSS method, and prove that the WR–HSS method is unconditionally convergent to the solution of the system of linear differential equations. In addition, we derive the upper bound of the contraction factor of the WR–HSS method in each iteration which is only dependent on the Hermitian part of the corresponding non-self-adjoint positive definite linear differential operator. Finally, the applications of the WR–HSS method to the unsteady discrete elliptic problem demonstrate its effectiveness and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a novel formulation for determining the short-circuit fault identity, that is the fault type, faulted line, and exact fault distance on it, by using available synchrophasor data. A simple and yet quite effective procedure is developed to model the fault area as a stand-alone sub-system. Thanks to phasor measurement units (PMUs), the proposed technique does not require the operating point and model of the portions being replaced. This greatly alleviates the complexity and technical problems involved in modeling the entire power system, as enforced by existing wide-area methods. A couple of effective theorems in Circuit Theory are exploited in a way as to make the prefault bus impedance matrix applicable in the postfault condition. The obtained fault equations are readily solved by the least-squares method to provide a closed-form solution for the fault distance. Two necessary and sufficient conditions are introduced to assess the fault location feasibility by any given set of PMUs. High accuracy is achieved since the calculations merely involve sound equations remaining after removing erroneous measurements of instrument transformers. The proposed method is successfully validated by more than 10 000 simulation cases conducted on the New England 39-bus and 118-bus test systems. [ABSTRACT FROM AUTHOR]

Inspired by the greedy randomized Kaczmarz method, we propose a probability criterion which can capture subvectors of the residual whose norms are relatively large. According to this probability criterion we select a submatrix randomly from the coefficient matrix, then average the projections of the current iteration vector onto each individual row of this chosen submatrix, constructing the greedy randomized average block Kaczmarz method for solving the consistent system of linear equations, which can be implemented in a distributed environment. When the size of each block is one, the probability criterion in the greedy randomized average block Kaczmarz method is a generalization of that in the greedy randomized Kaczmarz method. The greedy randomized Kaczmarz method is also a special case of the greedy randomized average block Kaczmarz method. Two kinds of extrapolated stepsizes for the greedy randomized average block Kaczmarz method are analyzed. The experimental results show the advantage of the greedy randomized average block Kaczmarz method over the greedy randomized Kaczmarz method and several existing randomized block Kaczmarz methods. [ABSTRACT FROM AUTHOR]

A technique for computing an ILU preconditioner based on the factored approximate inverse (FAPINV) algorithm is presented. We show that this algorithm is well-defined for H-matrices. Moreover, when used in conjunction with Krylov-subspace-based iterative solvers such as the GMRES algorithm, results in reliable solvers. Numerical experiments on some test matrices are given to show the efficiency of the new ILU preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, we consider the system of linear equations $A\tilde {x}=\tilde {b}$, where $A\in \mathbb {R}^{n \times n}$ is a crisp H-matrix and $\tilde {b}$ is a fuzzy n-vector. We then investigate the existence and uniqueness of a fuzzy solution to this system. The results can also be used for the class of M-matrices and strictly diagonally dominant matrices. Finally, some numerical examples are given to illustrate the presented theoretical results. [ABSTRACT FROM AUTHOR]

Let R denote the Galois ring of characteristic p s and cardinality p s h . In this paper, we determine the Smith normal forms of matrices over R , compute the number of the orbits of m × n matrices under the group GL m ( R ) × GL n ( R ) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices. [ABSTRACT FROM AUTHOR]

Artificial neural networks have the advantages such as learning, adaptation, fault-tolerance, parallelism and generalization. This paper is a scrutiny on the application of diverse learning methods in speed of convergence in neural networks. For this aim, first we introduce a perceptron method based on artificial neural networks which has been applied for solving a non-singular system of linear equations. Next two famous learning techniques namely, the steepest descent and quasi-Newton methods are employed to adjust connection weights of the neural net. The main aim of this study is to compare ability and efficacy of the techniques in speed of convergence of the present neural net. Finally, we illustrate our results on some numerical examples with computer simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose two preconditioned AOR it- erative methods to solve systems of linear equations whose coefficient matrices are Z-matrix. These methods can be considered as improvements of two previously presented ones in the literature. Finally some numerical experiments are given to show the effective- ness of the proposed preconditioners. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, nonlinear minimax problems with inequality constraints are discussed. Combined the norm-relaxed SQP method with the idea of strongly sub-feasible directions method, a new method of quasi-strongly sub-feasible directions (MQSSFD) with arbitrary initial point for the discussed problems is presented. At each iteration of the proposed algorithm, an improved search direction is obtained by solving a quadratic program (QP) which always has a solution, and a high-order correction direction is yielded via a system of linear equations (SLE) to avoid the Maratos effect. After finite iterations, the iteration point always get into the feasible set by introducing a new non-monotone curve search. Under some mild conditions including the weak Mangasarian–Fromovitz constraint qualification (MFCQ), the proposed algorithm possesses global convergence, and the superlinear convergence is obtained without the strict complementarity. Finally, some elementary numerical experiments are implemented and reported. [Copyright &y& Elsevier]

