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On the Axiomatization of

Svetoslav Markov Institute of and Informatics Bulgarian Academy of Sciences “Acad. G. Bonchev” st., block 8, 1113 Sofia, Bulgaria [email protected]

Abstract In our study we follow the algebraically natural approach of completing the IRn up to the set IRn involving im- We investigate some abstract algebraic properties of the proper intervals. Thus, starting from the above three ba- system of intervals with respect to the arithmetic operations sic operations/relations (1)–(3), we arrive to the system and the relation inclusion and derive certain practical con- (IRn, +, R, ∗, ⊆). In the next Section 2 we briefly recall from these properties. In particular, we discuss the above mentioned algebraic construction and the conse- the use of improper intervals (in addition to proper ones) quent quasivector spaces. It is to be noted that everything and of midpoint-radius presentation of intervals. This work said in this section for intervals is also true for the more gen- is a theoretical introduction to interval arithmetic involving eral case of convex bodies and briefly repeats already pub- improper intervals. We especially stress on the existence of lished materials, cf. [7], [8]. In Section 3 we concentrate special “quasi”-multiplications in interval arithmetic and on the system (IRn, +, R, ∗, ⊆) obtained by algebraic com- their role in relevant symbolic computations. pletion. Using the theoretical foundations given in Section 2 and some specific properties of intervals (distinct from those of general convex bodies) we derive formulae for the 1. Introduction operations/relations involved. We show that the familiar midpoint-radius presentation of intervals is a special case of the presentation of elements in a quasivector . Section In this work we discuss several algebraic properties of (IR, +, ×, ⊆) the system of intervals with the arithmetic operations ad- 4 is devoted to the system involving multi- dition and multiplication and the relation inclusion. Our plication of one-dimensional intervals. In the Conclusion aim is to out certain practical advantages of using im- we discuss various topics like computer implementation of proper intervals and midpoint-radius presentation of inter- interval arithmetic, symbolic computations, etc. vals. Denote by IR the set of all compact intervals on the real 2. Quasivector spaces line R and by IRn the set of all n- of intervals. For n A, B ∈ IRn, α, β ∈ Rn, γ ∈ R, one defines addition, The system (IR , +) is a commutative monoid (semi- multiplication by scalars and inclusion, resp., by: group with null) with cancellation law. There is no opposite operator in (IRn, +). The operator multiplication by the A + B = {α + β | α ∈ A, β ∈ B}, (1) scalar −1: ¬A =(−1) ∗ A = {−α | α ∈ A},A∈ IR, γ ∗ B = {γβ | β ∈ B}, (2) briefly called negation (that may be suspected for opposite), A +(¬A)=0 A ⊆ B ⇐⇒ (α ∈ A =⇒ α ∈ B), (3) is not an opposite operator, as is violated for certain A ∈ IRn. Thus IRn is not a group; however it where all operations/relations are understood component- can be embedded in a group. The algebraic construction wise. We thus obtain the system (IRn, +, R, ∗, ⊆) to be that converts an abelian monoid with cancellation law into discussed in the sequel. a group will be further refered as embedding construction. We shall refer to the definitions of the opera- Recall that this approach is used to pass from the monoid of tions/relations (1)–(3) as set-theoretic. These definitions are nonnegative reals (R+, +) to the set of reals (R, +). Thus, not suitable for computations with intervals. Our final aim it is natural instead of the original system (IRn, +, R, ∗, ⊆) is to derive computationally efficient expressions for these to consider the extended system (IRn, +, R, ∗, ⊆) obtained operations based on the intrinsic properties of intervals. by the embedding construction. 