System of Linear Equations

The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate x k+1 by projecting the current point x k onto a separating hyperplane generated by a given linear combination of the original hyperplanes or halfspaces. In and Echebest et al. ( ) acceleration schemes for solving systems of linear equations and inequalities respectively were introduced, within a PAM like framework. In this paper we apply those schemes in an algorithm based on oblique projections reflecting the sparsity of the matrix of the linear system to be solved. We present the corresponding theoretical convergence results which are a generalization of those given in . We also present the numerical results obtained applying the new scheme to two algorithms introduced by García-Palomares and González-Castaño (1998) and also the comparison of its efficiency with that of .

In this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric, sparse systems of linear algebraic equations. These problems occur in finite difference and finite element simulations of semiconductor devices, and fluid flow problems.We consider five iterative methods that appear to be the most promising for this class of problems: the biconjugate gradient method, the conjugate gradient squared method, the generalized minimal residual method, the generalized conjugate residual method and the method of orthogonal minimization. Each of these methods was tested using similar preconditioning (incomplete LU factorization) on a set of large, sparse matrices arising from finite element simulation of semiconductor devices. Results are shown where we compare the computation time and memory requirements for each of these methods against one another, as well as against a direct method that uses LU factorization to solve these problems.The results of our nu...

Several mesh-based techniques in computer graphics such as shape deformation, mesh editing, animation and simulation, build and solve linear systems. The most common method to build a linear system consists in traversing the topology (connectivity) of the mesh, producing in general a representation of the set of equations in form of a sparse matrix. Similarly, the solution of the system is achieved, by means of iterating over the set of equations in the default sequence of the vertices (unknowns). This paper presents a new algorithm, which optimizes the build of the linear system and its storage, and which allows the iteration over the set of equations in any arbitrary order. Additionally, our algorithm enables rapid modifications to the linear system, avoiding a complete rebuild.

When the equations of linear elasticity are solved by the standard Galerkin method the equations become stiff for nearly incompressible materials. This results in a perturbed numerical approximation. To avoid this problem a well known technique employs a saddle-point formulation of the equations. In this paper a new technique will be presented which is based on a least-squares (re)formulation of the problem. The original second order elasticity equations are rewritten as a first order system and the least-squares functional arises from this system. Numerical examples are presented to show the difference between this method and the Galerkin method. A drawback of the least-squares method is that the condition number of the corresponding system of equations is depending on the incompressibility of the material, which will also be shown.

When the equations of linear elasticity are solved by the standard Galerkin method the equations become stiff for nearly incompressible materials. This results in a perturbed numerical approximation. To avoid this problem a well known technique employs a saddle-point formulation of the equations. In this paper a new technique will be presented which is based on a least-squares (re)formulation of the problem. The original second order elasticity equations are rewritten as a first order system and the least-squares functional arises from this system. Numerical examples are presented to show the difference between this method and the Galerkin method. A drawback of the least-squares method is that the condition number of the corresponding system of equations is depending on the incompressibility of the material, which will also be shown.

A new method for the solution of pentadiagonal systems of linear equations is presented. The method is a generalization of ordinary odd-even elimination used for tridiagonal systems. Using n processors, an n X n pentadiagonal system can be solved using the new method (generalized odd-even elimination) in time proportional to log, n.

A new method for the solution of pentadiagonal systems of linear equations is presented. The method is a generalization of ordinary odd-even elimination used for tridiagonal systems. Using n processors, an n X n pentadiagonal system can be solved using the new method (generalized odd-even elimination) in time proportional to log, n.

Neuro-Fuzzy Modeling has been applied in a wide variety of fields such as Decision Making, Engineering and Management Sciences etc. In particular, applications of this Modeling technique in Decision Making by involving complex Systems of Linear Algebraic Equations have remarkable significance. In this Paper, we present Polak-Ribiere Conjugate Gradient based Neural Network with Fuzzy rules to solve System of Simultaneous Linear Algebraic Equations. This is achieved using Fuzzy Backpropagation Learning Rule. The implementation results show that the proposed Neuro-Fuzzy Network yields effective solutions for exactly determined, underdetermined and over-determined Systems of Linear Equations. This fact is demonstrated by the Computational Complexity analysis of the Neuro-Fuzzy Algorithm. The proposed Algorithm is simulated effectively using MATLAB software. To the best of our knowledge this is the first work of the Systems of Linear Algebraic Equations using Neuro-Fuzzy Modeling.

Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then the system is solved to retrieve either a key or a plaintext. Raddum and Semaev proposed new methods for solving such sparse systems. It turns out that a combinatorial MaxMinMax problem provides bounds on the average computational complexity of sparse systems. In this paper we initiate a study of a linear algebra variation of this MaxMinMax problem.

Solution of linear algebra systems may come out with "ill-condition" or "well-condition" based on input information and solution methods. The aim of this study is to determine and correction of problems that may come out from the solution of matrix equations by computers and to calculate . A x f  linear algebra.

Solution of linear algebra systems may come out with "ill-condition" or "well-condition" based on input information and solution methods. The aim of this study is to determine and correction of problems that may come out from the solution of matrix equations by computers and to calculate. A x f  linear algebra.

Numerical methods are used to approximate solutions of equations when exact solutions can not be determined via algebraic methods. They construct successive approximations that converge to the exact solution of an equation or system of equations. The aim of this paper is to find the roots of Non-linear Equations Numerically using Newton’s Raphson Method by A New Mathematical Technique. We followed the applied mathematical method using a new mathematical technique and we found the following some results: The New Mathematical Technique facilitates the process of finding the roots of non-linear equations of different degrees, the possibility of drawing these roots graphically in addition to the accuracy, speed and logicality of the solution and reduce errors compared to the numerical analytical solution manually.

ABS mthods are direct iteration methods for solving linear systems where the i-th iterate satisfies the first i equations, and therefore a system on m equations is solved in at most m ABS steps. In this paper, using a new rank two update of the Abaffian matrix, we introduce a class of ABS-type methods for solving full row rank linear equations, where the i-th iterate solves the first 2i equations. So, termination is achieved in at most ⌊(m + 1)/2⌋ steps. We also show how to decrease the dimension of the Abaffian matrix by choosing appropriate parameters.

The formulation of steady-state initialization problems for fluid systems is a non-trivial task. If steady-state equations are specified at the component level, the corresponding system of initial equations at the system level might be overdeter-mined, if index reduction eliminates some states. On the other hand, steady-state equations are not suffi-cient to uniquely identify one equilibrium state in the case of closed systems, so additional equations are required. The paper shows how these problems might be solved in an elegant way by formulating overdetermined initialization problems, which have more equations than unknowns and a unique solution, then solving them using a least-squares minimization algorithm. The concept is tested on a representative test case using the OpenModelica compiler.

This paper compares the efficiency of the preconditioned conjugate gradient (PCG) method and the LDLTfactorization for solving a sparse system of linear equations in large‐scale structural analysis with both single‐ and multiple‐load cases. In the PCG method we present an incomplete LDLT preconditioning method that can be used to efficiently solve a system of linear equations with a wide range of matrix conditions, system sizes, and computer memory limitations. The preconditioning method is very flexible, and we can easily balance the size of the preconditioning matrix and the level of the preconditioning that is appropriate for the problem to be solved. For comparison, we use an implementation of a row‐oriented sparse LDLT solution method.

In this article, the problem of model reduction of 2-D systems is studied via orthogonal series. The algorithm proposed reduces the problem to an overdetermined linear algebraic system of equations, which may readily be solved to yield the simplified model. When this model approximates adequately the original system, it has many important advantages, e.g., it simplifies the analysis and simulation of the original system, it reduces the computational effort in design procedures, it reduces the hardware complexity of the system, etc. Several examples are included which illustrate the efficiency of the proposed method and gives some comparison with other model reduction techniques.

