Two Iterative Methods for Solving Linear Interval Systems

Copyright © 2018 Esmaeil Siahlooei and Seyed Abolfazl Shahzadeh Fazeli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conjugate gradient is an iterative method that solves a linear system Ax = b, whereA is a positive definite matrix. We present this new iterative method for solving linear interval systems ?̃??̃? = b, where ?̃? is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of ?̃??̃? and b at every step while the norm is sufficiently small. In addition, we present another iterative method that solves ?̃??̃? = ?̃?, where ?̃? is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution ?̃? for linear interval systems, where ?̃??̃? = b; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.


Introduction
Solving system of linear equations is a well-known problem in linear algebra.Many practical problems are modeled as system of linear equations.These problems have been studied by many scientists and several methods have been proposed to solve them [1,2].But, in everyday life, measuring instruments just estimate values, and usually measured values are not accurate.Sometimes for more accuracy, intervals are used to represent the actual values.For example, the length of a metal rod is estimated between 24.221 and 24.222 centimeters, or temperature measured by a thermometer is between 27.6 and 27.7 Celsius.
There are many types of uncertainty and there are many different mathematical systems that calculate uncertainties [3], e.g., rough sets theory [4], fuzzy numbers [5], probability theory [6], interval valued numbers [7], and dual fuzzy numbers [8].In this paper, uncertainty is considered as interval numbers.Some articles used this model to solve their problems [9,10].
Using interval numbers in algebra was initially developed in the mid-1960s.In 1966, Moore presented his book on interval analysis [11].Then Hansen offered a solution on interval linear algebraic equations [12].Then many authors published their methods for solving linear interval systems, such as Neumaier, Abolmasoumi and Alavi, Nirmala, and Ganesan [13][14][15].Nowadays, interval analysis methods have been applied to engineering problems, such as dynamic response analysis [16,17], geotechnical structures [18], and control systems [19].
In this paper, we introduce two new iterative methods for solving a linear interval system of equations that is a linear system involving uncertain coefficients appearing as interval numbers.The solution of this system is an interval vector.
The present paper is organized as follows.The basic definitions related to interval numbers and linear interval systems are discussed in Section 3. In Section 3.1, we introduce two additional interval operations.These are inverse operations of "−" and "+." Section 3.2 recalls some properties of interval numbers and interval arithmetic.Also, it introduces some new properties on introduced operations.Next, we review conjugate gradient method and then present a new method for solving linear interval systems based on conjugate gradient method.Another new iterative method using steepest descent idea is proposed in Section 3.4.Section 4 shows the experimental results of the proposed methods and discusses the accuracy and efficiency of the new methods.

Interval Arithmetic
Interval numbers and arithmetic are explained in [7,11,13,20].We review main definition here.Given ,  ∈ R, where  ≤ , the real bounded set is called a proper interval.The set of all proper intervals on R is denoted by IR.In this paper, all elements in IR are shown with a hat, i.e., x.
The magnitude is defined to be the maximum value of || for all  ∈ x.Thus The width and midpoint of x are denoted by (x) and (x), respectively, and defined as An interval number x is defined as a subset of interval number ỹ when  ≥  and  ≤  and denoted by x ⊆ ỹ.Equality held when x = ỹ.Each real number  ∈ R can be viewed as a special interval number ã = [, ].The interval [, ] simply can be denoted by  without confusion.
For each binary operation * which is defined on R, a binary operation * is defined on IR as for x, ỹ ∈ IR.Note that we will use operator * for both interval and real numbers when there is no confusion.Therefore, basic interval arithmetic is defined in the following.Let x, ỹ ∈ IR; then x − ỹ = [ − ,  + ] , x × ỹ = [min (, , , ) , max (, , , )] .
An interval vector is a vector whose elements are interval numbers.Similarly, an interval matrix is a matrix whose elements are interval numbers.
Consider s ∈ IR  ,  ∈ R  , Ã ∈ IR × , and  ∈ R × .We say  ∈ s if each element of vector  belongs to corresponding element of s and  ∈ Ã if each element of matrix  belongs to the corresponding element of Ã.
Norm of interval vector s is defined in [21] as We define a square interval matrix as diagonally magnitude dominant or for simplicity diagonally dominant if Proof.Equation ( 13) directly follows (12); just set   = Mig(ã  ) and   = Mag(ã  ) when  ̸ = .If (13) holds then and Finally, General matrix form of linear interval systems can be written as follows: where Ã is an interval matrix, x is an interval vector solution, and b is an interval vector.
To solve a linear interval system, there exist some different approaches [22][23][24].Four of the well-known approaches are as follows: (1) Find an interval vector x such that, for all  ∈ x,  ∈ Ã and  ∈ b exist such that  =  (united solution set).
(2) Find an interval vector x such that, for all  ∈ x, for all  ∈ Ã,  ∈ b exists such that  =  (tolerable solution set).Equivalently, (3) Find an interval vector x such that, for all  ∈ x, for all  ∈ b,  ∈ Ã exists, such that  =  (controllable solution set).Equivalently, (4) Find an interval vector x such that multiplication Ãx is equal to b (exact solution set).
Our proposed method calculates a solution for Ãx = b in the sense of the exact solution, where Ã is a diagonally dominant interval matrix.
For ã, b ∈ IR * , the operation "⊖" is defined in [20,33] as For ã, b ∈ IR, we define the operation "⊘" as Note that this definition can handle many cases where 0 belongs to b, and just in three cases compute undefined or unbounded solution.This property lets us design algorithms with high stability and consistent when working with interval values that contain 0.
In the following two theorems, we show that operations "⊖" and "⊘" are inverse of operations "+" and "×," respectively [34].In cases that x * is not unique, x * is considered as the longest interval as possible.In the above example, x must be Proof.Suppose ã × x = b.Before starting the proof, note that multiplication result, b, depends on the sign of ã and x.Table 1 shows sign of b related to sign of ã and x and also shows formulas of multiplication result.
Suppose that, given ã, b ∈ IR, ã × x = b, and  ≤ 0 and 0 ∈ b.Now, from Table 1, it is observed that the following must occur: This means in this case x is uniquely calculated by inverse operations of "×."For other cases except cases with condition 0 ∈ ã, one can provide similar proofs.
For cases with condition 0 ∈ ã, there is an ambiguity.From Table 1, there are three cases in column 0 ∈ ã.They have the same conditions  ≤ 0 ≤ .The solution can be in forms 0 ≤ , 0 ∈ x, and 0 ≥ , respectively.The solution should be in form of 0 ∈ x, because it is the longest possible interval solution and includes both of the others.Now, find  and . and Then, If   ̸ = max{/, /}, then   > max{/, /}, and   > max{/, /} ≥ .This contradicts with  = max{  ,   }.Hence Similarly, and therefore, we have x =  ⊘ .