2.1. The embedding construction γ ∗ (a + b)=γ ∗ a + γ ∗ b, (7) α ∗ (β ∗ c)=(αβ) ∗ c, (8) Every abelian monoid (M,+) with cancellation law in- 1 ∗ a = a, (9) duces an abelian group (M, +), where M = M 2/ ∼ is (α + β) ∗ c = α ∗ c + β ∗ c, αβ ≥ 0. the difference (quotient) set of M consisting of all pairs if (10) (A, B) factorized by the congruence relation ∼:(A, B) ∼ The only difference between a vector (linear) space and (C, D) iff A + D = B + C, for A, B, C, D ∈ M. a quasivector space is contained in assumption (10), where Addition in M is defined by the relation (α + β) ∗ c = α ∗ c + β ∗ c is required to hold just for αβ ≥ 0, whereas in a the same (A, B)+(C, D)=(A + C, B + D). (4) relation is assumed to hold for all scalars α, β ∈ R (known The neutral (null) element of M is the class (Z, Z), Z ∈ as second distributive law). Thus a vector space is a special M. Due to the existence of null element in M,wehave quasivector space. (Z, Z) ∼ (0, 0). The opposite element to (A, B) ∈ M is Conjugate elements. From opp(a)+a =0we obtain opp(A, B)=(B,A). The mapping ϕ : M −→ M defined ¬opp(a) ¬ a =0, that is ¬opp(a)=opp(¬a). The el- for A ∈ M by ϕ(A)=(A, 0) ∈ M is an embedding of ement ¬opp(a)=opp(¬a) is further denoted by a − and monoids. We embed M in M by identifying A ∈ M with the corresponding operator is called dualization or conju- the equivalence class (A, 0) ∼ (A + X, X), X ∈ M; all gation. We say that a− is the conjugate (or dual) of a.In elements of M admitting the form (A, 0) are called proper the sequel we shall express the opposite element symboli- and the remaining (new) elements are called improper. The cally as: opp(a)=¬a−, minding that a+(¬a−)=0(to be set of all proper elements of M is ϕ(M)={(A, 0) | A ∈ briefly written as a ¬ a− =0). Using conjugate elements ∼ M} = M. the quasistributive law (10) can be written in the form Using the above construction the system (IRn, +) is em- n bedded into the group (IR , +) in a unique way. (α + β) ∗ cσ(α+β) = α ∗ cσ(α) + β ∗ cσ(β), (11) Multiplication by scalars “∗” is extended from R×IRn σ n wherein is the sign functional to R × IR by means of −,α<0; n σ(α)= γ ∗ (A, B)=(γ ∗ A, γ ∗ B),A,B∈ IR ,γ∈ R. (5) +,α≥ 0,α∈ R, ¬(A, B)=(−1) ∗ In particular, negation is extended by and the convention a+ = a has been made (for a proof (A, B)=(¬A, ¬B),A,B∈ IRn . see [8]). Expression (11) is valid for all values of α, β (not In the sequel we shall use lower case roman letters only for equally signed α, β) which allows efficient sym- to denote the elements of IRn, writing e. g. a = n bolic calculations. (A1,A2),A1,A2 ∈ IR . For example, negation is writ- ¬a =(−1) ∗ a a ¬ b a +(¬b) ten: ; below means . 2.3. A decomposition theorem Inclusion “⊆” is extended in IR by means of An element y with the property y ¬ y =0(equivalently (A, B) ⊆ (C, D) ⇐⇒ A + D ⊆ B + C, (6) y = y−) is called linear or distributive; an element z such n that ¬z = z (equivalently z + z− =0) is called centred or wherein A, B, C, D ∈ IR and inclusion of interval n- 0-symmetric. tuples is meant component-wise. As is immediately seen, under this extension the practically important properties Theorem (Decomposition theorem). (Q, +, R, ∗) is a a ⊆ b ⇐⇒ a+c ⊆ b+c for c ∈ IR and a ⊆ b ⇐⇒ γ ∗a ⊆ quasivector space. For every x ∈ Q there exist unique γ ∗b for γ ∈ R are preserved. The system (IRn, +, R, ∗, ⊆) y,z ∈ Q such that: i) x = y + z; ii) y ¬ y =0; iii) involving improper intervals is now completely defined. ¬z = z;iv)y = z =⇒ y = z =0. The proof, see [8], is based on the fact that any x ∈ Q 2.2. Quasivector space: definition can be written in the form: (IRn, +, R, ∗) The interval system is a quasi-vector x = y + z =(1/2) ∗ (x + x−)+(1/2) ∗ (x ¬ x). (12) space in the sense of the following definition [8]: Definition.Aquasi-vector space (over R), denoted Note that the first summand in (12) is linear, whereas the (Q, +, R, ∗), is an abelian group (Q, +) with a mapping second one is centred. The of all linear elements of Q (multiplication by scalars) “∗”: R × Q −→ Q, such that is denoted Q = {x ∈ Q | x ¬ x =0} and the subset of all for a, b, c ∈ Q, α,β,γ ∈ R: centred elements of Q is denoted Q = {x ∈ Q | x = ¬x}. Note that negation coincides with opposite in Q, and that the two vector spaces (Q, +, R, ·), (Q, +, R, ·) are negation coincides with identity in Q. m-, resp. n-dimensional and therefore they are isomorphic to Rm, resp. Rn. Hence any a ∈ Q is a direct sum of the Corollary 1. Every quasivector space Q is a direct sum m n form a =(a ; a ), a ∈ R , a ∈ R . As elements of Q of Q = {x ∈ Q | x ¬ x =0 } and Q = {x ∈ Q | x = the elements of Q are of the form (a ;0)and the elements ¬x}, symbolically Q = Q Q . of Q — of the form (0; a ), so that: (Q, +, R, ∗) Q = {x ∈ Q | x ¬ x =0} Clearly, with is a linear space. Indeed, conjugation in Q coincides with a =(a ; a )=(a ;0)+(0;a ). (15) identity. Substituting x = x− in (11), we obtain that the ∗ familiar second distributive law (α + β) ∗ c = α ∗ c + β ∗ c, Because of the presence of the special operation “ ” and holds for all real α, β as required in the definition of a linear its different meaning in the two spaces, we shall con- (Q, +, R, ·) space. tinue to make a distinction between the spaces , (Q, +, R, ·) To characterize Q = {x ∈ Q | x = ¬x}, note that , calling the first one linear, and the second one Q Q is a quasivector space of centred elements, to be briefly centred (quasivector); also the elements of will be called Q called centred quasivector space.Asx = ¬x is equivalent linear, and the elements of — centred. Let us recall that “∗” and “·” coincide in Q, but are distinct in Q,soQ will to x + x− =0, conjugation coincides with the opposite operator in Q. The centred quasivector space Q can be be always considered together with the two multiplications: (Q, +, R, ·, ∗) converted into a linear space by re-defining multiplication . by scalars as follows: Applying the Decomposition theorem we have that any finite dimensional quasivector space Q is a direct sum (Rm, +, R, ·) α · c = α ∗ cσ(α),c∈ Q . of two spaces — the linear space and the (13) n centred quasivector space (R, +, R, ·, ∗), symbolically: m n Now (Q, +, R, ·) is a linear space. Indeed, substituting (Q, +, R, ·, ∗)=(R , +, R, ·) (R , +, R, ·, ∗). Thus we (13) in (11), we obtain that the second distributive law (α + can write down the operations addition and multiplication β) · c = α · c + β · c is valid for all real α, β; furthermore, by scalars for intervals a =(a ; a ),b=(b ; b ) ∈ Q in properties (7)–(9) remain valid: γ ·(a+b)=γ ·a+γ ·b, α· the form: (β · c)=(αβ) · c, 1 · a = a. a+b =(a; a)+(b; b)=(a + b; a + b), For a better distinction between the two multiplica- (16) tions by scalars, we call “∗” quasi-vector multiplication by γ ∗ b = γ ∗ (b; b)=(γ ∗ b; γ ∗ b)=(γb; |γ|b), (17) scalars and “·”—linear multiplication by scalars. Note that m n the linear multiplication by scalars “·” defined by (13) coin- wherein a ,b ∈ R , a ,b ∈ R , γ ∈ R and in the last cides in Q with the quasi-vector one, as in Q relation (13) expression relation (14) is applied. (Rn, +, R, ·, ∗) n =3 becomes α · c = α ∗ c (due to c− = c). Therefore For example, in the space for , the quasi-multiplication of centred elements by scalars looks as (Q, +, R, ·)=(Q, +, R, ·) Corollary 2. The space follows: −2 ∗ (1, 2, 1) = (2, 4, 2); −1 ∗ (−1, 2, −2) = (Q, +, R, ·) · , with “ ” defined by (13), is a linear space. (−1, 2, −2). Multiplication by −1 (negation) coincides Formula (13) gives an expression in Q for the linear with identity. To change the signs of the components of multiplication by scalars in terms of the quasi-vector one. a centred element one should take the opposite (or dual) Conversely, in Q the quasi-vector multiplication by scalars and not negation, e. g. opp(−1, 2, −2) = (−1, 2, −2)− = is expressed by the linear one via (1, −2, 2). For negation, subtraction, opposite and dual in Q we have: α ∗ c = |α|·c, c ∈ Q. (14) ¬a = −1 ∗ (a; a)=(−a; a), (18) Naturally, the quasi-vector multiplication by scalars is in- a ¬ b = a +(−1) ∗ b =(a − b; a + b), (19) clusion isotone, that is a ≤ b =⇒ γ ∗ a ≤ γ ∗ b, γ ∈ R, opp (a)=¬a− = ¬(a ; a ) =(−a ; −a ), (20) as a ≤ b =⇒|γ|a ≤|γ|b (which is not true for the linear − (a)=a =(a; a) =(a; −a). multiplication by scalar). dual − − (21)

2.4. Presentation of elements 3. Presentation of intervals

