An Elimination Method to Solve Interval Polynomial Systems
aa r X i v : . [ c s . S C ] J un An Elimination Method to SolveInterval Polynomial Systems
Sajjad Rahmany, Abdolali Basiri and Benyamin M.-AlizadehSchool of Mathematics and Computer Sciences,Damghan University, Damghan, Iran.September 20, 2018
Abstract
There are several efficient methods to solve linear interval poly-nomial systems in the context of interval computations, however, thegeneral case of interval polynomial systems is not yet covered as well.In this paper we introduce a new elimination method to solve andanalyse interval polynomial systems, in general case. This methodis based on computational algebraic geometry concepts such as poly-nomial ideals and Gr¨obner basis computation. Specially, we use thecomprehensive G¨obner system concept to keep the dependencies be-tween interval coefficients. At the end of paper, we will state someapplications of our method to evaluate its performance.
Many computational problems arising from applied sciences deal with floating-point computation and so require to import polynomial equations containingerror terms in computers. This redounds polynomial equations to appearwith perturbed coefficients i.e the coefficients range in specific intervals andso these are called interval polynomial equations . Interval polynomial equa-tions come naturally from several problems in engineering sciences such ascontrol theory [35,36], dynamical systems [30] and so on. One of the mostimportant problems in the context of interval polynomial equations is to1nalyse and study the stability and solutions of an (or a system of) intervalpolynomial(s). More generally, the problem is to carry as much as possibleinformation out from an interval polynomial system. Many scientific worksare done in this direction using interval arithmetic [2], for instance compu-tation of the roots in certain cases [6,7], however they do not enable us toobtain the desired roots, at least approximately [6]. Another example consiststhose works which contain (the most popular) method to solve an intervalpolynomial equation by computing the roots of some exact algebraic poly-nomials, while it is hard to solve an algebraic equation of high degree whichhas its own complexity challenging problems [9,11,12,18,21]. There is also anew method described in [41] which counts the zeros of a univariate inter-val polynomial. In addition to numerical methods, there are some attemptsto combine numeric and symbolic methods to solve an interval polynomialsystem. In [8], Falai et al. state a modification of Wu’s characteristic setmethod for interval polynomial systems, and use numerical approximationto find an interval containing the roots. The essential trick in this work is toomit all the terms with interval coefficients containing zero, which permitsthe division of interval coefficients simply. This consideration may fail someimportant polynomials.In this paper, we try to use exact symbolic methods to facilitate analysinginterval polynomial systems thanks to algebraic elimination methods likeparametric computation techniques to analyse a parametric polynomial sys-tem which allows us to consider all exact polynomials arising from an intervalpolynomial. As we will state later, it is very important to keep the trace ofinterval coefficients during the computations. Roughly speaking we asso-ciate an auxiliary parameter to each interval coefficient provided that eachparameter ranges over its own related interval only. Nowadays there areimportant results [38,39], efficient algorithms [17,19,20,22,24,25] and power-ful implementations [33,34] in the context of parametric computations andanalysing parametric polynomial systems. We introduce the new concept interval Gr¨obner system for a system of interval polynomials using the con-cept of comprehensive Gr¨obner systems [39] which is used to describe alldifferent behaviours of a parametric polynomial system. Interval Gr¨obnersystem contains a finite number of systems where each one is a Gr¨obnerbasis for (non-interval) polynomial systems obtained from the main system.It is worth noting that against to [8], we don’t omit any interval coefficientand cover all possible cases for the exact coefficients arising from the inter-vals. We also design an algorithm to compute interval Gr¨obner system of an2nterval polynomial system.This paper is organized as follows. In Section 2 we state introductorydefinitions and recall interval arithmetic. In Section 3 we start to explaininterval polynomials and their related concepts. We next receive to Section 4which states the main idea behind this paper. To recall the concepts of com-putational algebraic tools we state Section 5 containing a brief introductionto Gr¨obner basis and comprehensive Gr¨obner system, together with theirrelated algorithms. After, we describe our elimination method for intervalpolynomial systems in Section 6. Finally, as some applications and example,we state the Section 7 which contains two applied examples.
In this section we recall the interval arithmetic and related concepts whichare needed for the rest of this text. The main references of this section are[16] and [26]. Let R denote the set of real numbers while R ∗ is used to showthe extended real numbers set i.e. R ∪ {−∞ , ∞} . Definition 2.1.
Let a, b ∈ R ∗ . We define kinds of real intervals defined by a and b as follows: Closed interval : [ a, b ] = { x | a ≤ x ≤ b } ( a, b = ±∞ )Left half open interval : ( a, b ] = { x | a < x ≤ b } ( b = ±∞ )Right half open interval : [ a, b ) = { x | a ≤ x < b } ( a = ±∞ )Open interval : ( a, b ) = { x | a < x < b } (1)It is worth noting that approximately all of existing texts in the subjectof interval computation deal with closed intervals. Most of times we denotethe intervals by capitals, and their lower (resp. upper) bounds by underbar(resp. overbar), as X = [X , X ]to denote closed intervals. However, as there are some applied problemsincluding non-closed intervals we preferred to consider all different types ofintervals. Remark 2.2.
Having all different kinds of intervals at once, we use the otion [ a, b, i, j ] where i, j ∈ { , } to denote the intervals in (1), as follows: [ a, b, i, j ] = [ a, b ] if i = j = 1( a, b ] if i = 0 , j = 1[ a, b ) if i = 1 , j = 0( a, b ) if i = j = 0 However when all of intervals come from one sort of presentation, we preferto use the (1) form.
