System theoretic based characterisation and computation of the least common multiple of a set of polynomials

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1 Linear Algebra and its Applications 381 (2004) 1 23 wwwelseviercom/locate/laa System theoretic based characterisation and computation of the least common multiple of a set of polynomials Nicos Karcanias a,, Marilena Mitrouli b a Control Engineering Centre, City University, Northampton Square, London EC1V OH, UK b Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece Received 16 March 2002; accepted 4 November 2003 Submitted by V Mehrmann Abstract The paper provides a system theoretic characterisation of the least common multiple (LCM) m(s) of a given set of polynomials P which leads to an efficient numerical procedure for the computation of LCM that avoids root finding and use of greatest common divisor (GCD) procedures The procedure that is presented also leads to the computation of the associated set of multipliers of P with respect to LCM The basis of the new characterisation and computational procedure are the controllability properties of a natural realization S(A,b,C)associated with the set P It is shown, that the coefficients of the LCM are defined by the properties of the controllable subspace of the pair (A, b), which also leads to the characterisation of associated multipliers An algorithmic procedure that exploits the companion structure of A is formulated and its numerical properties are investigated The new method provides a robust procedure for the computation of LCM and enables the computation of approximate values, when the original data are innacurate 2003 Elsevier Inc All rights reserved Keywords: Algebraic computations; Least common multiple; Polynomials; System theory 1 Introduction Two of the key problems of algebraic computations [3,6,10] are the computation of the greatest common divisor (GCD) and the computation of the least Corresponding author addresses: nkarcanias@cityacuk (N Karcanias), mmitrouli@ccuoagr (M Mitrouli) /$ - see front matter 2003 Elsevier Inc All rights reserved doi:101016/jlaa

2 2 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 common multiple (LCM) of a set of polynomials From the applications in Control Theory viewpoint the GCD is linked with the computation of zeros of representations whereas LCM is connected with the derivation of minimal fractional representations of rational models, which are essential for the study of a variety of algebraic design problems [10] We may always consider a set of polynomials in a degree ordered representation ie P ={p i (s), i k : p i (s) R[s], {p i (s)} =d i,d 1 = n d 2 = m d i,i = 3,,k} Such sets, with fixed the first two maximal degrees (n, m) may be represented by their coefficient vectors p i,wherep 1 R n+1, p i R m+1, i = 2, 3,,k and the set P by the vector p =[p t 1,pt 2,,pt k ]t R σ, σ = (n + 1) + (m + 1)(k 1), or by a point P in the projective space P σ 1 (R) Such a representation is linked to the generalised resultant [1] and allows the discussion of properties of sets P, when there is uncertainty in their coefficients The GCD and LCM problems are naturally interlinked [9], but they are of a different nature For k-sets of polynomials ordered by the two maximal degrees (n, m), those who have a nontrivial GCD (different than 1), belong to a proper variety of P σ 1 (R) Such variety is defined by the set of minors of the generalised resultant associated with P In this sense, the existence of a nontrivial GCD is a nongeneric property However, for every P set the corresponding LCM always exist It is such properties which make the GCD and LCM computational processes different In fact, the difference in nature of GCD and LCM computations is expressed by the fact that uncertainty in the coefficients of the set of polynomials results in coprimeness and thus we have a trivial GCD equal to 1 This is due to that the existence of nontrivial GCD is a nongeneric property [5,13,17] However, the same problem for LCM results always in a nontrivial solution with error in the coefficients and the degree For the case of two polynomials t 1 (s), t 2 (s) with LCM p(s) and GCD z(s), we have the standard identity that t 1 (s)t 2 (s) = z(s)p(s), which indicates the natural linking of the two problems For randomly selected polynomials, the existence of a nontrivial GCD is a nongeneric property [9,17], but the corresponding LCM always exists This suggests that there are fundamental differences between the two computational problems; such problems acquire a special dimension for engineering models, where uncertainty about the true value of the parameters on one hand and (rounding off) computational errors on the other, makes the computation of nongeneric values of invariants a very difficult task This creates the need for the development of procedures, which lead to meaningful approximate solutions Such computations are referred to as approximate, or robust algebraic computations [9] The case of GCD computation has taken a lot of attention [8,9,12] (and references there in) and algorithms dealing with the evaluation of exact and approximate solutions have been derived This paper deals with the corresponding problems of LCM computation, which has taken much less attention so far Existing procedures for the computation of LCM