The spaces involved in Corollary 2 are vector spaces. We see that any element of a finite quasivector space is If we want to efficiently compute within these spaces we represented in the form (15) as a sum of a linear and a cen- should assume them to be finite dimensional so that their tred component. This refers to arbitrary finite quasivector elements have finite presentation. Thus we shall assume spaces, such as quasivector spaces of zonotopes [9]. We next wish to interpret formulae (18)–(15) for the case of in- Let us check if the linear part (23) of a proper element terval vectors. To this we shall consider the presentation (A, 0) is a proper element. Clearly, (A, ¬A) is a proper of proper intervals. element if there exists X ∈ M such that (A, ¬A)=(X, 0), Intervals (interval vectors) are usually represented by that is A = X ¬ A. As we know this property is satisfied sets of real numbers. Two familiar presentations of inter- by intervals, indeed, we have X =(2a;0), where a is vals (interval vectors) are those by pairs of real numbers the midpoint of A. Hence, for the value of (23) we obtain (vectors) either in end-point or midpoint-radius form. Up to 1/2 ∗ X =(a;0). Summarizing we obtain for the proper now we intentionally avoided to represent intervals in any interval (a; a), a ≥ 0: (a; a)=(a, 0)+(0; a), which form. We now discuss the presentation of intervals by draw- is exactly of the form (15). ing consequences directly from their algebraic properties. Remarks. 1. Presentation (a; a)=(a, 0) + (0; a) corresponds (for a > 0) to the symbolic form a = a ± 3.1. The case of proper intervals a often used in engineering sciences. 2. Convex bodies with so-called Minkowski operations [7], [14] also form a IRn Proper intervals as elements of are pairs of the form quasivector space. However, for convex bodies the equation (A, 0) A ∈ IRn (A, 0) , where . Assume first that is a linear A = X ¬ A is not solvable in general and consequently, (A, 0) ¬ (A, 0) = 0 element, that is ; using (4) this means the linear part of a proper convex body may not be proper. A ¬ A =0 . As we know such properties possess exactly For example, in the case of two-dimensional convex bodies, A ∈ IRn Rn the point intervals , that is vectors from . As- if A is a (proper) triangle, then such X does not exist and (A, 0) (A, 0) = ¬ (A, 0) sume now that is centred, that is ; consequently the linear part of A is an improper element. A = ¬ A A ∈ IRn according to (4) this means , that is is a In the case of proper intervals (a ≥ 0)wehave centred interval (symmetric with respect to the origin). Recall that point intervals are represented as [α, α] and A ¬ B = {α − β | α ∈ A, β ∈ B}, (25) proper centred ones — as [−β,β], where β ≥ 0 is the radius and, in particular, ¬B = {−β | β ∈ B}. of the centred interval. Any proper interval is written in end- point form as [α − β, α + β],β≥ 0. MR-presentation of improper intervals. Formula (15) In midpoint-radius form (briefly: MR-form) a proper in- demonstrates the practical meaning of the decomposition terval A ∈ IRn is written as A =(a; a), a ≥ 0 and is theorem: there the linear (point) interval of the form (a ;0) interpreted as the set: corresponds to the midpoint of a and the centred interval (0; a) corresponds to the radius (error bound) of a.A n A =(a ; a )={ξ ∈ R ||ξ − a |≤a }, (22) negative radius a < 0 corresponds to improper interval (a; a). In the end-point form [a − a,a + a] of an im- wherein the module of a vector is meant component-wise, proper interval we have that the left end-point is greater than |(α , ..., α )| =(|α |, ..., |α |) i. e. 1 n 1 n . Using MR-form a the right one, a − a ≥ a + a. (a;0) point interval is written as and a proper centred one Let us check how improper one-dimensional elements of (0; a) a ≥ 0 a as , . The component is called the midpoint the form a =(0,A), A ∈ IR, are decomposed according a and the component — the radius or error (bound). to the Decomposition theorem. a Within the interpretation (22) the component is an For the linear and the centred part (summand) of a = Rn element of the vector space , whereas the component (0,A) we obtain respectively: a ≥ 0 is an element of the monoid (Rn)+. The relation between the MR-presentation (22) and the end-point pre- (1/2) ∗ (a + a−)=(1/2) ∗ (¬A, A), (26) sentation is given by (a ; a )=[a − a ,a + a ], resp. (1/2) ∗ (a ¬ a)=(1/2) ∗ (0,A¬ A). (27) [e−,e+]=((e+ + e−)/2; (e+ − e−)/2]. Let us see how proper elements of the form a =(A, 0), Clearly, the centred part (27) of the improper interval (0,A) A ∈ IR, are decomposed according to the Decomposition is an improper interval. Denoting A =(a ; a ), a ≥ 0,we theorem. For the linear and the centred parts (summands) have A ¬ A =(a ; a )+(−a ; a )=(0;2a ), hence the of a =(A, 0) we obtain respectively, cf. (12): value of (27) is opp(0; a )=(0;−a ). Let us check if the