Now we recall the interval arithmetic and discuss on the interval depen-dencies what will occur in the evaluation of interval expressions.There exist two equivalent ways to state interval arithmetic. The first isbased on the endpoints of intervals while the second considers each intervalas a subset of real numbers. Let [ a , b , i , j ] and [ a , b , i , j ] be two realintervals. Note that each real number a is considered as [ a, a, ,
1] which iscalled a degenerate interval. Four essential arithmetic operations are definedas follows:[ a , b , i , j ] + [ a , b , i , j ] = [ a + a , b + b , min( i , i ) , min( j , j )][ a , b , i , j ] − [ a , b , i , j ] = [ a − b , b − a , min( i , j ) , min( j , i )][ a , b , i , j ] × [ a , b , i , j ] = [ a k b ℓ , a k ′ b ℓ ′ , min( i k , j ℓ ) , min( i k ′ , j ℓ ′ )]where a k b ℓ and a k ′ b ℓ ′ are the minimum and maximum of the set { a a , a b , b a , b b } respectively, and finally[ a , b , i , j ] / [ a , b , i , j ] = [ a , b , i , j ] × [1 /b , /a , j , i ]provided that a > b < a = i = 0 or b = j = 0.Note in the above relations that all ambiguous cases ∞ − ∞ , ±∞ × ±∞±∞ and will induce the biggest possible interval i.e. R .As an easy observation, when an interval X = [ a, b, i, j ] with a = b contains zero, we can compute 1 /X as follows: • If a = 0 then 1 X = 1[ a, b, , j ] = [ 1 b , + ∞ , j, , If b = 0 then 1 X = 1[ a, b, i,
0] = [ −∞ , a , , i ] , • If ab < X as X = [ a, , i, ∪ [0 , b, , j ] we have1 X = 1[ a, , i, ∪ [0 , b, , j ] = [ −∞ , a , , i ] ∪ [ 1 b , + ∞ , j, . Now consider A and B are two intervals as two sets of real numbers. We canstate the above definitions of four essential arithmetic operations as A op B = { x | ∃ a ∈ A, b ∈ B : x = a op b } where op ∈ { + , − , × , / } . Remark 2.3.
Although interval arithmetic seems to be compatible with realnumbers arithmetic, but this affects distributivity of multiplication over addi-tion and the existance of inverse elements. More preciesly for each intervals
X, Y and Z , • X × ( Y + Z ) ⊆ X × Y + X × Z ,and if X is non-degenerated then • X × X = 1 , but ∈ X × X and • X + ( − X ) = 0 , but ∈ X + ( − X ) .Furthermore, if X contains some negative real numbers, then X n = X · · · X | {z } n times . To see this, let for instance X = [ a, b, i, j ] where a < and | a | < b . Then, X = [0 , b , , j ] while X × X = [ ab, b , min( i, j ) , j ] . To solve this inconsis-tency, we define the n , th power of an interval for non-negative integer n asfollows: [ a, b, i, j ] n = n = 0[ a n , b n , i, j ] 0 ≤ a [ b n , a n , j, i ] b ≤ , max( a n , b n ) , , c ] a < < b where c = (cid:26) i | b | < | a | j otherwise . f ( x, y ) = xx + y , X = [1 ,
2] and Y = [1 , f ( X, Y ) in two ways. The first way is to compute f ( X, Y ) as an usual evaluation using interval arithmetic: XX + Y = [1 , / [2 ,
5] = [1 / , . (2)However, one can manipulate the expression to see XX + Y = 11 + YX = 1 / [3 / ,
4] = [1 / , /
3] (3)Let us separate x and y first depending on f ( x, y ): we call y (resp. x ), a first (resp. second ) class variable of f as it appears one (resp. more thanone) time in the structure of f . As it can be easily seen, the answer of (3)is a narrower interval and in fact the exact value. The reason is that X is asecond class variable for (2) and so it brings dependency between two partsof the expression. This is while there is no dependency in (3) given that YX appears only one time, and so it is a first class variable. Dependency is one ofthe crucial points of this paper to find the solution set of interval polynomialsystems.Dependencies are the main reason that causes to appear an amount oferror by introducing larger intervals than the exact solution as all publicationsin this area try to avoid dependencies. Nevertheless it is possible to canceldependencies by considering X − X = 0 (see [16]) as well as X/X = 1 easily,while sometimes this becomes a difficult work.
Let R be the field of real numbers, considered as the ground field of compu-tations all over the current text and consider the set { x , . . . , x n } to be theset of variables. Definition 3.1.
Each polynomial of the form [ f ] = m X i =1 [ a i , b i , ℓ i , k i ] x α i · · · x α in n (4)6 s called an interval polynomial, where [ a i , b i , ℓ i , k i ] is a real interval for each i = 1 , . . . , m , and each power product x α i · · · x α in n is called a monomial wherethe powers are non negative integers. We denote the set of all interval poly-nomials by [ R ][ x , . . . , x n ] . Definition 3.2.
Let [ f ] be an interval polynomial as defined in (4). The set ofall polynomials arising from [ f ] for different values of intervals in coefficientsis called the family of [ f ] and is denoted by F ([ f ]) . More preciesly: F ([ f ]) = { m X i =1 c i x α i · · · x α in n | c i ∈ [ a i , b i , ℓ i , k i ] , i = 1 , . . . , m } Similar to the family of an interval polynomial, we can define the familyof a set of interval polynomials as follows:
Definition 3.3.
Let S := { [ f ] , . . . , [ f ] ℓ } be a set of interval polynomialswith F ([ f ] j ) = F j for each j = 1 , . . . , ℓ . We define the family of S to be theset F × · · · × F ℓ , denoted by F ( S ) . We now define the concept of solution set for an interval polynomial.
Definition 3.4.
For an interval polynomial [ f ] ∈ [ R ][ x , . . . , x n ] , we say that a = ( a , . . . , a n ) ∈ R n is a real solution or a real root of [ f ] , if there exists apolynomial p ∈ F ([ f ]) such that p ( a ) = 0 . Similarily, we say that a systemof interval polynomials has a solution, if the contained interval polynomialshave a common solution. Example 3.5.