Abstract: Contrary to linear difference equations, there is no general theory of difference equations of the form , with , of total degree and , where is a field of characteristic zero. This article concerns the following problem: given , G and , find an upper bound on the degree d of a polynomial solution , if it exists. In the presented approach the problem is reduced to constructing a univariate polynomial for which d is a root. The authors formulate a sufficient condition under which such a polynomial exists. Using this condition, they give an effective bound on d, for instance, for all difference equations of the form with quadratic G, and all difference equations of the form with G having an arbitrary degree. [Copyright &y& Elsevier]

The SST Decomposition Method for solving system of linear equations make it possible to obtain the values of roots of the system with the specified accuracy as the limit of the sequence of some vectors. In this topic we are going to consider vectors as fuzzy vectors. We have considered a numerical example and tried to find out solution vector ... in fuzzified form using method of SST Decomposition. [ABSTRACT FROM AUTHOR]

An efficient numerical algorithm for solving system of linear equations based on modified homotopy perturbation method (MHPM) is presented. We also study the convergence of the proposed algorithm. The convergent conditions of the proposed algorithm are constructed. The connection with the well-known algorithms such as successive over-relaxation (SOR) method is discussed. The proposed algorithm is compared with SOR method and those obtained by the original homotopy perturbation method. Some examples from the literature are given to illustrate that the proposed algorithm is outperform the results obtained by SOR and the original homotopy perturbation method. [ABSTRACT FROM AUTHOR]

Cloud computing economically enables customers with limited computational resources to outsource large-scale computations to the cloud. However, how to protect customers' confidential data involved in the computations then becomes a major security concern. In this paper, we present a secure outsourcing mechanism for solving large-scale systems of linear equations (LE) in cloud. Because applying traditional approaches like Gaussian elimination or LU decomposition (aka. direct method) to such large-scale LEs would be prohibitively expensive, we build the secure LE outsourcing mechanism via a completely different approach—iterative method, which is much easier to implement in practice and only demands relatively simpler matrix-vector operations. Specifically, our mechanism enables a customer to securely harness the cloud for iteratively finding successive approximations to the LE solution, while keeping both the sensitive input and output of the computation private. For robust cheating detection, we further explore the algebraic property of matrix-vector operations and propose an efficient result verification mechanism, which allows the customer to verify all answers received from previous iterative approximations in one batch with high probability. Thorough security analysis and prototype experiments on Amazon EC2 demonstrate the validity and practicality of our proposed design. [ABSTRACT FROM AUTHOR]

For the tridiagonal system of linear algebraic equations whose matrix is nonstrictly Jacobi diagonally dominant in columns we establish sufficient conditions for all components of the solution to be nonnegative. [ABSTRACT FROM AUTHOR]

In this paper, an improved feasible QP-free method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. In view of the computational cost, the most attractive feature of the new algorithm is that only one system of linear equations is required to obtain the revised feasible descent direction. Thereby, per single iteration, it is only necessary to solve three systems of linear equations with the same coefficient matrix. In particular, without the positive definiteness assumption on the Hessian estimate, the proposed algorithm is still global convergence. Under some suitable conditions, the superlinear convergence rate is obtained. [ABSTRACT FROM AUTHOR]

This note explains how Emil Artin's proof that row rank equals column rank for a matrix with entries in a field leads naturally to the formula for the nullity of a matrix and also to an algorithm for solving any system of linear equations in any number of variables. This material could be used in any course on matrix theory or linear algebra. [ABSTRACT FROM PUBLISHER]

Abstract: A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman–Morrison–Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman–Morrison–Woodbury inversion. Numerical experiments are provided to illustrate the theory. [Copyright &y& Elsevier]

The generalized inverse is a subject of practical significance for estimating the solution of a system of linear equations under a wide variety of conditions. While the literature has ample references on the underlying mathematical theory and properties of the generalized inverse, it lacks a qualitative discussion of the generalized inverse that would facilitate the understanding of its properties under varying constraints. This paper discusses and develops the generalized inverse using heuristic arguments that offer key insights into its practical uses. [ABSTRACT FROM AUTHOR]

The paper treats a linear overdetermined system which contains second order hyperbolic equations with two linear super singularities. The authors obtain some new integral representations of the solutions. [ABSTRACT FROM AUTHOR]

Abstract: In a Coalitional Resource Game (CRG for brief), agents form coalitions to pool their resources in order to achieve certain goals, requiring the expenditure of these resources. A particular coalition is said to be successful, if the common resources of its members enable to achieve a set of goals that satisfies all members of the coalition. It is known that when resources are consumable it is NP-complete to decide whether a given coalition is successful. In this paper, we show a connection of CRGs with sharable resources and max–min linear systems of inequalities. This correspondence leads to polynomial algorithms for checking whether a given CRG admits a successful coalition and for several other problems whose counterparts for CRGs with consumable resources are hard. On the other hand, we prove that some problems concerning the structure of successful coalitions are hard also in the case of sharable resources. [Copyright &y& Elsevier]

Abstract: Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss–Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is . In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods. [Copyright &y& Elsevier]