linear part (26) of an improper el- (1/2) ∗ (a + a−)=(1/2) ∗ (A, ¬A), (23) ement (0,A) is an improper element. Clearly, (¬A, A) (1/2) ∗ (a ¬ a)=(1/2) ∗ (A ¬ A, 0). (24) is an improper element if there exists X ∈ M such that (¬A, A)=(0,X), that is A = X ¬ A.Wehave It is immediately seen that the centred part (24) of a proper X =(2a;0), where a is the midpoint of a. Hence, for the interval is a proper interval. Denoting A =(a; a), a ≥ value of (26) we obtain opp(1/2 ∗ X)=(−a;0). We con- 0, using (19), we have A ¬ A =(a; a)+(−a; a)= clude that the linear part of any interval is always a (proper) (0; 2a), which gives a value of (24) equal to (0; a ). point interval. Summarizing we obtain for the improper interval the interval problem (28) reduces to two linear algebraic (0,A)=opp(a; a), a ≥ 0: (−a; −a)=(−a;0)+ problems, one for the midpoints and one for the radii: (0; −a), which is in agreement with (15). Therefore we can consider (15) as a generalization of the MR- Ax = b , (29) presentation of intervals. The difference is that the value |A| x = b. (30) of a in (15) can be negative. We can thus speak of neg- ative radii (or negative errors) corresponding to improper Assuming that the real matrices A and |A| are nonsingular intervals. we obtain for the solution of (29)–(30): x = A−1b, x = −1 −1 In the n-dimensional case a in (15) belongs to Rn and |A| b . We must assume |A| b ≥ 0, so that x ≥ 0. thus the components of a can have negative values corre- We thus obtain the following corollary: sponding to improper one-dimensional intervals. We can again speak of n-dimensional radii (or errors) whose com- Corollary. Given system (28), such that A ∈ Rn×n, ponents are not necessarily nonnegative. b =(b,b) ∈ IRn, assume that the real matrices A and |A| are nonsingular and |A|−1b ≥ 0. Then there exists a A, B ∈ IR (A, B) ∈ IR Proposition 1. Let and as unique interval solution x to (28). defined in Section 2.1. If A =(a; a),B =(b; b), then (A, B)=(a − b; a − b) . Sets of solutions. For the determination of solution sets Proof. We have of algebraic problems with uncertain parameters it is of practical significance [2], [3], [15] to be able to find the set (X, 0), if A = B + X, (A, B)= n (0,Y), if A + Y = B, {x ∈ R | A ∗ x ⊆ b}, (31)

n×n where X, Y ∈ IR. Minding that (X, 0) = X =(a − where A =(A ; A ) ∈ IR is an interval matrix (with n×n n×n n b ; a − b ), Y =(b − a ; b − a ), (0,Y)=opp(Y,0) = A =(aij ) ∈ R , A =(aij ) ∈ R ) and b ∈ IR is (a − b; a − b) we obtain (A, B)=(a − b; a − b). an interval vector. The multiplication in (31) is multiplication by scalars, IRn Improper interval vectors as elements of are pairs hence we have the equivalences of the form (A, B), where A, B ∈ IRn and at least one pair (Ai,Bi) is an improper interval. In principle, we can A ∗ x ⊆ b ⇐⇒ (A ; A ) ∗ x ⊆ (b ; b ) assume that a and a are vectors of different , ⇐⇒ (Ax; A|x|) ⊆ (b; b) as this is often needed in applications. For an example of ⇐⇒ | b − Ax|≤b − A|x|. m>n, one sometimes considers two-dimensional points (32) that contain errors only in one of the components, say the We know that the last equivalence is true for proper inter- ordinate. Alternatively, as an example of m

n×n where A =(aij ) ∈ R is a nonsingular matrix of reals 3.3. Inclusion and b ∈ IRn is an interval vector. We are interested in vectors x ∈ IRn satisfying (28); such x are called interval The system (IR, ⊆). Interval inclusion (3) as defined solutions (other authors call them algebraic solutions [1], p. for proper intervals A =(a; a),B =(b; b) ∈ IR is 65, or formal solutions [13], [15]). expressed in MR-form by: A ⊆ B ⇐⇒ | b −a|≤b −a. Denoting x =(x; x) ∈ IRn we have A ∗ x = We shall prove that a relation of exactly the same form is A ∗ (x; x)=(A ∗ x; A ∗ x)=(Ax; |A|x). Thus true for any intervals (proper or improper). Proposition 2. Interval inclusion (6) as defined for arbi- 4. Interval multiplication trary intervals a =(a; a),b =(b; b) ∈ IR is expressed in MR-form by: This section is devoted to the system (IR, +, ×, ⊆) in-