In this paper, we develop an approach for finding the cofactor, ad joint, determinant and inverse of a three by three matrix under the Cell Arrangements method using the coefficient matrix of a given systems of linear equation in three unknowns. The method takes out completely the seemingly daunting task in evaluating such matrices associated to the standard matrix method in solving simultaneous equation in three variable. Unlike the standard matrix method that goes through a lengthy process to obtain separately all the matrices necessary for the determination of the unknowns, the structural frame of the Cell Arrangement method comes in handy and are consistent with the results from systems that have unique solutions. This alternative approach provides all the vital hybrid matrices of the coefficient matrix needed in the determination of the unknowns of the system of equations in three variables. It is our view that by far, the Cell arrangement method is easy to work with and less p...

Nowadays, we face many equations in everyday life, where many attempts have been made to find their solutions, and various methods have been introduced. Many complex problems often lead to the solution of systems of equations. In mathematics, linear programming problems is a technique for optimization of a linear objective function that must impose several constraints on linear inequality. Linear programming emerged as a mathematical model. In this study, we introduce the category of ABS methods to solve general linear equations. These methods have been developed by Abafi, Goin, and Speedicato, and the repetitive methods are of direct type, which implicitly includes LU decomposition, Cholesky decomposition, LX decomposition, etc. Methods are distinguished from each other by selecting parameters. First, the equations system and the methods of solving the equations system, along with their application, are examined. Introduction and history of linear programming and linear programming problems and their application were also discussed.

We will consider Rohde's general form of {1}-inverse of a matrix A. The necessary and sufficient condition for consistency of a linear system Ax=c will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula x=A(1)c+(I-A(1)A)y for obtaining the general solution of the linear system. These results will be applied for finding the general solution of various homogenous and nonhomogenous linear systems as well as for different types of matrix equations.

This paper considers the solution of systems of algebraic equations that are expressed by a global rectangular system of equations, and a set of conditional equations that are expressed as disjunctions. These disjunctions are given by equations and inequalities, where the latter define the domain of validity of the equations. The solution of such a system is defined by variable values satisfying the rectangular equations, and exactly one set of equations for each of the disjunctions This paper addresses first the solution of systems of linear disjunctive equations. Using a convex hull representation of each of the disjunctions, it is shown that these equations can be converted into an MILP problem. A sufficient condition is presented under which this model is shown to be solvable as an LP problem. An extension to nonlinear disjunctive equations is presented by incorporating the proposed MILP formulation within a Newton iterative scheme. The application of the proposed algorithms is illustrated with several examples, including piecewise linear mass balances in process networks, and pipe networks with different flow regimes and check valves.

This paper considers the solution of systems of equations that are expressed by the two sets of equations: a global rectangular system of equations involving more variables than equations, and a set of conditional equations that are expressed as disjunctions. The set of disjunctions are given by equations and inequalities, where the latter define the domain of validity of the equations. In this way the solution of such a system is defined by variables x satisfying the rectangular equations, and exactly one set of equations for each of the disjunctions. This paper focuses mainly in the solution of systems of linear disjunctive equations. Using a convex hull representation of the disjunctions, the disjunctive system of equations is converted into an MELP problem. A sufficient condition is presented under which the model is shown to be solvable as an LP problem. The extension of the proposed method to nonlinear disjunctive equations is also discussed. The application of the proposed algorithms are illustrated with several examples.

We give a short overview over existing iterative algorithms for solving complex symmetric systems of linear equations, which can take advantage of this special structure. This type of system occurs when the method of moments is used to solve the Electric Field Integral Equation (EFIE). The relatively new method CSYM is described and the differences and advantages over the existing methods are pointed out, namely the minimal residual property, short term reccurrences relations and no potential serious breakdowns. We tested CSYM on matrices resulting from a discretisation of the EFIE to solve a scattering problem. The Method of Moments is applied in combination with a Multilevel Fast Multipole Method. When applied to systems having more than 100000 unknowns CSYM rapidly reduces the residual during the first iterations while solvers which do not minimize the residual in any sense are less efficient in this stage.

The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate x k+1 by projecting the current point x k onto a separating hyperplane generated by a given linear combination of the original hyperplanes or halfspaces. In and Echebest et al. ( ) acceleration schemes for solving systems of linear equations and inequalities respectively were introduced, within a PAM like framework. In this paper we apply those schemes in an algorithm based on oblique projections reflecting the sparsity of the matrix of the linear system to be solved. We present the corresponding theoretical convergence results which are a generalization of those given in . We also present the numerical results obtained applying the new scheme to two algorithms introduced by García-Palomares and González-Castaño (1998) and also the comparison of its efficiency with that of .