Let us find the solution set of [ − , − x + [1 , x + [1 ,
3] = 0 where all intervals are closed. When x ≥ , we have [ − , − x + [1 , x + [1 ,
3] = [ − x + x + 1 , − x + 2 x + 3] = [0 , So we must have ≤ − x + x + 1 and − x + 2 x + 3 ≤ which implies x ∈ [1 , . imilarly when x ≤ , we have [ − , − x + [1 , x + [1 ,
3] = [ − x + 2 x + 1 , − x + x + 3] = [0 , or equivalently ≤ − x + 2 x + 1 and − x + x + 3 ≤ which concludes x ∈ [ − . , − . . Thus the solution set of this interval polynomial is [ − . , − . ∪ [1 , . It is notable that using interval arithmetic in the well-known solution way ofa quadratic polynomial equation due to discriminant, we receive to [ − . , − . ∪ [0 . , . as the solution set that contains an amount of error. In this section we are going to describe the problems which may occur usingthe usual elimination method on a system of interval polynomials. To facili-tate the description, let us give an example. Consider the system containing f = [1 , x + x + 2 x , f = [1 , x + x + 1 , f = [3 , x + x + 4 x . Going to eliminate the variable x , we have f − f = [ − , x + 2 x − , f − f = [ − , x + 4 x − . One may now conclude that if we choose 0 from both intervals [ − ,
1] and[ − ,
3] then the system has no solution because 2 x − x − . However this case is impossible since both of intervals [ − ,
1] and [ − ,
3] cannot be zero at once! To see this, notice that [ − ,
1] comes from [1 , − [1 , − ,
1] being zero, [1 ,
4] must give some values in [1 , − ,
3] comes from [3 , − [1 ,
4] and so this interval can be zeroonly when [1 ,
4] gives some values in [3 ,
4] and this is a contradiction. This8imple linear system shows that the usual elimination method can cause awrong decision or appearing some extra values in the solution set. The mainreason is that we forgot the dependencies between [ − ,
1] and [ − ,
3] duringthe computation while they are both dependent on [1 ,
4] and so they aredependent.To solve this problem, we must keep the trace of each interval coefficient.In doing so, our idea is to use a parameter instead of each interval, to seehow new coefficients are built. For instance let us substitute [1 , ,
4] and[3 ,
4] by a , b and c as parameters in the above example. So we have˜ f = ax + x + 2 x , ˜ f = bx + x + 1 , ˜ f = cx + x + 4 x and doing elimination steps we have˜ f − ˜ f = ( a − b ) x + 2 x − , ˜ f − ˜ f = ( c − b ) x + 4 x − . Now one can conclude that under the assumption that a − b = 0, the coeffi-cient c − b can not be zero. The reason is that if c − b = 0 then a = c while1 ≤ a ≤ ≤ c ≤
4. Therefore using parameters prevent us takingwrong decisions about the solution set.The main question here is that how we can use elimination method whenthe coefficients contain some parameters. In fact as we will state in the nextsections, we do the elimination steps thanks to Gr¨obner basis and for theparametric case, we use the concept of comprehensive Gr¨obner system, tosee the simplest possible polynomials to solve. So our idea is to convertthe interval polynomial systems to a parametric polynomial system and usethe parametric algorithmic aspects, of course with some modifications, tosolve the parametric system by dividing the solution set into finitely manycomponents. At the end, we convert the result to see the solution set of theinterval polynomial system.
In this section we recall the concepts and notations of ordinary and para-metric polynomial rings. Let K be a field and x , . . . , x n be n (algebraicallyindependent) variables. Each power product x α · · · x α n n is called a monomial9here α , . . . , α n ∈ Z ≥ . Because of simplicity, we abbreviate such monomi-als by x α where x is used for the sequence x , . . . , x n and α = ( α , . . . , α n ).We can sort the set of all monomials over K by special types of total orderingsso called monomial orderings, recalled in the following definition. Definition 5.1.
The total ordering ≺ on the set of monomials is called amonomial ordering whenever for each monomials x α , x β and x γ we have: • x α ≺ x β ⇒ x γ x α ≺ x γ x β , and • ≺ is well-ordering. There are infinitely many monomial orderings, each one is convenient fora special type of problems. Among them, we point to pure and graded reverselexicographic orderings denoted by ≺ lex and ≺ grevlex as follows. assume that x n ≺ · · · ≺ x . We say that • x α ≺ lex x β whenever α = β , . . . , α i = β i and α i +1 < β i +1 for an integer 1 ≤ i < n . • x α ≺ grevlex x β if n X i =1 α i < n X i =1 β i breaking ties when there exists an integer 1 ≤ i < n such that α n = β n , . . . , α n − i = β n − i and α n − i − > β n − i − . It is worth noting that the former has many theoretical importance while thelatter speeds up the computations and carries fewer information out.Each K − linear combination of monomials is called a polynomial on x , . . . , x n over K . The set of all polynomials has the ring structure with usual poly-nomial addition and multiplication, and is called the polynomial ring on x , . . . , x n over K and denoted by K [ x , . . . , x n ] or just by K [ x ]. Let f bea polynomial and ≺ be a monomial ordering. The greatest monomial w.r.t. ≺ contained in f is called the leading monomial of f , denoted by LM( f )and the coefficient of LM( f ) is called the leading coefficient of f which ispointed by LC( f ). Further, if F is a set of polynomials, LM( F ) is defined10o be { LM( f ) | f ∈ F } and if I is an ideal, in ( I ) is the ideal generated byLM( I ) and is called the initial ideal of I . We are now going to remind theconcept of Gr¨obner basis of a polynomial ideal which carries lots of usefulinformation out about the ideal. Definition 5.2.