volving multiplication of intervals. Multiplication of one- a ⊆ b ⇐⇒ | b − a |≤b − a . (33) dimensional proper intervals A, B ∈ IR is defined by the set-theoretic expression: Proof. As in Section 2.1. define a, b by a =(X, Y ), b = (U, V ), where X =(x; x), Y =(y; y), U =(u; u), A × B = {αβ | α ∈ A, β ∈ B}. (37) V =(v; v) are intervals from IR. According to (6): As before we are looking for a computationally efficient (X, Y ) ⊆ (U, V ) ⇐⇒ X + V ⊆ Y + U. (34) expression of (37). It has been proved [4] that multiplication (37) is extended uniquely in IR in such a way that : In MR-presentation (34): reads: a ⊆ b =⇒ c × a ⊆ c × b, a, b, c ∈ IR. (38) (x − y; x − y) ⊆ (u − v; u − v) Hence, we can assume that the system (IR, +, ×, ⊆) is ⇐⇒ (x + v ; x + v ) ⊆ (y + u ; y + u ) well-defined. To obtain a formula of MR type we first define ⇐⇒ | y + u − x − v|≤y + u − x − v a functional “relative error” κ as follows. ⇐⇒ | u − v − (x − y)|≤u − v − (x − y), For a ∈ IR any non-centred interval (a =0) can be written as: a =(a; a)=|a|∗(σ(a); a/|a|), wherein σ(α)= (α) which according to Proposition 1 is equivalent to (33). sign . Examples: (10; 1) = 10 ∗ (1; 0.1); (−2; 4) = 2 ∗ Remark. From (33) we see that the case a proper and b (−1; 2). improper is impossible (in this case we have b − a < 0 Thus we can write: a =(a; a)=|a|∗(σ(a); κ(a)), and (33) cannot be satisfied). where κ(a)=a/|a| is the relative error in a. When mul- tiplication is considered the functional κ(a) plays important Lattice operations. Consider now the lattice operations roles. Note that the condition κ(a) < 1 means that a does x ∨ y =sup (x, y) x ∧ y =inf (x, y) ⊆ ⊆ , ⊆ ⊆ in the case not contain zero (in the sense of (3). The case of proper in- x, y ∈ IR x ⊆ y sup (x, y)=y .For we have ⊆ etc., but tervals has been fully discussed in [5]. We next extend the the general case is more involved. The proof of the next definition of the functional κ in IR. proposition is rather technical and will be omitted. Intervals from IR.Fora =(a; a) ∈ IR we define κ x ⊆ y y ⊆ x Proposition 3. If neither nor then by: denoting ε = sign(x − y)(x − y) we have for x ∨ y = |a|/|a|,a =0; κ(a)= sup⊆(x, y) and x∧y =inf⊆(x, y), respectively, the values: ∞,a=0,

x+y+ε|x−y| |x−y|+x+y Condition κ(a) < 1 means that a does not contain zero ( 2 ; 2 ), in the sense of (6), whereas κ(a) ≤ 1 means that a may x+y−ε|x−y| −|x−y|+x+y contain zero only as an end-point. ( 2 ; 2 ). To write an MR-formula for interval multiplication de- note by E(a) condition κ(a) ≤ 1 and by C(a, b) condition Clearly, the values of −|x − y| + x + y may be nega- tive, that is the interval x∧y =inf⊆(x, y) may be improper. κ(a) > 1 ≥ κ(b) or (κ(a) ≥ κ(b) > 1,a b ≥ 0). Hence, the lattice (IR, ⊆) is complete, whereas (IR, ⊆) is not. Note that E(b) means κ(b) ≤ 1 and C(b, a) means Inclusion isotonicity is preserved in IR with respect to κ(b) > 1 ≥ κ(a) or (κ(b) ≥ κ(a) > 1,a b ≥ 0). (16)–(17), i. e. The following MR-expression is equivalent to the re- a ⊆ b ⇐⇒ a + c ⊆ b + c (35) spective end-point formula given by E. Kaucher [4]: a × b =(a; a) × (b; b)= a ⊆ b ⇐⇒ γ ∗ a ⊆ γ ∗ b. (36)  (39)  (ab +σ(a)σ(b)ab; |a|b +|b|a), E(a)&E(b);  if Remark. A generalization of (33) for the n-dimensional (b + σ(b)σ(a)b) ∗ (a; a), if C(a, b); case a, b ∈ IRn is straightforward. The same holds true for  (a + σ(a)σ(b)a) ∗ (b; b), C(b, a);  if Proposition 3 regarding the lattice operations. 0, if κ(a) ≥ 1,κ(b) ≥ 1,ab ≤ 0. Remark. The two intermediate cases involve multipli- of the midpoint-radius presentation of intervals. Let us cation by scalars; these cases can be viewed as a single case summarize our arguments in favor of using the extended by a, b interchanging places, as in the FORTRAN-like algo- interval systems (IRn, +, R, ∗, ⊆), (IR, +, ×, ⊆) involv- rithm below. ing improper intervals (IR-systems). The advantages of using IR-systems instead of the respective IR-systems An algorithm for interval multiplication. (IRn, +, R, ∗, ⊆), (IR, +, ×, ⊆) can be compared with the IF E(a) and E(b) advantages of using the familiar real vector space R n and THEN a × b =(a b + σ(a )σ(b )a b ; |a |b + |b |a ) real field R instead of the respective systems (R+)n, R+, ELSE IF κ(a) ≥ 1,κ(b) ≥ 1,a b ≤ 0 THEN a × b =0 involving only nonnegative reals. Recall that the system ELSE IF C(a, b) (IRn, +) is a group, so an equation of the form a + x = b THEN a × b =(b + σ(b )σ(a )b ) ∗ (a ; a ) is always solvable, whereas it is not always solvable in ELSE a × b =(a + σ(a )σ(b )a ) ∗ (b ; b ) (IR, +). Moreover, (IRn, ⊆) is not a complete