3 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) rely on the standard factorisation of polynomials, computation of a minimal basis of a special polynomial matrix [2] and use of algebraic identities, GCD algorithms and numerical factorisation of polynomials [7] This paper presents an alternative approach to the computation of LCM, which is based on standard system theory concepts and avoids root finding, as well as use of the algebraic procedure and GCD computation The new characterisation of LCM leads to an efficient computational procedure based on properties of controllability of a linear system, associated with the given set of polynomials, and also provides a procedure for the computation of the associated set of polynomial multipliers linked to LCM For a given set of polynomials P a natural realization S(A,b,C) is defined by inspection of the elements of the set P It is shown that the degree r of LCM is equal to the dimension of the controllable subspace of the pair (A, b), whereas the coefficients of LCM express the relation of A r b with respect to the basis of the controllable space The companion form structure of A simplifies the computationof controllability properties and leads to a simple procedure for defining the associated set of polynomial multipliers of P with respect to LCM A special feature of the algorithmic procedure is that a number of possibly difficult steps are substituted by simple closed formulae derived from the special structure of the system An overall algorithmic procedure is formulated for computing LCM and multipliers, which is based on standard numerical linear algebra procedures The numerical aspects of the algorithm are investigated and demonstrated by a number of examples The developed results provide a robust procedure for the computation of LCM and enable the computation of approximate values, when the original data have some numerical inaccuracies In such cases the method computes an approximate LCM with degree smaller than the generic degree (see Example 43 of Section 4 of the paper) In fact, a generic set of polynomials is coprime and thus their LCM is their product The existence of an LCM with degree different than that of the product of polynomials occurs only when the given set of polynomials is not coprime In fact, an LCM with degree less than the degree of the product of all polynomials implies that at least two polynomials in the set have a nontrivial GCD and this is a nongeneric property Therefore, this degree with value less than the degree of the product is a nongeneric property for the set of all k-number polynomials (ordered by the (n, m) degrees Throughout the paper if a property is said to be true for i k, k Z +, this means it is true for all 1 i k The proof of the results are given in Appendix A 2 Problem statement, definitions and preliminary results Consider the set of polynomials P ={p i (s), i k : p i (s) R[s], {p i (s)}=d i }, where R[s] denotes the ring of polynomials over R and { } denotes the degree function By LCM {P} p(s) and GCD {P} z(s) we shall denote the LCM and GCD

4 4 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 of the set P respectively Without loss of generality, we may assume the polynomials p i (s) to be monic, ie p i (s) = s d i + a1 i sdi 1 + +ad i i 1 s + ai d i, i k, (21) where aj i denotes the coefficient a j of the ith polynomial of the set We define the rational vector associated with P p 1 (s) 1 p 2 (s) 1 g P (s) = Rk (s) (22) p k (s) 1 as the associated rational representative (ARR) of P Clearly if the p i (s) are not monic, then we can always write g P (s) = diag{c 1,,c k } g P (s), (23) where g P (s) corresponds to a monic set The problem we consider here is the computation of the LCM of P using the system based properties of the minimal linear system associated with g P (s) [3,6] This is aimed as an alternative to the algebraic procedures [7] based on algebraic properties and use of GCD, or alternative algebraic procedures [2] For the transfer function matrix g P (s) we may always define a left, right coprime matrix fraction description (MFD) [6] g P (s) = D l (s) 1 n l (s) = n r (s)d r (s) 1, (24) where D l (s) R k k [s],d r (s) R[s] For the system represented by g P (s) we have the following properties: Proposition 21 The ARR g P (s) of P has the following properties: (i) g P (s) has no finite zeros and it is strictly proper (ii) If ν = min{d i : d i = {p i (s)}, i k}, then g P (s) has an infinite zero of order ν (iii) If p(s) is an LCM of P, then p(s) is defined (modulo c R, c /= 0) by p(s) = D r (s) = D l (s) (25) (iv) If S(A,b,C ) is any realization of g P (s), then it is minimal, if and only if the following conditions hold true p(s) = si A =D r (s) = D l (s) (26) The above result provides a procedure for computing LCM as the characteristic polynomial of a minimal realization An efficient procedure for such a computation is summarised by the following remark

5 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) Remark 21 If S(A,b,C ) is a balanced minimal realization of g P (s) [11], then p(s) = si A provides an efficient, realization based computational procedure for LCM Although a balanced realization provides a stable numerical procedure, it has the disadvantage that it transforms the original data and this may lead to creation of additional numerical instabilities In the following, we shall work with a realization that is not minimal, which however is based on the original data and thus avoids introduction of additional numerical errors Such a realization is defined below: Proposition 22 Let g P (s) R k (s) be the ARR of P We may define a state space realization of g P (s) as follows: (a) For every p i (s) as in (21), we define the phase variable realization of p i (s) 1 as: p i (s) 1 = c t i (si A i) 1 b i, (27a) where A i = R d i d i, (27b) ad i i ad i i 1 a2 i a1 i c t i =[1, 0,,0, 0] R1 d i, b i =[0, 0,,0, 1] t R d i (b) Then g P (s) has the following phase variable realization g P (s) = C P (si A P ) 1 b P, (28a) where C P R k σ,a P R σ σ,b P R σ,σ = k i=1 d i and C P = bl-diag{,c t i,}, A P = bl-diag{,a i,}, b 1 b P = b i (28b) b σ The above realization is the phase variable realization [6] of g P (s) and S(A P,b P,C P ) will be called the normal system associated with P and shall be denoted in short by S P The advantage of such realization is that it is defined by