Let I be a polynomial ideal of K [ x ] and ≺ be a monomialordering. The finite set G ⊂ I is called a Gr¨obner basis of I if for each nonzero polynomial f ∈ I , LM( f ) is divisible by LM( g ) for some g ∈ G . Using the well-known Hilbert basis theorem (See [4] for example), it isproved that each polynomial ideal possesses a Gr¨obner basis with respectto each monomial ordering. There are efficient algorithms also to computeGr¨obner basis. The first and the most simplest one is the Buchberger al-gorithm which is devoted in the same time of introduction of Gr¨obner basisconcept while he most efficient known algorithm is the Faug`ere’s F algorithm[10] and another signature-based algorithms such as G V [13] and GVW [14].It is worth noting that Gr¨obner basis of an ideal is not unique necessarily. Tohave unicity, we define the reduced Gr¨obner basis concept. As an importantfact the reduced Gr¨obner basis of an ideal is unique up to the monomialordering.
Definition 5.3.
Let G be a Gr¨obner basis for the ideal I w.r.t. ≺ . Then G is called a reduced Gr¨obner basis of I whenever each g ∈ G is monic, i.e. LC( g ) = 1 and none of the monomials appearing in g is divisible by LM( h ) for each h ∈ G \ { g } . One of the most important applications of Gr¨obner basis is its help tosolve a polynomial system. Let f = 0... f k = 0be a polynomial system and I = h f , . . . , f k i be the ideal generated by f , . . . , f k . We define the affine variety associated to the above system orequivalently to the ideal I to be V ( I ) = V ( f , . . . , f k ) = { α ∈ K n | f ( α ) = · · · = f k ( α ) = 0 } where K is used to denote the algebraic closure of K . Now let G be aGr¨obner basis for I with respect to an arbitrary monomial ordering. As an11nteresting fact, I = h G i which implies that V ( I ) = V ( G ). This is the keycomputational trick to solve a polynomial system. Let us continue by anexample. Example 5.4.
We are going to solve the following polynomial system: x − xyz + 1 = 0 y + z − xy + z = 0 By the nice properties of pure lexicographical ordering, the reduced Gr¨obnerbasis of the ideal I = h x − xyz + 1 , y + z − , xy + z i ⊂ Q [ x, y, z ] has theform G = { g ( z ) , x − g ( z ) , y − g ( z ) } w.r.t. z ≺ lex y ≺ lex x , where g ( z ) = z − z + 5 z − z − z − z + 4 z − z + 4 z − g ( z ) = 2 z − z + 11 z + 2 z − z − z + 2 z − z + 4 z ++7 z − z − z + 11 z + 2 z − g ( z ) = z − z + z + 2 z + z − z − z + 2 z − z − z + 1 . This special form of Gr¨obner basis for this system allows us to find V ( G ) bysolving only one univariate polynomial g ( z ) and putting the roots into thetwo last polynomials in G . Suppose now that the same system of Example 5.4 is given as followswith parametric coefficients on parameters are a, b and c : ax − ( a − b + 1) xyz + 1 = 0 y + c z − a + b + c ) xy + z = 0The solutions of this system depend on the values of parameters apparently aswe can see that the system has no solutions whenever a = 0 and b = 1 whileit converts to the system of Example 5.4 for a = 1 , b = 1 and c = − K t into a finite number of partitions, for which thegeneral form of polynomials contained in assigned Gr¨obner basis is known.12et K be a field and a := a , . . . , a t and x := x , . . . , x n be the sequencesof parameters and variables respectively. We call K [ a ][ x ], the parametricpolynomial ring over K , with parameters a and variables x . This ring is infact the set of all parametric polynomials as m X i =1 p i x α i where p i ∈ K [ a ] is a polynomial on a with coefficients in K , for each i . Definition 5.5.
Let I ⊂ K [ a ][ x ] be a parametric ideal and ≺ be a monomialordering on x . Then the set G ( I ) = { ( E i , N i , G i ) | i = 1 , . . . , ℓ } ⊂ K [ a ] × K [ a ] × K [ a ][ x ] is said a comprehensive Gr¨obner system for I if for each ( λ , . . . , λ t ) ∈ K t and each specialization σ ( λ ,...,λ t ) : K [ a ][ x ] → K [ x ] P mi =1 p i x α i m X i =1 p i ( λ , . . . , λ t ) x α i there exists an ≤ i ≤ ℓ such that ( λ , . . . , λ t ) ∈ V ( E i ) \ V ( N i ) and σ ( λ ,...,λ t ) ( G i ) is a Gr¨obner basis for σ ( λ ,...,λ t ) ( I ) with respect to ≺ . Becauseof simplicity, we call E i and N i the null and non-null conditions respectively. Remark that, by [39, Theorem 2.7], every parametric ideal has a compre-hensive Gr¨obner system. Now we give an example from [20] to illustrate thedefinition of comprehensive Gr¨obner system.
Example 5.6.
Consider the following parametric polynomial system in Q [ a, b, c ][ x, y ] : Σ : ax − b = 0 by − a = 0 cx − y = 0 cy − x = 0 Choosing the graded reverse lexicographical ordering y ≺ x , we have the fol-lowing comprehensive Gr¨obner system: i E i N i { } { } { a − b , a c − b , b c − a ,ac − a, bc − b }{ bx − acy, by − a } { a − b , a c − b , b c − a , { b } ac − a, bc − b }{ cx − y, cy − x } { a, b } { c }{ x, y } { a, b, c } { } For instance, for the specialization σ (1 , , for which a , b and c , σ (1 , , ( { bx − acy, by − a } ) = { x − y, y − } is a Gr¨obner basis of σ (1 , , ( h Σ i ) . It is worth noting that if V ( E i ) \ V ( N i ) = ∅ for some i , then the triple( E i , N i , G i ) is useless, and so it must be omitted from the comprehensiveGr¨obner system. In this case we say that the pair ( E i , N i ) is inconsistent . It iseasy to see that inconsistency occurs if and only if N i ⊂ p h E i i and so we needto an efficient radical membership test to determine inconsistencies. In [19,20] there is a new and efficient algorithm to compute comprehensive Gr¨obnersystem of a parametric polynomial ideal which uses a new and powerfulradical membership criterion. Therefore we prefer to employ this algorithmso called PGB algorithm in our computations. Another essential trick whichis used in [20] is the usage of minimal Dickson basis which reduces the contentof computations in
PGB . Before explain it, let us recall some notations whichare used in the structure of
PGB . Let ≺ x and ≺ a be two monomial orderingson K [ x ] and K [ a ] respectively. Let also ≺ x , a be the block ordering of ≺ x and ≺ a , comparing two parametric monomials by ≺ x , breaking tie by ≺ a .For a parametric polynomial f ∈ K [ a ][ x ], we denote by LM x ( f ) (resp. byLC x ( f )) the leading monomial (resp. the leading coefficient) of f when it isconsidered as a polynomial in K ( a )[ x ], and so LC x ( f ) ∈ K [ a ]. Definition 5.7.
