Properties of Interval Vector-Valued Arithmetic Based on gH-Difference

Mathematical Computation March 2015, Volume 4, Issue 1, PP.7-12

Properties of Interval Vector-Valued Arithmetic Based on gH-Difference Juan Tao 1, Zhuhong Zhang 2, # 1. Department of Mathematics, College of Science, Guizhou University, Guiyang, Guizhou 550025, China 2. Department of Information and Communication Engineering, College of Big Data and Information Engineering, Guizhou University, Guiyang, Guizhou 550025, China #

Email: sci.zhzhang@gzu.edu.cn

Abstract Although the arithmetic rules of interval number can be in parallel extended for interval vector, many of their properties are invalid in that maybe two interval vectors are incomparable. This work investigates some arithmetic properties in the interval vector-valued space, after developing several operational rules between interval vectors. Relying upon the generalized interval vector-valued Hukuhara difference and derivative, some important properties and especially the associative and distributive laws are derived by means of the versions of width vector and vector-valued partial order. Finally, one property of such Hukuhara derivative is acquired for interval vector-valued functions. These will be applied to interval algebraic equations, interval programming, interval dynamical systems and so forth.

Keywords: Generalized Hukuhara Difference; Generalized Hukuhara Derivative; Interval Arithmetic; Interval Vector-valued Space

1 INTRODUCTION Interval analysis is a fundamental tool for studying many engineering problems with uncertain bounded parameters, such as interval dynamical systems and engineering optimization design. Moore proposed the interval arithmetic theory for the first time in 1966[1]. Since then, many researchers have made great contributions to apply the interval theory to some problems, such as interval methods of interval equations[2-4], interval programming[5] and so on. In the interval arithmetic, it is well known that the Minkowski’s addition of two interval numbers A and B is not an invertible operation. However, the inversion of addition is very important in the interval analysis when solving interval differential equations. An extensively adopted inversion of addition is Hukuhara difference (H-difference, write AΘB) proposed by Hukuhara[6], but such H-difference does not always exist. A necessary but not sufficient condition, which AΘB is meaningful, is that A and B satisfy w(A)  w(B), where w(A) represents the width of A. After that, Stefanini[7-9] proposed the concept of generalized Hukuhara difference (gH-difference, write AΘgB). From then on, interval-valued differential equations have been preliminarily investigated [10-13].

In the present work, some arithmetic rules of interval number are extended to probe into some properties of interval vector. Especially, the properties of associative and distributive laws between interval vectors are acquired, relying upon a partial order relation between vectors. In addition, the concept of interval vector-valued derivative is also given, while an important property is acquired for interval vector-valued function. These will help us resolve interval programming and interval dynamical systems.

2 PRELIMINARIES Interval arithmetic as an important fundamental tool of interval dynamical systems involves in five algebraic rules for interval number- addition, subtraction, interval multiplication, division and scalar multiplication. In other words, for two any given bounded and closed intervals A and B, these operations are usually defined as follows[14]: -7www.ivypub.org/mc

A + B = {a + b︱a  A, b  B}; A – B = A + (–B) = {a – b︱a  A, b  B};

(1) (2)

A·B = AB = {a b︱a  A, b  B};

(3)

A  B = A B –1, B –1 = { 1b ︱b  B}, 0  B;

(4)

k A = {k a︱a  A}, k  R.

(5)

In general, A minus A does not equal 0 in (2), but the cancellation law of interval number holds, namely A + C = B + C iff A = B. Since A – A  0, many properties of the real number theory can not be extended to interval analysis. To overcome the drawback of the above subtraction, a new subtraction rule, originally proposed by Stefanini in 2008, can help us study interval number.

Definition 1[9]. The subtraction of A minus B, so-called gH-difference, is defined by A Θg B = C,

(6)

where C satisfies that A = B + C iff w(A)  w(B) , or that B = A + (-1)C iff w(A)  w(B).

In the above definition, w(A) represents the width of interval A. Notice that C as in (6) always exists. Since gHdifference has no restriction to intervals A and B, it can derive out many useful properties. For example, Stefanini provided the following properties[7]:  (a) A Θg A = 0;  (b) A Θg (A+B) = -B, (A+B) Θg B = A;  (c) A Θg B = (–B) Θg (–A) = – (B Θg A);  (d) if w(B)  w(A), then A + (B Θg A) = B; conversely, B – (B Θg A) = A. In addition, one can derive the following associative law of interval number based on gH-difference.