We study systems of String Equations where block variables need to be assigned strings so that their concatenation gives a specified target string. We investigate this problem under a multivariate complexity framework, searching for tractable special cases such as systems of equations with few block variables or few equations. Our main results include a polynomial-time algorithm for size-2 equations, and hardness for size-3 equations, as well as hardness for systems of two equations, even with tight constraints on the block variables. We also study a variant where few deletions are allowed in the target string, and give XP algorithms in this setting when the number of block variables is constant.

Solving large scale system of Simultaneous Linear Equations (SLE) has been (and continue to be) a major challenging problem for many real-world engineering and science applications. Solving SLE with singular coefficient matrices arises from various engineering and sciences applications [1]-[6]. In this paper, efficient numerical procedures for finding the generalized (or pseudo) inverse of a general (square/rectangle, symmetrical/unsymmetrical, non-singular/singular) matrix and solving systems of Simultaneous Linear Equations (SLE) are formulated and explained. The developed procedures and its associated computer software (under MATLAB [7] computer environment) have been based on "special Cholesky factorization schemes" (for a singular matrix). Test matrices from different fields of applications have been chosen, tested and compared with other existing algorithms. The results of the numerical tests have indicated that the developed procedures are far more efficient than the existing algorithms.

We design a new model of preconditioner for systems of linear equations. The convergence properties of the proposed methods have been analyzed and compared with the classical methods. Numerical experiments of convection-diffusion equations show a good im- provement on the convergence, and show that the convergence rates of proposed methods are superior to the other modified iterative methods.

This paper discusses a 1D one-dimensional mathematical model for the thermoelasticity problem in a two-layer plate. Basic equations in dimensionless form contain both temperature and displacement. General solutions of homogeneous equations (displacement and temperature equations) are assumed to be a linear combination of Trefftz functions. Particular solutions of these equations are then expressed with appropriately constructed sums of derivatives of general solutions. Next, the inverse operators to those appearing in homogeneous equations are defined and applied to the right-hand sides of inhomogeneous equations. Thus, two systems of functions are obtained, satisfying strictly a fully coupled system of equations. To determine the unknown coefficients of these linear combinations, a functional is constructed that describes the error of meeting the initial and boundary conditions by approximate solutions. The minimization of the functional leads to an approximate solution to the prob...

Linear Programming problems and System of Linear equations have many applications in various science and engineering problems like network analysis, operations research etc. In general Linear Programming Problem (LPP) and the system of linear equations contain crisp parameters that is real numbers or complex numbers as their coefficients and constants, but in real life applications, LPP and system of equations may contain the constrains or the parameters as uncertain. These uncertain values are not the exact real numbers but vary within some range of values, the values may vary within an interval or can be considered as fuzzy number. In this paper, we have developed a new Ranking function (which converts the fuzzy number into crisp) to solve a fully fuzzy LPP and System of equations. Unlike the previous ranking functions, the proposed ranking function uses fuzzy number itself improving the accuracy of the solution. The ranking function is derived by replacing the non-parallel sides of the trapezoidal fuzzy number with non-linear functions. Various numerical examples are included and compared with the pre-existing methods.

We derive an explicit determinantal formula for the least squares solution of an over-determined system of linear equations. From this formula it follows that the least squares solution lies in the convex hull of the solutions to the square subsystems of the original system. We extend this result and prove that this geometric property holds for a more general class of problems which includes the weighted least squares and I,,-norm minimization problems.

The estimation of the axial and radial component distributions of the blood velocity during its flow in a diverging artery is attempted. To this scope part of the Navier-Stokes equations and the continuity equation were treated analytically. Several flow characteristics, such as the back flow near the internal artery wall surfaces are investigated. Analytical results obtained by the proposed methodology are also included.