Some Properties of the Interval Arithmetic Operations.
In this section, first we recall some basic properties of interval arithmetic.For ã, b, c, d ∈ IR the following rules hold [35]: For ã, b, c ∈ IR we have In relation (35), equality holds when  =  or  ×  ≥ 0.
and using definition of ⊘ we have Define B = ãb and C = ãc; then b = B ⊘ ã and c = C ⊘ ã.So, and hence (40) holds.

Modified Conjugate Gradient Method for Linear Interval
Systems.Conjugate gradient method is one of the most useful techniques in solving iterative methods for solving linear system of equations, whose matrix is symmetric and positive definite.The conjugate gradient method was proposed by Hestenes and Stiefel in 1952 as an iterative method for solving linear systems with positive definite coefficient matrices [36].The conjugate gradient algorithm for linear system of equations can be briefly described as follows.
Consider system  =  with solution .Start with an initial estimate  0 of .At step  use   to new estimate  +1 of  which is closer to .At each step residual   =  −   is computed and used as a measure of the goodness of the estimate   .The algorithm is detailed below for solving  = , where A is a symmetric, positive definite matrix.
We generalize this method for linear interval systems when the interval matrix Ã is symmetric, Ã = Ã for all 1 <= ,  <= , and b is an interval vector.There are articles describing and analyzing conjugate gradient method such as [37][38][39].These articles illustrated which operator was used in original role, or acted in inverse operator.The new algorithm is obtained from original conjugate gradient method by modifying and replacing all operations with related interval operations; also all operators acting in the role of inverse operators are replaced by inverse of interval operators.In addition, we use interval vector norm instead of real vector norm.
In the first modification on conjugate gradient, we change some operation "−," with inverse operation "⊖" whenever needed.In Step 2 of Algorithm 1, "−" is applied for computing difference of  and  ×  0 .In Step 7 operation "−" is used, but at proof of conjugate gradient method in [39],   is replaced with expression ( +1 +(  ×   )).So, the operation "+" is summed over  +1 and   ×   .This means "−" in Step 7 is used for inverse of "+." Therefore we use the operation "⊖" instead of "−" in steps 2 and 7.
The second modification is similar to the first modification.We change operation "/" with inverse operation "⊘."In steps 5 and 8 of Algorithm 1 operation "/" is used as inverse of multiplication in proof of conjugate gradient [39].
The third modification, norm(r  ) 2 , is used instead of      .In real linear algebra, expression      equals square of norm2 of   , and value of      is related to amount of error.We replace this with norm(r  ) 2 that has the same meaning.The interval modified conjugate gradient method is the result of these modifications and is shown in Algorithm 2.