lattice, (IR, ⊆) A compact end-point formula is published in [11]. A whereas is, so that lattice operations induced by ⊆ tool for visualizing the operation multiplication (39) using the relation “ ” always exist. In particular, the lattice op- (a, b) the above algorithm can be downloaded from [16]. To run eration “meet” inf⊆ is well-defined, whereas it is not (IR, ⊆) the program the .NET Framework Redistributable Package always available in (two proper intervals may not (dotnetfx.exe) should be installed. The visualization tool intersect). offers the possibility to move continuously one of the argu- The IR-systems incorporate the respective IR-systems, ments a, b in the product a × b by dragging the correspond- so everything that can be done in an IR-system, can be also ing point by the mouse. This allows to demonstrate various performed in the respective IR-system using improper inter- properties of the product, like its continuity (which is not vals. An user not familiar with improper intervals may not obvious from the conditional expression (39)), morphism notice that the IR-system handles them — unless certain fi- w. r. t. conjugation, etc. nal results cannot be expressed by proper intervals — then In the simple case of centred intervals we should have the IR-system may produce improper intervals, whereas the (0; a) ⊆ (0; b)=⇒ (0; c)×(0; a) ⊆ (0; c)×(0; b) for any IR-system will issue a message for non-existing solutions. (0; c), or, equivalently a ≤ b =⇒ c×a ≤ c×b for all real Computations in the extended IR-systems are in no way a, b, c. This property should hold for the case when a and more complicated or time consuming than computations b take possibly negative values (as we know such a prop- with proper intervals. On the contrary, the correspond- erty does not hold for the familiar multiplication). This is ing formulae may be simpler as nonnegativity tests may be why it is interesting to find a relation between the “quasi- avoided. Formulae (16), (17) hold for intervals from IR n multiplication” of radii and the familiar multiplication of and thus, in particular, for proper intervals. This means that reals. Substituting a = b =0in (39) we obtain: (16), (17) provide MR-expressions for the set-theoretic op-  erations addition (1) and multiplication by scalars (2) in- ab, a ≥ 0,b≥ 0,  if volving proper intervals. Note that there are no simpler for- a×b = −ab, a ≤ 0,b<0,  if (40) mulae than (16), (17) for the operations addition (1) and 0, a>0,b<0 a<0,b>0. if or multiplication by scalars (2) for proper intervals; we men- tion this in order to dispel the myth that the arithmetic using Some examples: (−2)×(−3) = −6, 2×(−3) = 0. improper intervals is more involved than the arithmetic for Apart of being inclusion isotone, we have for the quasi- proper intervals. A close examination of formula (39) leads multiplication: −(a×b)=(−a)×(−b), which is also not us to a similar conclusion: this expression is computation- true for the familiar multiplication. ally as efficient as the respective expressions for proper in- Clearly the system of the radii (R, +, ×) is not a tervals. a · b = ring, but if we define a multiplication by An axiomatic introduction to interval arithmetic neces- (a) (b)(|a|×|b|) (R, +, ·) sign sign , then system is a ring, cf. sarily uses improper intervals and has certain methodolog- [6]. Therefore the system of readiuses can be consid- ical advantages. Such an introduction shows which are the ered as a ring of reals endowed additionally with a quasi- basic concepts. We have shown that the operations/relations (R, +, ·, ×) multiplication (40), i. e. . (1)–(3) and (37) are in the bases of the whole theory. The axiomatic approach passing through improper in- 5. Conclusions tervals demonstrates the different algebraic nature of the spaces of midpoints and radii. Many authors have pointed This work is a theoretical introduction to interval arith- out practical arguments regarding the need of a differenti- metic. We show that the rigorous theory naturally in- ated approach in the presentation of midpoints and radii in volves improper intervals and points out the basic role the computational practice, cf. [10] (p. 29–30), [12]. [17]. The general idea is that for the computer presentation of [4] Kaucher, E., Interval Analysis in the Extended Interval the radii we normally need a much smaller number of dig- Space IR. Computing Suppl. 