Lemma 1. (A + B) Θg C = A + (B Θg C) iff w(C)  w(A+B) and w(C)  w(B).

Proof. Write D = (A + B) Θg C and E = B Θg C. If w(C)  w(A + B) and w(C)  w(B), one can derive A + B = C + D and B = C + E, i.e., A + C + E = C + D, and hence D = A + E. Conversely, if it is false that w(C)  w(A + B) and w(C)  w(B), three cases are included: (i) w(A + B) < w(C) and w(B) < w(C ), (ii) w(C)  w(A + B) and w(B) < w(C ), (iii) w(A + B) < w(C) and w(C)  w(B). Case (i): one can imply C = A + B + (–1)D and C = B + (–1)E, namely A + (–1)D = (–1)E. Again since D = A + E, we have A + (–1)(A + E) = (-1)E, and hence A + (–1)A = 0. This yields contradiction because of A + (–1)A  0. Case (ii): one can get A + B = C + D and C = B + (–1)E. So, E + (–1) E = 0.

This results in contradiction. Case (iii) does not hold, due to w(A+B) < w(B). Summarily, the above conclusion is true.  We next study the distributive law of interval number under certain assumptions. For two interval numbers B and C, write B = [ b , b ] and C = [ c , c ]. Introduce a partial order relation between B and C, namely B  C  b  c and b c .

Lemma 2. The following distributive law holds: A(B Θg C) = AB Θg AC,

(7)

if one of the following conditions holds:  (a) w(B)  w(C), and 0  C  B, B  C  0 or symmetric B and C;  (b) w(B)  w(C), and 0  B  C, C  B  0 or symmetric B and C. Proof. Write D = B Θg C and E = AB Θg AC. (a) it follows from w(B)  w(C) that B = C + D. As related to one of the other conditions of (a), B and C can ensure that C and D have the same sign or are symmetric. Thereby, AB = AC + AD, and hence w(AB)  w(AC). This yields AB = AC + E. Similarly, following condition (b), one can also prove that -8www.ivypub.org/mc



the conclusion is true.

3 INTERVAL VECTOR ARITHMETIC AND BASIC PROPERTIES Let  stand for the n-dimensional real interval vector space, i.e.,  = {X = (X1, X2, …, Xn) | Xi = [ xi , xi ] , 1  i  n}.

We say that X is a degenerate interval vector if x i = xi with 1  i  n. X is said to be symmetric if x i = - xi with 1  i  n. Let A, B, C and D be four interval vectors in  with A = (A1, A2, …, An), B = (B1, B2 ,…, Bn), C = (C1, C2, …, Cn) and D = (D1, D2, …, Dn). We naturally extend the above algebraic operation rules for interval number, (+,Θg, ·), into the following rules for interval vector:  (a) A + B = (A1 + B1, A2 + B2,…, An + Bn);

(8)

 (b) A Θgv B = (A1 Θg B1, A2 Θg B2, ..., An Θg Bn);

(9)

 (c) AB = in1 AiBi;

(10)

 (d) kA = (kA1, kA2, …, kAn), kR.

(11)

Stefanini[7] acquired some properties based on the width of interval number, e.g., see section 2. However, it is almost impossible to extend them to the n-dimensional interval vector space if adopting the version of the width of interval vector, i.e., w(A)=maxiw(Ai). Therefore, we introduce the version of width vector: W(A) = (w(A1), w(A2), …, w(An)),

(12)

and analyse the properties of interval vector relying upon the following partial order relation[15], namely for X, Y, X  Y  Xi  Yi (1  i  n), X  Y  Y  X,

(13)

X < Y  X  Y, X  Y,

(14)

X || Y  X  Y or X  Y,

(15)

where Xi  Yi  x i  y i and xi  yi with 1  i  n.

Based on the above order relations and (12), we notice that the operation of Θ gv is similar to that of Θ g, and meanwhile A Θgv B is always meaningful for any A and B in . As we know, there exists only one relation between W(A) and W(B): W(A)  W(B), W(A)  W(B) or W(A) ╫ W(B), where the last one represents that W(A) and W(B) are not comparable. Write C = A Θgv B. We easily imply that A = B + C iff W(A)  W(B), and B = A + (-1)C iff W(A)  W(B). Thereby, when W(A)  W(B) or W(A)  W(B), C is easily computed. However, when the latter one takes place, C can only be computed by (9).