Interval Steepest Descent Method for Linear Interval
Systems.First, we define derivative on interval functions.In differential algebra, the derivative of () with respect to (1) function CG (, ,  0 ,  ) ⊳ Input: measures the sensitivity to change value of () with respect to .In mathematical terms According to this, the derivative of an interval function (x) with respect to x is denoted as   (x) or   x(x) or (/x)(x) and is defined by the following formula: Immediately from definitions ( 44) and ( 45) and the properties discussed, one can obtain the following properties of derivative and partial derivative of interval functions: Now consider the linear interval system Ãx = b when Ã ∈ IR × is a symmetric interval matrix, and b ∈ IR  is a known interval vector.Let Φ(x) be an interval function The gradient of Φ(x) with respect to x is the vector of partial derivatives as and then, from (52) Assume x * is the solution of linear interval system Ãx = b; then Then The derivative of Φ(x + α) with respect to α is If   Ãx + α  Ã ⊖   b equals zero, then one can obtain α as Now α moves Φ(x) to Φ(x + α) with respect to direction  when x is fixed.So Φ(x+ α) ⊆ Φ(x) and also (Φ(x+ α)) ≤ (Φ(x)).
In the following, we assume that x0 is a given vector as initial guess.

Results and Discussion
In this section, some numerical examples show how our iterative methods can solve interval systems.Accuracy of produced solutions is measured by a kind of norm of residual vector where x is the produced solution in th iteration.The norm used in this section is then maximum of magnitude of all residual vector elements.We examine our methods with a famous system and then examine them with random matrices with several dimensions.Our results illustrate the effectiveness of our algorithms.−14, 14] [−9, 9] Let x0 = [0, 0, 0]  be initial estimation and  = 10 −3 be tolerance.Interval modified conjugate gradient method finds the estimation x at iteration 13.However, interval steepest descent finds solution ỹ at iteration 4 when x and ỹ are found at 26 ℎ and 11 ℎ iterations, respectively.
Tables 2 and 3 show values x and ỹ with their maximum of the magnitude of the residual vector at  ℎ iteration, respectively.
The method presented in [13,15,25,26,30] could not solve this system where 0 belongs to their elements to the right-hand side vector, because their methods cannot resolve divide-by-zero issues.
This system is solved in [22,40], and solutions obtained by them are We considered as initial solution and  = 10 −3 as tolerance.
The generalized conjugate gradient method, after 7 iterations obtained, is The results of these experiments are shown in Figure 1.This figure shows the convergence of the two proposed methods.As it can be seen, the interval steepest descent method has faster convergence than interval modified conjugate gradient method.
This example illustrates that our methods can solve systems where some of their elements have 0 in their intervals, without producing divide-by-zero error.
which has the exact solution Example 10.We examined the two proposed methods with a number of random linear interval systems Ãx = b when Ã ∈ IR × is a random interval symmetric diagonally dominant matrix.Tables 4 and 5 show the number of iterations needed by our presented methods to find the solution with tolerances  = 10 −3 and  = 10 −7 for random interval systems with various dimensions.Note that there are infinite random matrices with dimension  × , and number of iterations just does not depend on .It depends on , the coefficient matrix, and right-hand side vector.However, each row of these tables just shows number of iterations for random linear systems.These matrices are not special matrices, and these tables show a good view of dependency of dimension and approximation of number of iterations.These tables also show needed iterations in interval steepest descent are less than the interval modified conjugate gradient method.

Conclusion
In this paper, we presented two operators "⊖" and "⊘" as inverse operators of "+" and "×" in interval arithmetic.The proposed operation "⊘" can solve many equations in the form ãx = b, where 0 ∈ b, without concerning divide-by-zero.
Using these two operators, we proposed two iterative methods for computing exact solution of linear interval systems.The first method, based on the conjugate gradient method, replaced real operations with interval operations.The second method uses steepest descent idea to solve linear interval systems.We presented the interval function Φ(x).Then using analysis of Φ(x), it was proved that the second method is convergent.
Our proposed methods are using operations "⊖" and "⊘," solving systems, specially systems which have 0 belonging to some of their elements without the divide-by-zero issues.This is the main advantage of our methods.Also, our results showed the efficiency and fast convergence of the proposed methods.In addition, one can control and improve the accuracy of the solution, by control of the tolerance parameter .
The important application of interval number and interval analysis is in the modeling of uncertain values.There are many uncertain values in fields of engineering sciences that can be modeled in interval numbers.Uncertainty in some of these problems can be modeled in interval numbers, for example, the interval methods applied to linear state feedback control of uncertain values [41], reliability optimization under parameter uncertainty [42], decision-making under uncertainty [43], and perturbation analysis [44].Our next research is to consider the applicability of our method to some of the engineering problems.

Figure 1 :
Figure 1: Tightness of solutions and convergences of our methods.

Example 9 .
Consider Example 8 with a different b

Table 1 :
Interval multiplication in detail.
derivative with respect to one variable when all other variables are fixed.The partial derivative  with respect to x is denoted as   x (x) and is defined as Let x ∈ IR  and (x) = (x 1 , x2 , . . ., x ) be a IR  → IR function.The partial derivative of a multivariable interval function is a

Table 2 :
Intermediate results of interval modified conjugate gradient method.

Table 3 :
Intermediate results of interval steepest descent method.

Table 4 :
Iterations needed in interval steepest descend method solving random linear interval systems.

Table 5 :
Iterations needed in interval modified conjugate gradient method solving random linear interval systems.