A new method for solving interval and fuzzy equations: Linear case

A new approach to the solution of interval and fuzzy equations based on the generalized procedure of interval extension called “interval extended zero” method is proposed. The central for the proposed approach is the treatment of the interval zero as an interval centered around zero. It is shown that such proposition is not of heuristic nature, but is the direct consequence of the interval subtraction operation. Some methodological problems concerned with this definition of interval zero are discussed. It is shown that the resulting solution of interval linear equations based on the proposed method may be naturally treated as a fuzzy number. An important advantage of a new method is that it substantially decreases the excess width effect. On the other hand, we show that it can be used as a reliable practical tool for solving linear interval and fuzzy equations as well as the systems of them. The features of the method are illustrated by the example of the solution of the well known Leontief input-output problem in the interval setting.     

Linear interval equations: Computing enclosures with bounded relative or absolute overestimation is NP-hard

It is proved that for every δ>0, if there exists a polynomial-time algorithm for enclosing solutions of linear interval equations with relative (or absolute) overestimation better than δ, then P=NP. The result holds for the symmetric case as well.     

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.     

Linear programming with interval right hand sides

In this paper, we study general linear programs in which right hand sides are interval numbers. This model is relevant when uncertain and inaccurate factors make difficult the assignment of a single value to each right hand side. When objective function coefficients are interval numbers in a linear program, classical criteria coming from decision theory (like the worst case criterion) are usually applied to determine robust solutions. When the set of feasible solutions is uncertain, we identify a class of linear programs for which these classical approaches are no longer relevant. However, it is possible to compute the worst optimum solution. We study the complexity of this optimization problem when each right hand side is an interval number. Then, we exhibit some duality relationships between the worst optimum solution problem and the best optimum solution to the dual problem.     

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.     

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.     

Linear matrix equations and root clustering

Given A ε ?^{n + n} and ? ? ?, we search for a criterion assuring that the spectrum of A is clustered in ?, σ(A)? ? One approach to root clustering is the linear matrix equation, whose half plane version dates back to Lyapunov. The existing literature deals with an algebraic region defined by a single polynomial. In this paper, we construct a novel linear matrix equation related to the intersection of algebraic regions. This considerably enlarges the family of regions with root clustering criteria.     

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 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.     

Linear difference equations and generalized continued fractions

In one of his papers [5] Gautschi presents an algorithm for determining the minimal solution of a second-order homogeneous difference equation. The method is based on the connection between the existence of a minimal solution of such a difference equation and the convergence of a certain continued fraction. In the present paper, these results are generalized. For this purpose we use the concept of generalized continued fraction. The resulting algorithm is suitable for solving (n+1)-th order recursions (n≥1) for which there existn independent solutions that are dominated by each solution that does not belong to the space spanned by thesen solutions.     

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 convexity of a system of linear interval equations

A necessary and sufficient condition for a convex solution set to a system of linear interval equations is presented. The relation between the convexity and feasibility of an optimization problem is also discussed. An extended Leontief system is proposed as an application of the results.     

A symmetric iterative interval method for systems of nonlinear equations

An iterative method for nonlinear systems of equations is presented that is based on the idea of symmetric methods known from linear systems. Due to the use of interval arithmetic the convergence to a solution can be proved under relatively weak conditions provided an initial inclusion of that solution is known. The concept of symmetry leads to a reduction of computation time compared to some well-known methods.     

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     

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.     

Linear oscillation of second-order nonlinear difference equations with damped term

In this paper, we shall establish relations between the oscillation of second-order nonlinear difference equations with damped term and the oscillation of their linear limiting equations.