In general, (A+B)ΘgvC is not equivalent to A+(BΘgvC). For example, we take A = ([1,2], [3,4]), B = ([1,2], [3,4]) and C = ([1,4], [3,6]). Hence (A+B)ΘgvC = ([0,1], [2,3]) and A+(BΘgvC) = ([–1,2], [1,4]). Whereas, by means of the order relation of interval vector, the following associative law holds under certain weak assumptions: Theorem 1. (A + B) Θgv C = A + (B Θgv C) iff W(C)  W(A+B) and W(C)  W(B).

Proof. Write D = (A + B) Θgv C and E = B Θgv C. If W(C)  W(A+B) and W(C)  W(B), one can derive A + B = C + D and B = C + E, i.e., A + C + E = C + D, and hence D = A + E. Conversely, if D = A + E, then through (8) and (9) we gain (Ai + Bi) Θg Ci = Ai + (Bi Θg Ci), 1  i  n, and then Lemma 1 follows w(Ci)  w(Ai + Bi) and w(Ci)  w(Bi). This yields W(C)  W(A+B) and W(C)  W(B).  Theorem 2. (A+B)Θgv(C+D) = (AΘgvC)+(BΘgvD), provided that one of the following conditions holds:  (a) W(A)  W(C), W(B)  W(D);  (b) W(A)  W(C), W(B)  W(D).

Proof. Write E = (A + B) Θgv (C + D), F = A Θgv C, and G = B Θgv D. Under condition (a), one can derive A = C + F and B = D + G, i.e., A + B = (C + F) + (D + G). Furthermore, it follows from such assumption that W(A + B)  W(C + D). Therefore, this yields A + B = C + D + E, and hence E = F + G, i.e., the conclusion is true. On the other hand, -9www.ivypub.org/mc

under condition (b), one can show W(A + B)  W(C + D), and thus C + D = A + B + (-1)E. Further, such condition follows C = A + (–1)F and D = B + (–1)G, and therefore C + D = A + B + (–1)(F + G). Summarily, one gets E = F + G. This hints that the conclusion is true.  Additionally, Minkowski’s subtraction satisfies A – B = A + (–B), but it is not true that A Θgv B = A + (Θgv B). For example, take A = ([1, 2], [3, 4]) and B = ([2, 4], [3, 5]), it is to see that A Θgv B = ([–2, –1], [–1, 0]) and A + (Θgv B) = ([-3, 0], [-2, 1]). However, the following property is true.

Theorem 3. AΘgvC = (AΘgvB)+(BΘgvC), only when one of the following conditions is true:  (a) W(A)  W(B)  W(C);  (b) W(A)  W(B)  W(C).

Proof. Write D = A Θgv C, E = A Θgv B, and F = B Θgv C. If W(A)  W(B)  W(C), one can derive that A = C + D, A = B + E and B = C + F. i.e., C + D = C + F + E. Therefore, this yields D = E + F. On the other hand, if W(A)  W(B)  W(C), we obtain that C = A + (–1)D, B = A + (–1)E and C = B + (–1)F, i.e., A + (–1)D = A + (–1)E + (–1)F. Thus, this hints D = E + F. Summarily, the conclusion is true.  For the conventional interval arithmetic, the interval distributive law of multiplication to addition or subtraction does not always hold. At the same time, such distributive law is also false for multiplication to gv-difference. For example, when taking A = ([1, 2], [1, 2]), B = ([0, 2], [1, 3]) and C = ([–2, –1], [–1, 0]), we obtain A(B Θgv C) = ([2, 6], [2, 6]) and AB Θgv AC = ([4, 5], [3, 6]). However, relying upon the partial order relation of interval vector, we get the following property: Theorem 4. The following distributive law holds: A(BΘgvC) = ABΘgAC,

(16)

if one of the following conditions holds:  (a) W(B)  W(C), and one of 0  C  B, B  C  0 and symmetric B and C;  (b) W(B)  W(C), and one of 0  B  C, C  B  0 and symmetric B and C. Proof. Condition (a) or (b) yields that Bi and Ci satisfies one of the conditions in Lemma 2 through (12) and (13).

Thereby, Lemma 2 follows Ai (Bi Θg Ci) = AiBi Θg AiCi with 1  i  n, and accordingly the conclusion holds by means of (10).  We next develop the relation between scalar multiplication and gv-difference: Theorem 5. For any given kR and B, C  , if B || C, then the following formula always holds: k(B Θgv C) = kB Θgv kC.