By the above notations, let P ⊂ K [ a ][ x ] be a set of para-metric polynomials and G ⊂ P . Then, G is called a minimal Dickson basisof P denoted by M DBasis ( P ) , if: • For each p ∈ P , there exist some g ∈ G such that LM x ( g ) | LM x ( p ) and • For each two distinct polynomials in G as g and g , none of LM x ( g ) and LM x ( g ) divides another. PGB to compute a minimal Dickson basis for P is only when P is a Gr¨obner basis for h P i itself w.r.t. ≺ x , a and P ∩ K [ a ] = { } .In this situation, it suffices by Definition 5.7 to omitt all polynomials p from P for which there exists a p ′ ∈ P such that LM x ( p ′ ) | LM x ( p ).The PGB algorithm as is shown below, uses
PGB-main algorithm tointroduce new branches in computations.
Algorithm 1
PGB procedure PGB ( P, ≺ a , ≺ x ) E, N := { } , { } ; ≺ x , a :=The block ordering of ≺ x , ≺ a Return
PGB-main ( P, E, N, ≺ x , a ); end procedure The main work of
PGB-main is to create all necessary branches andimport them in comprehensive Gr¨obner system at output. In this algorithm A ∗ B is defined to be the set { ab | a ∈ A, b ∈ B } .15 lgorithm 2 PGB-main procedure
PGB-main ( P, E, N, ≺ x , a ) G := The reduced Gr¨obner basis for P w.r.t. ≺ a , x if ∈ G thenReturn ( E, N, { } ); end if G r := G ∩ K [ a ]; if IsConsistent ( E, N ∗ G r ) then P GB := { ( E, N ∗ G r , { } ) } ; else P GB := ∅ ; end ifif IsConsistent ( G r , N ) then G m := MDBasis ( G \ G r ); elseReturn ( P GB ); end if h := lcm( h , . . . , h k ), where h i = LC x ( g i ) and g i ∈ G m ; if IsConsistent ( G r , N ∗ { h } ) then P GB := P GB ∪ { ( G r , N ∗ { h } , G m ) } ; end iffor i = 1 , . . . , k do P GB := P GB ∪ PGB-main ( G \ G r , G r ∪{ h i } , N ∗{ Q i − j =1 h j } , ≺ a , x ) end forend procedure As it is shown in the algorithm, it computes first a Gr¨obner basis ofthe ideal h P i over K [ a , x ] i.e. G , before performing any branches based onparametric constraints, according to [20, Lemma 32] as follows: Lemma 5.8.
By the notations used in the algorithm, for each specialization σ ( λ ,...,λ t ) if ( λ , . . . , λ t ) ∈ V ( G r ) \ V ( Y g ∈ G \ G r LC x ( g )) then σ ( λ ,...,λ t ) ( G ) is a Gr¨obner basis for σ ( λ ,...,λ t ) ( h P i ) . After this, the algorithm computes a minimal Dickson basis i.e. G m andcontinues by taking a decision for each situation that one of the leading16oeffiecients of G m is zero. By this, PGB-main constructs all necessarybranches to import in comprehensive Gr¨obner system. All over the algo-rithm, when it needs to add a new branch ( E i , N i , G i ) into the system, thealgorithm IsConsistent is used as follow to test the consistency of para-metric conditions ( E i , N i ). Algorithm 3
IsConsistent procedure
IsConsistent ( E, N ) f lag :=false; for g ∈ N while f lag =false doif g / ∈ p h E i then f lag :=true; end ifend forReturn f lag ; end procedure The main part of this algorithm is radical membership test. The powerfultrick which is used in [19,20] to radical membership check is based on linearalgebra methods tackling with a probabilistic check. We refer the reader to[20, Section 5] for more details.
In this section we introduce the new concept of interval Gr¨obner system andits related definitions and statements.Now we state the following proposition as an immediate consequence ofDefinition 3.4. Recall that for a polynomial system S ⊂ R [ x , . . . , x n ] thevariety of S is the set of all complex solutions of S , denoted by V ( S ). Proposition 6.1.
A system [ S ] of interval polynomials has a solution if andonly if there exists a polynomial system S in F ([ S ]) with V ( S ) = ∅ . There is an efficient criterion due to the well-known Hilbert Nullestelen-satz theorem which determines if V ( S ) = ∅ by Gr¨obner basis: V ( S ) = ∅ ifand only if the Gr¨obner basis of h S i does not contain any constant. Note17hat there are infinitely many polynomial systems in F ([ S ]) for an intervalpolynomial system [ S ] and so it is practical impossible to check all of them byNullestelensatz theorem. Nevertheless, we give a finite partition on the set ofall polynomial systems arising from [ S ] using the concept of comprehensiveGr¨obner system. Definition 6.2.