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We demonstrate a method using linear congruence equations to generate solutions to the N-Queens problem. There are only a few papers in the literature generating solutions for every N. Our method generates solutions for every N, and the number of solutions produced by our method is larger than the number of solutions given in these papers.

Linear and non-linear bending analysis of laminated plates with different boundary conditions is presented using generalized differential quadrature (GDQ) method. Governing equations are based on the first-order shear deformation theory (FSDT) and von Kármán-type of geometric non-linearity. GDQ technique and Newton-Raphson method are employed to solve the system of non-linear equations.

We introduce GPLS (Genetic Programming for Linear Systems) as a GP system that finds mathematical expressions defining an iteration matrix. Stationary iterative methods use this iteration matrix to solve a system of linear equations numerically. GPLS aims at finding iteration matrices with a low spectral radius and a high sparsity, since these properties ensure a fast error reduction of the numerical solution method and enable the efficient implementation of the methods on parallel computer architectures. We study GPLS for various types of system matrices and find that it easily outperforms classical approaches like the Gauss–Seidel and Jacobi methods. GPLS not only finds iteration matrices for linear systems with a much lower spectral radius, but also iteration matrices for problems where classical approaches fail. Additionally, solutions found by GPLS for small problem instances show also good performance for larger instances of the same problem.

In this paper, we describe some new variants and applications of the wavelet algebraic multigrid method. This method combines the algebraic multigrid method (a well known family of multilevel techniques for solving linear systems, without use of knowledge of the underlying problem) and the discrete wavelet transform. These two techniques can be combined in several ways, obtaining different methods for solution of linear systems; these can be used alone or as preconditioners for Krylov iterative methods. These methods can be applied for solution of linear systems with shifted matrices of the form A À hI, whose efficient solution is very important for implicit ODE methods, unsteady PDEs, computation of eigenvalues of large sparse matrices and other important problems.

In this paper, we develop an approach for finding the cofactor, ad joint, determinant and inverse of a three by three matrix under the Cell Arrangements method using the coefficient matrix of a given systems of linear equation in three unknowns. The method takes out completely the seemingly daunting task in evaluating such matrices associated to the standard matrix method in solving simultaneous equation in three variable. Unlike the standard matrix method that goes through a lengthy process to obtain separately all the matrices necessary for the determination of the unknowns, the structural frame of the Cell Arrangement method comes in handy and are consistent with the results from systems that have unique solutions. This alternative approach provides all the vital hybrid matrices of the coefficient matrix needed in the determination of the unknowns of the system of equations in three variables. It is our view that by far, the Cell arrangement method is easy to work with and less p...

Previously, the authors proposed a general method for finding particular solutions for overdetermined systems of partial differential equations (PDE), where the number of equations is greater than the number of unknown functions. In this paper, an algorithm for finding solutions for overdetermined PDE systems is proposed, where the authors use a method for finding an explicit solution for overdetermined algebraic (polynomial) equations. Using this algorithm, some overdetermined PDE systems can be solved in explicit form. The main difficulty of this algorithm is the huge number of polynomial equations that arise, which need to be investigated and solved numerically or explicitly. For example, the overdetermined hydrodynamic equations obtained earlier by the authors give at least 10 million such equations. However, if the equations are solved explicitly, then it is possible to write out the solution of the hydrodynamic equations in a general form, which is of great scientific interest.

This paper compares the efficiency of the preconditioned conjugate gradient (PCG) method and the LDLTfactorization for solving a sparse system of linear equations in large‐scale structural analysis with both single‐ and multiple‐load cases. In the PCG method we present an incomplete LDLT preconditioning method that can be used to efficiently solve a system of linear equations with a wide range of matrix conditions, system sizes, and computer memory limitations. The preconditioning method is very flexible, and we can easily balance the size of the preconditioning matrix and the level of the preconditioning that is appropriate for the problem to be solved. For comparison, we use an implementation of a row‐oriented sparse LDLT solution method.