(17)

Proof. Write D = B Θgv C and E = kB Θgv kC. If W(B)  W(C), one can see that W(kB)  W(kC). This follows that B = C + D and kB = kC + E. Hence, we have kC + E = kB = kC + kD. This implies E = kD. In addition, if W(B)  W(C), one can see that C = B + (–1)D and W(kB)  W(kC). This shows that kC = kB + (–1)kD and kC = kB + (–1)E, and hence kD = E. Thus, the conclusion holds.  Example 1. Theorem 1 can be applied to solve the following linear interval vector equation: A + X = B,

(18)

where A and B belong to  and X is an interval vector solution to be determined satisfying (18). We notice that there exists a solution for (18) iff W(A)  W(B), namely X = B Θgv A. In fact, if W(A)  W(B), we can obtain W(A)  W(A + B) and A + (B Θgv A) = (A + B) Θgv A = B through Theorem 1. Conversely, if equation (18) has a solution, then we have Ai + Xi = Bi, with 1  i  n, where Xi is the i-th interval number of interval vector X. If it is false that W(A)  W(B), then we can see W(A) > W(B) or W(A) ╫ W(B). In such case, there exists i0, 1  i0  n, such that w( Ai )  w( Bi ) , and hence w( Ai )  w( Bi )  w( Ai  X i ) . This yields contraction.  0

0

0

0

0

0

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4 THE DERIVATIVE OF INTERVAL VECTOR-VALUED FUNCTION AND BASIC PROPERTIES Stefanini and Bede[10] gave the concept of generalized Hukuhara derivative (gH-derivative) of interval-valued function in terms of the concept of gH-difference. Naturally, in the case of interval vector, we can also develop a similar concept (write it as gv-derivative). In other words, for an interval vector-valued function F: [a, b], the gv-derivative at x0 is defined as follows: F (x0) = lim 1h [F(x0 + h) Θgv F(x0)]. h 0

(19)

If F (x0)   exists and satisfies (19), we say that F is gv-differentiable at x0 and that F( x0) is the gv-derivative of F(x) at x0. In order to study the properties of the gv-derivative, write F(x) = ( F1(x), F2(x), …, Fn(x)) and |F(x)| = (|F1(x)|, |F2(x)|, …, |Fn(x)|), where Fi(x)= [ F i ( x), Fi ( x)] with 1  i  n. As we know through (9), one can see that if F is gv-differentiable, then one can imply F i(x) = [mini{ F i ( x), Fi ' ( x) }, maxi{ F i ( x), Fi ' ( x) }], with 1  i  n. '

'

(20)

Further, introduce the version of midpoint vector, M(F(x)) = (m(F1(x)), m(F2(x)), …, m(Fn(x))),

(21)

where m(Fi(x)) is the midpoint of interval number Fi(x). Relying upon (12) and (21), F(x) can be represented by its midpoint and width vectors, i.e., F(x) = <M(F(x)), W(F(x))>. Hence, if F is gv-differentiable, then the gv-derivative of F(x) satisfies F (x) = <[M(F(x))], |[W(F(x))]|> .

Through the concept of gH-derivative, Stefanini and Bede developed some properties, e.g., (a) (kF)  = kF  and (b) (F+G)  F +G , where k  R, F and G are two interval-valued functions. By means of the gv-derivative, one can analogously show that (a) and (b) are true for interval vector-valued function. Moreover, the following property holds: Theorem 6. (F+G) = F+G, if both w(Fi(x)) and w(Gi(x)) are simultaneously increasing or decreasing functions with 1  i  n.

Proof. It follows from the representation of F(x) and F(x) that F+G = < M(F(x))+M(G(x)), W(F(x))+W(G(x))> (F+G) = < [M(F(x))+M(G(x))], |[W(F(x))+W(G(x))]|>.

For a given subscript i, if both w(Fi(x)) and w(Gi(x)) are simultaneously increasing or decreasing, then |[w(Fi(x)) + w(Gi(x))]| = |[w(Fi(x))]| + |[w(Gi(x))]|, as [w(Fi(x))]and [w(Gi(x))] have the same sign. Therefore, we have (F+G) = <[M(F(x))+M(G(x))], |[W(F(x))]|+|[W(G(x))]|> =<[M(F(x))], |[W(F(x))]|> + <[M(G(x))], |[W(G(x))]|> = F +G .