Let [ S ] = { [ f ] , . . . , [ f ] ℓ } be a system of interval polynomi-als. We define the ideal family of [ S ] , denoted by IF ([ S ]) to be the set IF ([ S ]) = {h p , . . . , p ℓ i | ( p , . . . , p ℓ ) ∈ F ([ S ]) } Theorem 6.3.
Let [ S ] be a system of interval polynomials and ≺ be a mono-mial ordering on R [ x , . . . , x n ] . Then • The set of initial ideals { in ( I ) | I ∈ IF ([ S ]) } is a finite set, and • For each set of ideals of IF ([ S ]) with the same initial ideal, there existsa set of parametric polynomials which induces the ideals by differentspecializations.Proof. To prove this theorem, we use the concept of comprehensive Gr¨obnersystem. Suppose that S ∗ is obtained by replacing each interval coefficientby a parameter. Note that if an interval appears in t ≥ t distinct parameters to it. It is easy to check that each elementof IF ([ S ]) is the image of S ∗ under a suitable specialization. On the otherhand by [39, Theorem 2.7] S ∗ has a finite comprehensive Gr¨obner system as G = { ( E , N , G ) , . . . , ( E k , N k , G k ) } , where for each specialization σ thereexists a 1 ≤ j ≤ k such that LM( σ ( S ∗ )) = LM( G i ). It is worth noting thatalthough there is a finite number of branches in G , we can also remove thosespecializations with complex values, and also those with values out of theassigned interval. Thus for each I ∈ IF ([ S ]) there exists an 1 ≤ i ≤ k with in ( I ) = h LM( G i ) i and this finishes the proof.What is explained in the proof of Theorem 6.3 yields to extend the conceptof comprehensive Gr¨obner system for interval polynomials. Definition 6.4.
Let [ S ] ⊂ [ R ][ x , . . . , x n ] be a system of interval polynomialswith t interval coefficients, and ≺ be a monomial ordering on R [ x , . . . , x n ] .Let also that G = { ( E , N , G ) , . . . , ( E k , N k , G k ) } be a set of triples ( E i , N i , G i ) ∈ R [ h , . . . , h t ] × R [ h , . . . , h t ] × R [ h , . . . , h t ][ x , . . . , x n ]18 here { h , . . . , h t } is the set of parameters assigned to each interval coeffi-cient. Then we call G an interval Gr¨obner system for [ S ] denoted by G ≺ ([ S ]) if for each t − tuple ( a , . . . , a t ) of the inner values of interval coefficients thereexists an ≤ i ≤ k such that: • For each p ∈ E i , p ( a , . . . , a t ) = 0 , • There exist some q ∈ N i such that q ( a , . . . , a t ) = 0 , and • σ ( G i ) is a Gr¨obner basis for h σ ([ S ]) i with respect to ≺ , where σ is thespecialization h j a j for j = 1 , . . . , t . Theorem 6.5.
Each interval polynomial system possesses an interval Gr¨obnersystem.Proof.
Let S ∗ be the parametric polynomial system obtained by assigningeach interval coefficient to a parameter. As mentioned in the proof of The-orem 6.3, G ≺ ([ S ]) is the same comprehensive Gr¨obner system of S ∗ whereeach parameter is bounded to give values from its assigned ideal. On theother hand it is proved that each system of parametric polynomials has acomprehensive Gr¨obner system, which terminates the proof.We give now an easy example to illustrate what described above. Example 6.6.
Consider the interval polynomial system [ S ] = (cid:26) [ − , xy + [0 , y + [3 ,
5) = 0[ − , xy + [1 , y = 0 (5) To obtain a parametric polynomial system, we assign the intervals [ − , , [0 , , [3 , , [ − , and [1 , by h , . . . , h respectively. Then we observe the parametric polyno-mial system S ∗ = { h xy + h y + h , h xy + h y } ⊂ R [ h , . . . , h ][ x, y ] Using the lexicographic monomial ordering y ≺ x we can compute a compre-hensive Gr¨obner system for h S ∗ i which contains about triples. Howeversome of them are admissible only for some values of parameters out of theirassigned interval. For instance the triple ( { } , { h , h , h , h } , { h } ) is notacceptable in this example, since h ∈ [1 , and so it can not be zero. Byremoving such triples, there remains only one shown in the following table.Therefore the following table shows G ≺ ([ S ]) . i N i G i { h h } { h , h , h } { }{ h h } { h , h } { }{ h h h h } { h h − h h } { }{ h h h } { h , h } { }{ h h h } { h } { }{ h h h } { h } { h y + h , h h x − h h h }{ h h } { h , h h − h h } { h + h xy }{ h xy + h y + h , − h h y − h h y + h h y, { h h h h ( h h − h h ) } {} − h h h x + h h h x + h h h y + h h h , − h h h y − h h + h h h } Table 1: Interval Gr¨obner system of System (5)
In tis section we state our algorithm so called
IGS to compute intervalGr¨obner system for an interval polynomial system. This algorithm is basedon the
PGB algorithm with some extra conditions for the definition of con-sistency. To begin let [ S ] = { [ f ] , . . . , [ f ] ℓ } ⊂ [ R ][ x , . . . , x n ] be a system ofinterval polynomials, where for each 1 ≤ j ≤ ℓ ,[ f ] j = m j X i =1 [ a ij , b ij ] x α ij, · · · x α ij,n n and ( α ij, , . . . , α ij,n ) ∈ Z n ≥ , for each i . As it is mentioned in Theorem 6.3, weassign to each interval coefficient [ a ij , b ij ] a parameter h ij to convert [ S ] toa parametric polynomial system S ∗ . The following proposition describes therelations between comprehensive Gr¨obner systems of S ∗ and interval Gr¨obnerbases of [ S ]. Proposition 6.7.
