On the determination of the interval hull of linear interval equations

We present new methods for solving special classes of linear interval equations by means of suitable selection systems. First we prove some theorems characterizing the extremal points of the solution set. For invers-stable interval matrices and systems whose solution set are contained in one orthant we give algorithms to select such systems the solutions of which generate the interval hull.     

A Farkas-type theorem for linear interval equations

We give a Farkas-type necessary and sufficient condition for a system of linear interval equations to have a nonnegative solution, and derive a consequence of it.     

A two-sequence method for linear interval equations

It is shown that only two matrix sequences are to be constructed to solve a system of linear interval equations with an inverse stable, strongly regular interval matrix.     

Enclosing solutions of linear interval equations is NP-hard

We show that if the conjectureP≠NP is true, then there does not exist a general polynomial-time algorithm for enclosing the solution set of a system of linear interval equations.     

The AOR method for solving linear interval equations

This paper is motivated by the paper [7], where the SOR method for solving linear interval equations was considered. It is known that sometimes the AOR method for systems of linear (“point”) equations converges faster than the SOR method. We give some sufficient conditions for the convergence of the interval AOR method for the same class of interval matrices which are considered in [7].     

Enclosing solutions of overdetermined systems of linear interval equations

A method for enclosing solutions of overdetermined systems of linear interval equations is described. Several aspects of the problem (algorithm, enclosure improvement, optimal enclosure) are studied.     

Solution of linear equations with a symmetrically skyline-stored nonsymmetric matrix

Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetrically skylined-stored nonsymmetric coefficient matrix are presented. Such systems arise in various computations, among which are the finite element discretization in conjunction with incremental continuum mechanics, or space-time finite elements for dynamical systems. These routines can be used for constrained systems without prearranging. The feature of partial decomposition is installed and its application to the analysis of singular matrices is discussed.     

Solution of linear equations with skyline-stored symmetric matrix

Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetric, skyline-stored coefficient matrix are presented. Such systems arise in a variety of applications, notably the numerical discretization of conservative physical systems by finite differences or finite element techniques. The routines can be used for processing constrained systems without need for prearranging equations. The application to ‘superelement’ condensation of large-scale systems is discussed.     

Solvability of interval linear equations and data analysis under uncertainty

Consideration was given to the problem of recognizing the solvability (nonemptiness of the solution set) of an interval systems of linear equations. A method based on the so-called recognizing functional of the solution set was proposed to solve it. A new approach to data processing under interval uncertainty based on the unconstrained maximization of the recognizing functional (“maximal consistency method”) was presented as an application, and its informal interpretations were described     

An estimate of the absolute value and width of the solution of a linear system of equations with tridiagonal interval matrix by the interval sweep method

We consider linear systems of algebraic equationsSu=f with tridiagonal interval matrixS and interval vector f An interval version of the sweep method allows us to find an interval vector u=(u₁, u₂,..., u _n )^T that contains the united set of solutions of the system. In the paper we present estimates of the absolute value and the width of the intervals u _i ,i=1, 2,...,n under certain assumptions on the elements of the matrixS that do not include the traditional condition of diagonal dominance. The width estimates are three orders of magnitude narrower, and the assumptions on the system’s coefficients are weaker than those in works published so far.     

Solution of a sparse linear system by using digraph

In this work, a method is given to compute solution of a sparse system of linear equations using Cramer's rule. The solution is computed with the help of digraph approach given in Chen, wherein only the non-zero entries are used. Only integer entries including right hand side are considered. We show that efficiency with which a sparse linear system can be analyzed by digital computer using digraph approach as a tool depends largely upon the efficiency with which 1-factors and 1-factorial connections are generated. Finally, the usefulness of digraph approach is discussed. The program is experimented by applying on three examples.

     

The controllability of an interval linear discrete system

A linear discrete control system with variable interval coefficients is considered. We investigate the controllability of the system, i.e. the possibility of steering its trajectory bundle from one given slab to another for a finite number of steps using the appropriate control. The necessary and sufficient conditions of controllability in the form of solvability of a linear programming problem are obtained. The optimal plan for the problem gives by the control that steers the trajectory bundle of the system from the initial slab into the minimal neighborhood of a finite slab.     

On the solution of a quasi-tridiagonal system of linear equations

A method is described for solving a system of linear algebraic equations which is almost tridiagonal. Numerical results are presented for a number of test problems and some comparisons are made with the results obtained from algorithms proposed by other authors. A possible extension of the technique is briefly outlined.     

Solution of systems of linear algebraic equations with preliminary flattening

A method of improving computing properties of matrices of systems of linear algebraic equations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 144–149, September–October, 1999.     

Iterative methods for systems of equations with interval coefficients and linear form

We introduce iterative methods for systems of equations with interval coefficients and linear form by suitable matrix splittings. When compared to the iterative methods for systems amenable to iteration introduced in [1], improved convergence and inclusion properties can be proved under suitable conditions. The method can also be used in the solution of specific nonlinear systems of equations by interval arithmetic methods.     

Solution of linear equations in remote sensing and picture reconstruction

The Moore-Penrose generalized inverse is utilized to obtain two general optimal solutions of a given system of linear equations. These solutions involve two matricesW andV. It is shown that available information regarding the desired solution and/or residual vector can be incorporated inW andV. Several of the known results in published literature are shown to be special cases of the optimal solutions given here.     

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.     

An observer-canonical state-representation for a system of linear variable difference equations

This short paper presents a technique to convert a well-formed system of linear variable difference equations to a corresponding system of state equations. The technique applies to both causal and noncausal systems. The resulting state-representation is in the observer version of the Luenberger canonical form.     

Linear interval equations

Necessary and sufficient criteria are given for the existence and uniqueness of solutions of linear interval equations. Explicit formulas are given for the solution set when the solution set is convex. Necessary and sufficient conditions are given for the convexity of the solution set.