Example 2. Let F(x) = ([– x + 1, x + 2], [– x2 + 1, x2 + 2]) and G(x) = ([x + 1, 2x + 2], [x2 + 1, 2x2 + 2]), where x[0,1]. It is easy to see that W(F(x)) = (2x + 1, 2x2 + 1) and W(G(x)) = (x + 1, x2 +1). On the other hand, we can obtain (F(x)+G(x)) = ([0, 3], [0, 6x]) and F (x)+G (x) = ([–1, 1], [–2x, 2x]) + ([1, 2], [2x, 4x]), and thus (F + G)  = F +G .

5 CONCLUSIONS After listing some existing properties of gH-difference for interval number, we first study the associative and distributive laws based on interval number. We then concentrate on investigating the algebraic properties of gvdifference in the real interval vector-valued space after defining the arithmetic rules of interval vector-addition, gvdifference, multiplication and scalar multiplication. Depending on the versions of width vector and vector-valued partial order relation, some important properties of interval vector are derived, e.g., the associative law about - 11 www.ivypub.org/mc

addition and gv-difference and the distributive law of multiplication to gv-difference. These are helpful for studying interval programming and dynamical systems. Finally, the concept of gv-derivative for a given interval vectorvalued function is defined, and accordingly a basic property is acquired. These properties will be adopted to solve interval dynamical systems in our future work.

ACKNOWLEDGMENT This work is supported by Doctoral Fund of Ministry of Education of China (20125201110003) and NSFC (61065010).

REFERENCES [1]

Moore, Ramon E. Interval analysis. Englewood Cliffs: Prentice-Hall, 1966

[2]

Neumaier, Arnold. Interval methods for systems of equations. Cambridge: Cambridge university press, 1990

[3]

Nedialkov, Nedialko S., Kenneth R. Jackson, and John D. Pryce. An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE. Reliable Computing 7.6 (2001): 449-465

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Lin, Youdong, and Mark A. Stadtherr. Validated solutions of initial value problems for parametric ODEs. Applied Numerical Mathematics 57.10 (2007): 1145-1162

[5]

Lu, H. W., et al. Numerical solutions comparison for interval linear programming problems based on coverage and validity rates." Applied Mathematical Modelling 38.3 (2014): 1092-1100

[6]

Hukuhara, Masuo. Integration des applications mesurables dont la valeur est un compact convexe. Funkcial. Ekvac 10 (1967): 205-223

[7]

Stefanini, Luciano. A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy sets and systems 161.11 (2010): 1564-1584

[8]

Stefanini, Luciano. A generalization of Hukuhara difference for interval and fuzzy arithmetic. Working Paper EMS Series, University of Urbino, www.repec.org, 2008

[9]

Stefanini, Luciano. A generalization of Hukuhara difference. Soft Methods for Handling Variability and Imprecision 48 (2008) : 203-210

[10] Stefanini, Luciano, and BarnabĂĄs Bede. Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis: Theory, Methods & Applications 71.3 (2009): 1311-1328 [11] Malinowski, Marek T. Interval Cauchy problem with a second type Hukuhara derivative. Information Sciences.213 (2012): 94-105 [12] Chalco-Cano, Yurilev, et al. Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets and Systems 219 (2013): 49-67 [13] Lupulescu, Vasile. Hukuhara differentiability of interval-valued functions and interval differential equations on time scales. Information Sciences 248 (2013): 50-67 [14] Moore, Ramon E., R. Baker Kearfott, and Michael J. Cloud. Introduction to interval analysis. Siam, 2009 [15] Guerra, Maria Letizia, and Luciano Stefanini. A comparison index for interval ordering. Foundations of Computational Intelligence (FOCI), 2011 IEEE Symposium on. IEEE, (2011): 53-58

AUTHORS 1

2

Received her BSc degree from Guizhou

MSc degree from Guizhou University in

University in 2002. Received her MSc

1998. Received his Ph.Dâ&#x20AC;&#x2122;s degree from

degree from University of Science and

Chongqing University in

Technology Beijing in 2005. Her current

current

interest includes interval dynamics.

evolutionary

computation,

optimization,

uncertain

J. Tao, born in 1976. Ph.D candidate.

Z.H. Zhang, born in 1966. Received his

research

control theory and visual neural networks.

- 12 www.ivypub.org/mc

2004. His

interests

include immune

programming,