Using the above notations, let [ G ] and G be an intervalGr¨obner basis for [ S ] and a comprehensive Gr¨obner basis for S ∗ respectivelyw.r.t. the same monomial ordering. Then for each ( E, N, G ) ∈ [ G ] , thereexists ( E ′ , N ′ , G ′ ) ∈ G such that V ( E ) \ V ( N ) ⊂ V ( E ′ ) \ V ( N ′ ) and G, G ′ have the same initial ideal.Proof. This comes from Definitions 6.4 and 5.5.20ccording to the above proposition, to compute an interval Gr¨obner basisfor [ S ], it is enough to compute a comprehensive Gr¨obner basis for S ∗ , anduse a criterion to omit those triples ( E, N, G ) lying in
G \ [ G ], we call them redundant triples. Remark 6.8.
Note that for a triple ( E, N, G ) in G \ [ G ] , the intersection of V ( E ) \ V ( N ) with the cartesian product of interval coefficients is empty. We are now going to present a criterion to determine the elements of
G \ [ G ]. This criterion of course is based on the answer of this question: How can we sure that a system of polynomials E ⊂ R [ a , . . . , a t ] has a realroot in the interval [ α , β ) × · · · × [ α t , β t ) ? In the case for which h E i is zero dimensional, this question is answeredtotally thanks to some efficient computational tools like Sturm’s chain byisolating the real roots. However in the case of positive dimensional, thereexist only some algorithm to isolate the real roots. Among them there existsan algorithm which determines whether a multivariate polynomial systemhas real root or not.Because of the reasons declared above, we convert the above key questionto the problem of determining whether a polynomial system has a real rootor not. Theorem 6.9.
Let E ⊂ R [ a , . . . , a t ] be a finite polynomial set. Let also F = E ∪ { a i + ( a i − β i ) b i − α i | i = 1 , . . . , t } ⊂ R [ a , . . . , a t , b , . . . , b t ] where b j ’s are algebraic independent by a i ’s and [ α i , β i ) be a real interval foreach i = 1 , . . . , t . Then the system E = 0 has a solution in [ α , β ) × · · · × [ α t , β t ) if and only if the system F = 0 has a real solution.Proof. Let E = 0 has a solution ( γ , . . . , γ t ) ∈ [ α , β ) × · · · × [ α t , β t ). Letalso η i = r α i − γ i γ i − β i for each i = 1 , . . . , t . It is easy to see that γ i + ( γ i − β i ) η i − α i = 021hich implies that ( γ , . . . , γ t , η , . . . , η t ) is a solution of F = 0.Conversely, suppose that there exists ( γ , . . . , γ t , η , . . . , η t ) ∈ R t whichis a solution of F = 0, i. e. f ( γ , . . . , γ t ) = 0 for each f ∈ F and γ i + ( γ i − β i ) η i − α i = 0, for each i = 1 . . . , t . It is enough to show that γ i ∈ [ α i , β i ).In doing so, we see that γ i = α i + β i η i η i = ( β i − α i ) η i η i + α i . Indeed, 0 ≤ η i η i < α i ≤ ( β i − α i ) η i η i + α i | {z } γ i < β i which finishes the proof. Remark 6.10.
Note that for the intervals [ α, ∞ ) and ( −∞ , β ] we can usethe auxiliary polynomials a − α − b and a − β + b respectively. Using Theorem 6.9 and the Remarks 6.8, 6.10 we can determine theelements of
G \ [ G ] exactly (see the notations of Proposition 6.7). Corollary 6.11.
Let ( E, N, G ) ∈ G and [ α , β ) , . . . , [ α t , β t ) be t real inter-vals. Then ( E, N, G ) is redundant if and only if the system F = 0 has noreal roots, where F = E ∪ { a i + ( a i − β i ) b i − α i | i = 1 , . . . , t } ∪ { Y g ∈ N ( c g g − }⊂ R [ a , . . . , a t , b , . . . , b t , c g : g ∈ N ] . Proof.
The proof comes from Theorem 6.9 and this fact that if Q g ∈ N ( c g g −
1) = 0 then there exists a g ∈ N for which c g g − g = 0.The above corollary is the criterion which determines all redundant triples,and so tackling this criterion with PGB algorithm we can design our newalgorithm to compute interval Gr¨obner systems. We design now the
IGS algorithm by its main procedure. 22 lgorithm 4
IGS procedure
IGS ( S [ ] , ≺ x )Assign a , . . . , a t to interval coefficients and name it S ∗ ; ≺ a := an arbitrary monomial ordering on a , . . . , a t ; E, N := { } , { } ; ≺ x , a :=The block ordering of ≺ x , ≺ a Return
PGB-main ( P, E, N, ≺ x , a , L ); \\ L is the ordered set of inter-val coefficients which is needed to check consistency end procedure The
PGB-main algorithm is the same which which is used in
PGB algorithm. We only change the definition of consistency as below.
Definition 6.12.
Let [ α , β ) , . . . , [ α t , β t ) be t real intervals and E, N ⊂ R [ a , . . . , a t ] . The pair ( E, N ) is called consistent if it is not redundant, orequivalently, [ V ( E ) \ V ( N )] ∩ [ α , β ) × · · · × [ α t , β t ) = ∅ . According to the above definition, we change the
IsConsistent algo-rithm to
Interval-IsConsistent , which checks the consistency for radicalmembership and redundancy determination.23 lgorithm 5
Interval-IsConsistent procedure
Interval-IsConsistent ( E, N, [ α , β ) , . . . , [ α t , β t )) test :=false; f lag :=false; if ( E, N ) is not redundant then test :=true; end ifif test then f lag :=false; for g ∈ N while f lag =false doif g / ∈ p h E i then f lag :=true; end ifend forend ifReturn f lag ; end procedureRemark 6.13. It is worth noting that redundant triples will be omitted beforethe algorithm goes to continue with them. This property causes that
IGS returns less triples than
PGB . In this section we state the ability of interval Gr¨obner system to solve para-metric polynomial systems for which the parameters range over specific in-tervals, which may appear, for instance, when analysing fuzzy polynomialsystems. In doing so, we can use the same
IGS algorithm, after convertingan interval polynomial system to a parametric one. To give an example, wepoint at resolution of fuzzy polynomial systems which converts to solve aparametric polynomial system with parameters range over [0 ,
1] (see [1]).
