Automata–based Method for Solving Systems of Linear Constraints in {0,1}

We consider a finite state automata based method of solving a system of linear Diophantine equations with coefficients from the set {-1,0,1} and solutions in {0,1}.

An Algorithm to Decide Feasibility of Linear Integer Constraints Occurrng in Decision Tables

Abstract-To detect errors in decision tables one needs to decide whether a given set of constraints is feasible or not. This paper describes an algorithm to do so when the constraints are linear in variables that take only integer values. Decision tables with such constraints occur frequently in business data processing and in nonnumeric applications. The aim of the algorithm is to exploit. the abundance of very simple constraints that occur in typical decision table contexts. Essentially, the algorithm is a backtrack procedure where the the solution space is pruned by using the set of simple constrains. After some simplications, the simple constraints are captured in an acyclic directed graph with weighted edges. Further, only those partial vectors are considered from extension which can be extended to assignments that will at least satisfy the simple constraints. This is how pruning of the solution space is achieved. For every partial assignment considered, the graph representation of the simple constraints provides a lower bound for each variable which is not yet assigned a value. These lower bounds play a vital role in the algorithm and they are obtained in an efficient manner by updating older lower bounds. Our present algorithm also incorporates an idea by which it can be checked whether or not an (m - 2)-ary vector can be extended to a solution vector of m components, thereby backtracking is reduced by one component.

On the direct parallel solution of systems of linear equations: New algorithms and systolic structures

The paper presents two new algorithms for the direct parallel solution of systems of linear equations. The algorithms employ a novel recursive doubling technique to obtain solutions to an nth-order system in n steps with no more than 2n(n −1) processors. Comparing their performance with the Gaussian elimination algorithm (GE), we show that they are almost 100% faster than the latter. This speedup is achieved by dispensing with all the computation involved in the back-substitution phase of GE. It is also shown that the new algorithms exhibit error characteristics which are superior to GE. An n(n + 1) systolic array structure is proposed for the implementation of the new algorithms. We show that complete solutions can be obtained, through these single-phase solution methods, in 5n−log2n−4 computational steps, without the need for intermediate I/O operations.

Finite field computation technique for exact solution of systems of linear equations and interval linear programming problems

An error-free computational approach is employed for finding the integer solution to a system of linear equations, using finite-field arithmetic. This approach is also extended to find the optimum solution for linear inequalities such as those arising in interval linear programming probloms.

Differential resultants of super essential systems of linear OD-polynomials

The sparse differential resultant dres(P) of an overdetermined system P of generic nonhomogeneous ordinary differential polynomials, was formally defined recently by Li, Gao and Yuan (2011). In this note, a differential resultant formula dfres(P) is defined and proved to be nonzero for linear "super essential" systems. In the linear case, dres(P) is proved to be equal, up to a nonzero constant, to dfres(P*) for the supper essential subsystem P* of P.

A parallel method for solving pentadiagonal systems of linear equations

A new parallel approach for solving a pentadiagonal linear system is presented. The parallel partition method for this system and the TW parallel partition method on a chain of P processors are introduced and discussed. The result of this algorithm is a reduced pentadiagonal linear system of order P \Gamma 2 compared with a system of order 2P \Gamma 2 for the parallel partition method. More importantly the new method involves only half the number of communications startups than the parallel partition method (and other standard parallel methods) and hence is a far more efficient parallel algorithm.

Алгоритмы Решения Систем Линейных Диофантовых Уравнений в Дискретных Областях

Abstract.The algorithms for computation of minimal supported set of solutions for systems of linear Diophantine homogeneous equations over set of natural numbers and basis of systems of linear Diophantine homogeneous and inhomogeneous equations in ring and field of remainders on modulo of a number.

Points-to Analysis as a System of Linear Equations

We propose a novel formulation of the points-to analysis as a system of linear equations. With this, the efficiency of the points-to analysis can be significantly improved by leveraging the advances in solution procedures for solving the systems of linear equations. However, such a formulation is non-trivial and becomes challenging due to various facts, namely, multiple pointer indirections, address-of operators and multiple assignments to the same variable. Further, the problem is exacerbated by the need to keep the transformed equations linear. Despite this, we successfully model all the pointer operations. We propose a novel inclusion-based context-sensitive points-to analysis algorithm based on prime factorization, which can model all the pointer operations. Experimental evaluation on SPEC 2000 benchmarks and two large open source programs reveals that our approach is competitive to the state-of-the-art algorithms. With an average memory requirement of mere 21MB, our context-sensitive points-to analysis algorithm analyzes each benchmark in 55 seconds on an average.

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Lower semicontinuity of the feasible set mapping of linear systems relative to their domains

This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.

Comment: Eigenvalue sensitivity method for optimal stabilisation of linear systems

In a letter RauA proposed a new method for designing statefeedback controllers using eigenvalue sensitivity matrices. However, there appears to be a conceptual mistake in the procedure, or else it is unduly restricted in its applicability. In particular the equation — BR~lBTK = A/.I, in which K is a positive-definite symmetric matrix.

Stabilization of linear and nonlinear dynamical systems using an observer-controller configuration

In this paper the problem of stabilization of systems by means of stable compensations is considered, and results are derived for systems using observer�controller structures, for systems using a cascade structure, and for nonlinear systems

Comments on a decentralized control of linear multivariable systems

A strongly connected decentralized control system may be made single channel controllable and observable with respect to any channel by decentralized feedbacks. It is noted here that the system example considered by Corfmat and Morse to illustrate this fact is already single channel controllable and observable, with respect to one of the channels. An alternate example which fits into the situation is presented in this item.