Solving Interval Linear Equations with Modi(cid:12)ed Interval Arithmetic

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.


Introduction
Computing solution of the usual linear system of equations is easy, but obtaining the solution of interval linear equations is much more complicated. In fact, Obtaining solution of interval linear equations is a challenging problem in interval analysis. This problem was first considered by [1]. Some researchers worked on uncertain linear system. They proposed several methods for solving linear equations under uncertainty. Some of them focused on fuzzy linear equations [2]. Some others found an enclosure of solutions of interval linear system ( [3]- [6]).
There are several methods for solving linear equations. Gaussian elimination method is one of them. An interval version of Gaussian elimination method can be obtained from simply replacing each real arithmetic step by the corresponding interval arithmetic step. Firstly, we review some definitions.
An interval number X is generally represented as [X, X] where X ≤ X. If X = X, then X will be degenerate.
If A and A are two matrices in R m×n and A ≤ A, then the set of matrices is called an interval matrix, and the matrices A and A are called its bounds. Center and radius matrices have been defined as A special case of an interval matrix is an interval vector which is a one-column interval matrix Interval arithmetic is defined in [7].
Let A I and b I be n × n interval matrix and interval vector,respectively. For an interval system of linear algebraic equations A I x = b I , the solution set is defined as If A I is regular, then for a matrix A ∈ A I and any vector b ∈ b I an ordinary linear system Ax = b has the unique solution.
The solution set S is a non-convex polyhedral set [1]. The main objective is to find interval solution of linear interval system that is to determine The narrowest interval vector containing the set S is called the interval hull of S. This set is generally not an interval vector. It is usually difficult to describe S. Because S is generally so complicated in shape [8], it is usually impractical to use it. Instead, it is a common practice to seek the interval vector x I containing S that the narrowest possible interval components. This interval vector is called the hull of the solution set or simply the hull. In other words, we need to imbed the solution set S into the minimal box in R n . This problem is known to be NP-hard [9] and complicated from computational viewpoint for large-scale systems. Therefore, many researchers generally compute only outer bounds for the hull ( [4], [5], [7], [10]- [15]). Some iterative approaches were established at this context as well as direct numerical methods that provide over bounding of x * ([6], [15]).
Suppose we solve interval linear system of equations by Gaussian elimination method with existing interval arithmetic. If we replace the obtained solution in interval equations of system, then the width of obtained interval is more than the width of the right hand side interval. Therefore, if closeness of obtained interval to the right hand side interval is important, then we can use modified interval arithmetic for solving interval linear system of equations. In this paper, we solve interval linear system of equations by Gaussian elimination with modified interval arithmetic and we show that resulted interval of substituting solution obtained through modified interval arithmetic cannot give rise to excess width than the existing interval arithmetic.

Review of Interval Arithmetic
In this section, we recall the existing and modified interval arithmetic.

Existing Interval Arithmetic
In this section, we review the existing interval arithmetic. If • ∈ {+, −, ×, ÷} denotes any one of these operations for arithmetic on real numbers x and y, then the corresponding operation for existing arithmetic on interval numbers X and Y is Therefore, this definition produces the following rules for generating the endpoint of X • Y .
we exclude division by an interval containing 0.

Modified Interval Arithmetic
In this section, we recall the modified interval arithmetic [16]. Let * ∈ {⊕, ⊖, ⊗, ⊘} be any one of the operations for arithmetic on real numbers x and y, then the corresponding operation for modified arithmetic on interval numbers X and Y is where k = min{(m(X) * m(Y )) − α, β − (m(X) * m(Y ))} and m(·) is the midpoint of interval. α and β are the endpoint of the interval X • Y .

Gaussian Elimination Method
There are several methods for solving linear equations that can be labeled as Gaussian elimination. An interval version of any of them can be obtained from one using ordinary real arithmetic by simple replacing each real arithmetic step by the corresponding interval arithmetic step. One standard method involves converting the coefficient matrix to the upper triangular matrix and hence the interval solution vector computed using Gaussian elimination contains the set S. Suppose the elimination procedure does not fail because of division by an interval containing zero. The solution X I computed by using interval Gaussian elimination contain S [8].
Unfortunately, Using an interval version of classic Gaussian elimination generally does not yield a suitable algorithm because of -rounding errors in interval computations.
-division by an interval containing zero.
-dependence among generated intervals. Now, Let X EI and X M I be solution obtained through solving interval linear equations A I x = b I with the existing and modified interval arithmetic, respectively. Also, let b EI and b M I be intervals resulted from replacing X EI and X M I in equations of system, respectively. We define To demonstrate the effectiveness of the proposed approach in solving the interval systems of linear equations, the section presents a comparative study on the relative performance of the proposed method with the other approach. Some interval computations have been done by the interval toolbox INTLAB v6 [17]. Solutions of this system with existing and modified interval arithmetic are given in Table 1 on page 6. From Table 1, we conclude the solution obtained through modified interval arithmetic is closer to the right hand side intervals of system than existing interval arithmetic and hence the norm of error will be smaller than that one. Exact solution and the solution resulted from the existing and modified interval arithmetic is shown in Fig. 1 [4,6].
Solutions of this system with existing and modified interval arithmetic are given in Table 2 on page 6.
By considering Table 2, the solution resulted from modified interval arithmetic is closer to the right hand side intervals of system than the existing interval arithmetic and hence the norm of error will be smaller.

Conclusion
There are several methods for solving linear equations. Gaussian elimination method is one of them. An interval version of Gaussian elimination method can be obtained from simply replacing each real arithmetic step by the corresponding interval arithmetic step. In this study, interval Gaussian elimination method based on modified interval arithmetic has been considered. If we solve interval linear system of equations by the modified interval arithmetic, the replacing solution in interval system of equations, the interval width is less than the interval width previously obtained by existing interval arithmetic. Also, the closeness of interval width in system of equations to the right hand side is important so modified interval arithmetic is more effective for solving interval linear system of equations.