Example 7.1. ([1, Example 4.3] resolved) Consider the following system offuzzy polynomials: (cid:26) (0 . , . , . , . , , x + (0 . , . , . y (1 . , . , .
75) = (0 . , , . x + (1 . , . , . xy o solve this system, the general way is to decompose it into four paramet-ric polynomial systems. Here we solve only the third and fourth systems.Consider the third system as follows: (3) : f = ( − h ) x + ( h ) y + − hf = ( − + h ) x + ( − h ) y − + hf = ( − + h ) x + (2 − h ) xy − hf = (1 − h ) x + (1 + h ) xy − h Computing an interval Gr¨obner basis w.r.t. y ≺ lex x , we have only a triple ( { } , { h − } , { } ) which means that we have no solutions whenever h =1 . As our algorithm deals with the values of h ∈ [0 , , we must considerthe case h = 1 seperately. In this case, we find a solution ( h = 1 , x = − . , y = 0 . regarding the sign of variables.The fourth system is: (4) : f = ( − h ) x + ( − h ) y + − hf = ( − + h ) x + ( h ) y − + hf = ( − + h ) x + (1 + h ) xy − hf = (1 − h ) x + (2 − h ) xy − h where h ∈ [0 , , x ≤ and y ≤ . Computing an interval Gr¨obner basis, wesee only one triple ( { } , { h − h + 3 } , { } ) for which its Gr¨obner basis equalsto { } . Therefore the above system has no solution for h ∈ [0 , . Note thatour method deals with [0 , here and so we must check the system for h = 1 separately. By this, the system has no solution again by considering the signof x and y . and so this system has no solution.Note that another way to inform that these system have or have not anyreal solutions (and not the exact form of solutions), is to check their con-sistency by what we have stated in Algorithm 6.1, since all of variables haveinterval form in these systems. In doing so we must add three auxiliary poly-nomials h +( h − h , x +˜ x and y − ˜ y to the System (3) and h +( h − h , x +˜ x and y + ˜ y to the System (4), where ˜ h, ˜ x and ˜ y are some extra real variables.The result of course is that (3) is consistent however (4) is not. One of the interesting problems in the context of interval polynomials is the
Divisibility Problem stated as follows (See [32]):25 or an interval polynomial [ f ] ∈ [ R ] { x , . . . , x n } and a real polynomial g ∈ R [ x . . . . , x n ] , determine whether there is a polynomial p ∈ F ([ f ]) suchthat g is a factor of p . In [32] there is stated an efficient method based on linear programmingtechniques and nice properties of polytopes. In the sequel we exalin ourmethod to solve this problem by interval Gr¨obner basis. To continue, letus say that g i-divides [ f ] whenever the answer of the above problem is yes (Note that the letter ”i” stands for interval). By this conception we can nowexplain the following criterion based on interval Gr¨obner basis. Theorem 7.2.
Using the above notations, let [ S ] = { [ f ] , g } ⊂ [ R ] { x , . . . , x n } and [ G ] be a reduced interval Gr¨obner system for [ S ] with respect to a mono-mial ordering ≺ . Then g i-divides [ f ] if and only if there exists a triple ( E, N, G ) ∈ [ G ] where G = { g } and V ( E ) \ V ( N ) = { } .Proof. It is easy to see that g i-divides [ f ] if and only if there exists anspecialization σ for which g | σ ([ f ]) and of course σ ([ f ]) = 0, by the statementof divisibility problem. This implies that σ ([ S ]) can be expressed only by { g } and therefore there exists a pair of parametric sets ( E, N ) such that(
E, N, { g } ) ∈ [ G ]. Example 7.3.
We are going to solve the divisibility problem for [ f ] = [ − , x + [ − , y + [1 / , xy ∈ [ R ] { x, y } and g = x − √ y ∈ R [ x, y ] . By computing a reduced interval Gr¨obner basisfor [ S ] = { [ f ] , g } , we find the triple ( { c √ a + b } , { } , { x − √ y } ) where a, b and c denote inner values of the intervals [ − , , [ − , and [1 / , . This means that if the values of a, b and c satisfy the equation c √ a + b = 0 , then there exists some p ∈ F ([ f ]) such that g | p . For in-stance, by evaluating a = 1 / , b = − and so c = 2 √ / we find p = 1 / x − y +2 √ / xy ∈ F ([ f ]) which is divided by g (Note that p = 1 / x +3 √ y ) g ).Therefore g i-divides [ f ] . Remark 7.4.
We can use Theorem 7.2 for further aims. Let f, g ∈ R [ x , . . . , x n ] where g does not divide f . Then one can use Theorem 7.2 to find a polyno-mial ˜ f with the same coefficients of f which contain a few perturbation and | ˜ f . In doing so, one can convert f to an interval polynomial [ f ] by puttingthe interval [ c − ǫ, c + ǫ ] instead of the coefficient c , for each coefficient c appearing in f . Then using Theorem 7.2 one can increase the ǫ up enoughuntil g i-divides [ f ] with the desired precision. In the current paper we have introduced the concept of interval Gr¨obner sys-tem as a novel computational tool to analyse interval polynomial systems.We have further designed a complete algorithm to compute it using the ex-isting methods to analyse parametric polynomial systems. This concept cansolve some important problems from the applied and engineering researchfields such as solving fuzzy polynomial system, as devoted in this paper.
Acknowledgement
The authors would like to express their sincere thanks to Professor DeepakKapur, who devoted his time and knowledge during the preparation of thispaper. The third author would also like to give an special thanks to professorPrungchan Wongwises for her kindness on posting her Ph. D. thesis for him.
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