System of Linear Equations with Interval Coefficients (original) (raw)

Enclosing the solution set of overdetermined systems of interval linear equations

2014

We describe two methods to bound the solution set of full rank interval linear equation systems Ax = b where A ∈ IRm×n, m ≥ n is a full rank interval matrix and b ∈ IRm is an interval vector. The methods are based on the concept of generalized solution of overdetermined systems of linear equations. We use two type of preconditioning the m × n system: multiplying the system with the generalized inverse of the midpoint matrix or with the transpose of the midpoint matrix. It results an n × n system which we solve using Gaussian elimination or the method provided by J. Rohn in [8]. We give some examples in which we compare the efficiency of our methods and compare the results with the interval Householder method [11]. Mathematics Subject Classification (2010): 65G06.

Solvability of interval linear equations and data analysis under uncertainty

Automation and Remote Control, 2012

Consideration was given to the problem of recognizing the solvability (nonemptyness 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.

Computing Enclosures of Overdetermined Interval Linear Systems

Cornell University - arXiv, 2013

This work considers special types of interval linear systems-overdetermined systems. Simply said these systems have more equations than variables. The solution set of an interval linear system is a collection of all solutions of all instances of an interval system. By the instance we mean a point real system that emerges when we independently choose a real number from each interval coefficient of the interval system. Enclosing the solution set of these systems is in some ways more difficult than for square systems. The main goal of this work is to present various methods for solving overdetermined interval linear systems. We would like to present them in an understandable way even for nonspecialists in a field of linear systems. The second goal is a numerical comparison of all the methods on random interval linear systems regarding widths of enclosures, computation times and other special properties of methods.

Solving Interval Linear Systems Is NP-Hard Even When All Inputs Are Known With the Same Accuracy

2013

It is known that in general, solving interval linear systems is NP-hard. There exist several proofs of this NP-hardness, and all these proofs use examples with intervals of different width-corresponding to different accuracy in measuring different coefficients. For some classes of interval linear systems with the same accuracy, feasible algorithms are known. We show, however, that in general, solving interval linear systems is NP-hard even when all inputs are known with the same accuracy.

Interval estimations of solution sets to real-valued systems of linear or non-linear equations

Reliable computing, 2002

This is a second paper devoted to present the Modal Interval Analysis as a framework where the search of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this second paper is to show that the modal intervals are a suitable tool to approach problems where logical references appear, for example, to find interval estimates of a special class of generalized sets of solutions of real-valued linear and non-linear systems, the UE-solution sets.

A Comparison of some Methods for Solving Linear Interval Equations

SIAM Journal on Numerical Analysis, 1997

Certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined. We extend a proposed technique of Hansen and Rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix A = [A, A] is an H-matrix; when A is centered about a diagonal matrix, these bounds are the smallest possible (i.e., the bounds are then the solution hull). Hansen's scheme also computes the solution hull when the linear interval system Ax = b = [b, b] is such that A is inverse positive and b = −b = 0. Earlier results of others also imply that, when A is an M-matrix and b ≥ 0, b ≤ 0, or 0 ∈ b, interval Gaussian elimination (IGA) computes the hull. We also give a method of computing the solution hull inexpensively in many instances when A is inverse positive, given an outer approximation such as that obtained from IGA. Examples are used to compare these schemes under various conditions.

Detecting Unsolvability of Interval Linear Systems

In this paper we deal with detection of unsolvability of interval linear systems. Various methods based on existing algorithms or on existing sufficient conditions are developed. The methods are tested on a large variety of random systems and the results are visualized. The two strongest sufficient conditions are proved to be equivalent under a certain assumption. The topic of detecting solvability is also touched upon.

Interval Linear Algebra and Computational Complexity

Springer Proceedings in Mathematics & Statistics, 2017

This work connects two mathematical fields-computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebraregularity and singularity, full column rank, solving a linear system, deciding solvability of a linear system, computing inverse matrix, eigenvalues, checking positive (semi)definiteness or stability. We discuss these problems and relations between them from the view of computational complexity. Many problems in interval linear algebra are intractable, hence we emphasize subclasses of these problems that are easily solvable or decidable. The aim of this work is to provide a basic insight into this field and to provide materials for further reading and research.

Complexity and computability of solutions to linear programming systems

International Journal of …, 1980

Through key examples and constructs, exact and approximate, complexity, computability, and solution of linear programming systems are reexamined in the light of Khachian's new notion of (approximate) solution. Algorithms, basic theorems, and alternate representations are reviewed. It is shown that the Klee-Minty example has never been exponential for (exact) adjacent extreme point algorithms and that the Balinski-Gomory (exact) algorithm continues to be polynomial in cases where (approximate) ellipsoidal "centeredcutoff" algorithms (Levin, Shor, Khachian, Gacs-Lovasz) are exponential. By "model approximation," both the Ktee-Minty and the new J. Clausen examples are shown to be trivial (explicitly solvable) interval programming problems. A new notion of computable (approximate) solution is proposed together with an a priori regularization for linear programming systems. New polyhedral "constraint contraction" algorithms are proposed for approximate solution and the relevance of interval programming for good starts or exact solution is brought forth. It is concluded from all this that the "imposed problem ignorance" of past complexity research is deleterious to research progress on "computability" or "efficiency of computation."