Solving fully interval linear systems of equations using tolerable solution criteria (original) (raw)
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Method for the Solution of Interval Systems of Linear Equations
International Journal of Advances in Applied Sciences, 2013
In this paper, we discuss the solution of interval system of linear equations and proposed a new method for handling this type of system of linear equations. In this model we consider the coefficient matrix and the right vector as interval. Example problems are given to have the efficiency and powerfulness of the proposed method.
Reliable Computing, 2006
Consider the systems of linear interval equations whose coefficients are affine-linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in structure mechanics. A direct method for computing a tight enclosure for the solution set is proposed in this paper. It is proved that for systems with real matrix and interval right-hand vector the method generates the hull of the solution set. For such systems an explicit formula for the hull is also given. Finally some numerical examples are provided to demonstrate the usefulness of the method in structure mechanics.
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.
Solving Interval Linear Equations with Modified Interval Arithmetic
British Journal of Mathematics & Computer Science, 2015
Gaussian elimination method is one of the widely used methods for solving linear equations. An interval version of Gaussian elimination method has been used by simply replacing each real arithmetic step by the corresponding interval arithmetic step. Two interval arithmetics technique has been considered for modified interval arithmetics as well as several existing interval arithmetics. In this paper, modified interval arithmetic has been introduced based on two interval arithmetics technique. If we solve interval linear system of equations by existing interval arithmetic method the replacing solution in interval system of equations, the interval width is more than the interval width of right hand side intervals. On the other hand, applying modified interval arithmetic the interval width is less than interval width than previously obtained by existing interval arithmetic. Moreover, the closeness of interval width in system of equations to the right hand side is important so modified interval arithmetic is more effective and efficient for solving interval linear system of equations.
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.
Solving Linear Systems Using Interval Arithmetic Approach
2012
in this paper we discuss various classes of solution sets for linear interval systems of equations, and interval linear programming problems. And their properties, in this model we let the coefficient matrix and the right vector hands and the cost coefficient are interval. Interval methods constitute an important mathematical and computational tool for modeling real-world systems (especially mechanical) with bounded uncertainties of parameters, and for controlling rounding errors in computations. They are in principle much simpler than general probabilistic or fuzzy set formulation, while in the same time they conform very well to many practical situations. Linear interval systems constitute an important subclass of such interval models, still in the process of continuous development.
A method for outer interval solution of parametrized systems of linear equations
Consider the systems of linear interval equations whose coefficients are affine-linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in structure mechanics. A direct method for computing a tight enclosure for the solution set is proposed in this paper. It is proved that for systems with real matrix and interval right-hand vector the method generates the hull of the solution set. For such systems an explicit formula for the hull is also given. Finally some numerical examples are provided to demonstrate the usefulness of the method in structure mechanics.
Global Journal of Mathematical Sciences, 2014
The paper is a modificationofNguyen and Revol"s method for the solution set to the linear interval system. The presented methoddoes not require solving Kahan"s arithmetic which may be a hindrance to that of Nguyen and Revol"s method as Nguyen and Revol"s method relies mainly on interval data inputs.Our method under consideration first advances solutionusing real floating point LU Factorization to the real point linear system and then solves a preconditioned residual linear interval system for the error term by incorporating Rohn"s method which does not make use of interval data inputs wherein, the use of united solution set in the sense of Shary comes in handy as a tool for bounding solution for the linear interval system. Special attention is paid to the regularity of the preconditioned interval matrix. Numerical exampleis used to illustrate the algorithm and remarks are made based on the strength of our findings.
Subsquares Approach – A Simple Scheme for Solving Overdetermined Interval Linear Systems
Lecture Notes in Computer Science, 2014
In this work we present a new simple but efficient scheme-Subsquares approach-for development of algorithms for enclosing the solution set of overdetermined interval linear systems. We are going to show two algorithms based on this scheme and discuss their features. We start with a simple algorithm as a motivation, then we continue with a sequential algorithm. Both algorithms can be easily parallelized. The features of both algorithms will be discussed and numerically tested.