Coarrays have been part of the Fortran standard since Fortran 2008 and provide a syntactic extension of Fortran to support parallel programming, often called Coarray Fortran (CAF). Although MPI is the de facto standard for parallel programs running on distributed memory systems and little scientific software is written in CAF, many scientific applications could benefit from the use of CAF. We present the migration from MPI to CAF of the libraries PSBLAS and MLD2P4 for the solution of large systems of equations using iterative methods and preconditioners. In this paper we describe some investigations for implementing the necessary communication steps in PSBLAS and MLD2P4 and provide performance results obtained on linear systems arising from discretization of 2D and 3D PDEs.

This paper studies a semi-linear system of equations in R N , which comes from a mathematical model for a new tax system proposed in Chile's 2014 Tax Reform. The system of equations involves a non negative coefficients matrix and simultaneously relates the unknown vector with its positive part, and hence the nonlinear nature of the problem. In addition to find appropriate conditions for the existence and the uniqueness of a solution, in this paper an algorithm is proposed to obtain it by solving at most N linear systems of size N × N. The proof of the results is based on a monotony method with respect to the usual lexicographic order in R N .

We c haracterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of a determining if a system of linear equations is feasible and b computing the rank of an integer matrix, as well as other problems, are complete under logspace reductions. As an important part of presenting this classi cation, we show that the exact counting logspace hierarchy" collapses to near the bottom level. We review the de nition of this hierarchy below. We further show that this class is closed under NC 1-reducibility, and that it consists of exactly those languages that have logspace uniform span programs introduced by Karchmer and Wigderson over the rationals. In addition, we contrast the complexity of these problems with the complexity o f determining if a system of linear equations has an integer solution.

A system of equations in the l-calculus is a set of formulas of A (the equations) together with a finite set of variables of ,4 (the unknowns). A system Y is said to be P-solvable (fiq-solvable) iff there exists a simultaneous substitution with closed I-terms for the unknowns that makes the equations of 9' theorems in the theory fi (&). A system 9' can be viewed as a set of specifications (the equations) for a finite set of programs (the unknowns) whereas a solution for Y yields executable codes for such programs. A class G of systems for which the solvability problem is effectively decidable defines an equational programming language and a system solving algorithm for G defines a compiler for such language. This leads us to consider separation-like systems (SL-systems), i.e. systems with equations having form x& = z, where x is an unknown and z is a free variable which is not an unknown. It is known that the /I (/?q)-solvability problem for SL-systems is undecidable. Here we show that there is a class of SL-systems (NP-regular SL-systems) for which the /?-solvability problem is NP-complete. Moreover, we show that any SL-system Y can be transformed into an NP-regular SL-system Y". This transformation consists of adding abstractions to the LHS occurrences of the RHS variables of 9. In this sense NP-regular SL-systems isolate the source of undecidability for SL-systems, namely: a shortage of abstractions on the LHS occurrences of the RHS variables. NP-regular SL-systems yield an equational programming language in which unrestrained self-application is handled, constraints on executable code to be generated by the compiler can be specified by the user and (properties of) data structures can be described in an abstract way. However, existence of executable code satisfying a specification in such language is an *This paper is a revised version of [7, Part 21.

A system of equations in the I-calculus is a set of formulas of ,4 (the equations) together with a finite set of variables of A (the unknowns). A system 9' is said to be /?-solvable (&solvable) iff there exists a simultaneous substitution with closed L-terms for the unknowns that makes the equations of Y theorems in the theory p (/?q). A system Y can be viewed as a set of specifications (the equations) for a finite set of programs (the unknowns) whereas a solution for Y yields executable codes for such programs. A class G of systems for which the solvability problem is effectively decidable defines an equational programming language and a system solving algorithm for G defines a compiler for such language. This leads us to consider separation-like systems (SL-systems), i.e. systems with equations having form xfi = z, where x is an unknown and z is a free variable which is not an unknown. We show that the b (/3q)-solvability problem for SL-systems is undecidable. On the other hand we are able to define a class of SL-systems (regular SL-systems) for which the /&solvability problem is decidable in Polynomial Time. Such class yields an equational programming language in which self-application is handled, constraints on executable code to be generated by the compiler can be specified by the user and (properties of) data structures can be described in an abstract way.

The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are thereby deduced. An algorithm and a numerical example are constructed over the quaternions to validate the results of this paper.