2 (1980) 33–49. its than for the midpoints — maximum 2–3 decimal digits. From computational point of view this makes the MR-form [5] Kulpa, Z., S. Markov, On the inclusion properties of in- faster than the end-point form and should be preferred as a terval multiplication: A diagrammatic study, BIT 43(4), basic form in a computer implementation; end-point form 741–810, 2003. of I/O data can be easily available as a secondary optional [6] Markov, S. On the Algebraic Properties of Errors, presentation. PAMM (Proc. Appl. Math. Mech.) 1, 2002, 506-507. As another argument in favor of improper intervals we mention the increasing number of practical problems, such [7] Markov, S., On Quasilinear Spaces of Convex Bodies as solution sets of linear and nonlinear problems, functional and Intervals, Journal of Computational and Applied ranges, etc., and numerical methods that require such inter- Mathematics 162 (1), 93-112, 2004. vals, cf. [2], [3], [13], [15], [18]. [8] Markov, S., Quasivector Spaces and their Rela- There is an ongoing discussion on the implementation of tion to Vector Spaces, EJMC (Electronic Journal symbolic interval arithmetic in computer systems. on Mathematics of Computation) 2-1 (2005), 1–21. Hopefully this work offers convincing motivation in favor (http://www.cin.ufpe.br/ejmc) to an implementation involving improper intervals and MR- presentation of intervals. Expression (11) is an instructive [9] Markov, S., On the Quasivector Space of Zonotopes example that shows a possible way of overcoming problems in the Plane, PAMM (Proc. Appl. Math. Mech.) 5 (1), related to symbolic computations with conditional formulas 2005, 711–712. like (10), typical for interval arithmetic. Note that there is no similar formula in the monoid IR,sowehavetousea [10] Neumaier, A., Interval Methods for Systems of Equa- space involving improper intervals to this end. tions, Cambridge University Press, 1990. Note that when end-point presentation [α − β,α + [11] Popova, E., Multiplication distributivity of proper and β],β≥ 0 is used, then the point and centred intervals can- improper intervals. Reliable computing 7(2) (2001) not belong to spaces of different dimensions (as then we 129–140. shall not know what is, say, α + β. Clearly, the notation n IR is closely related to the end-point presentation. How- [12] Rump, S. M., Fast and Parallel Interval Arithmetic, ever, as we have seen, the midpoint and radius vectors need BIT 39 (1999), 3, 534–554. not be of same . For example, one often consid- ers intervals that have two-dimensional midpoint compo- [13] Sainz M. A., Gardenyes E., L. Jorba, Formal Solution nent, and one-dimensional radius component (errors in one to Systems of Interval Linear or Non-Linear Equations. of the two coordinates only). Different dimensions of the Reliable computing, 8(3), 189211, 2002. spaces of linear and centred elements, as prescribed by the [14] Schneider, R. Convex Bodies: The Brunn-Minkowski Decomposition theorem, are possible within the midpoint- Theory. Cambridge Univ. Press, 1993. radius presentation. [15] Shary, S., A new technique in systems analysis under interval uncertainty and ambiguity, Reliable computing References 8 (2002) 321–418. [16] Surilov, R., S. Markov, A visualization software [1] Fiedler, M., Nedoma, J., Ramik, J., Rohn, J., Zimmer- tool for interval operations, freely downloadable from: mann, K., Linear Optimization Problems with Inexact http://www.math.bas.bg/˜bio/IntervalOperations.exe Data, Springer, 2006. [17] Van Emden, M. H., On the Significance of Digits in [2] Goldsztejn, A., G. Chabert, On the approximation of Interval Notation, Reliable Computing 10, 1, 45–58. linear AE-solution sets. In: 12th GAMM IMACS In- ternational Symposion on Scientific Computing, Com- [18] Wolff von Gudenberg, J., V. Kreinovich, A Full puter Arithmetic and Validated Numerics, Duisburg, -Based Calculus of Directed and Undirected Germany (2006), submitted. Intervals: Markov’s Interval Arithmetic Revisited, Nu- merical Algorithms 37, 2004, 417–428. [3] Goldsztejn, A., G. Chabert, A Generalized Interval LU- decomposition for the Solution of Interval Linear Sys- tems, submitted to LNCS.