6 6 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 inspection of the set P without any numerical operations A key property of this realization is defined below: Proposition 23 The normal system S(A P,b P,C P ) associated with the set P is always observable, but not always controllable [6] For the case where P has polynomials which are pairwise coprime, then (A P,b P ) is also controllable The above results provide the means for the computation of LCM based on the properties of the natural system S P The study of these properties lead to a new system theoretic characterisation of LCM and is considered next 3 LCM characterisation in terms of properties of the natural system The controllability properties of S(A P,b P,C P ) are investigated next and are linked to the characterisation of LCM Theorem 31 Consider the normal system S(A P,b P,C P ) of P and let p(s) be the LCM of P The following properties hold true: (i) If r is the dimension of the controllable subspace of the pair (A P,b P ), then r = {p(s)} (ii) Let x = Px be a coordinate transformation that reduces (A P,b P,C P ) triple to the controllability normal form (Ã P, b P, C P ) Ã P = PA P P 1 Ā P Ā12 b P =, b P = Pb P =, O Ā 0 P C p =[ C P, C P ]=C PP 1, (31) where (Ā P, b P ) is controllable Then, (a) The monic LCM of P is defined by p(s) = si Ā P (b) A right coprime MFD of g P (s) is defined by g P (s) = C P adj(si Ā P ) b P / si Ā P =n(s) p(s) 1 (32) The above result establishes the link of LCM to the properties of the controllable space of (A P,b P ) pair This space Q P is defined as the column space of the controllability matrix [6] Q(A P,b P ) =[b P,A P b P,,A σ 1 P b P ] (33)

7 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) and the number r = [p(s)] is defined by r = rank Q(A P,b P ) = dim Q P (34) Lemma 31 The rank r of Q(A P,b P ) is the minimal number for which the set {Q r }={b P,A P b P,,A r 1 P b P } is independent whereas Ar P b P is dependent on the set {Q r } From the above it follows that determining a basis of Q P does not require computation of the whole controllability matrix; such a basis is determined as follows: Remark 31 A basis for Q P is constructed by a searching procedure that tests the rank of matrices r i = rank Q i (A P,b P ),where Q i (A P,b P ) =[b P,A P b P,,A i P b P ], i = 0, 1, 2,,σ 1 (35) The smallest index j for which r j 2 <r j 1 = r j = defines the rank of Q(A P,b P ) and thus Q r 1 (A P,b P ) is a basis of Q P Note The test for computing the possible increase of the rank at each step may become numerical This may provide the means to introduce the LCM with a certain accuracy defined from the data set The basis matrix for Q P defined by the columns of Q r 1 (A P,b P ) is referred to as natural basis of Q P and its significance for the computation of LCM is shown below: Theorem 32 Let Q P be the controllable subspace of S P The monic LCM of P is linked to Q P as shown below: (i) p(s) is the characteristic polynomial of the restriction Ā P of the map A P on the A P - invariant subspace Q P (ii) The monic LCM of P is the r-degree polynomial p(s), r = dim Q P,p(s)= s r + c r s r 1 + +c 2 s + c 1 the coefficients of which express the dependence of A r P b P with respect to the natural basis {Q r 1(A P,b P )} of Q P ie A r P b P = c 1b P c 2 A P b P c r A r 1 P b P (36) The above results provide a system theoretic characterisation of LCM and a simple algorithmic procedure for its computation A problem that is closely linked to LCM computation of the set P ={p i (s), i k} is the computation of the set of multipliers R ={r i (s), i k} of P with respect to the LCM These are defined by the factorisation problems p(s) = p i (s)r i (s), r i (s) R[s] (37)

8 8 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 Remark 32 The factorisation problem defined by (37) may be reduced to a standard numerical linear algebra problem and an efficient procedure for its computation has been suggested in [9] as an alternative to the Euclidean division The computation of LCM, as described by (36) leads also to the following characterisation of LCM: Corollary 31 Let Q = Q r (A, b) =[b,ab,,a r 1 b,a r b], where r = rank Q(A, b) The row echelon form of Q is given by c c 2 Q =, (38) c r O 0 where c i define the LCM as p(s) = s r + c r s r 1 + +sc 2 + c 1 Remark 33 If Q =[b,ab,,a r 1 b] is the reduced controllability matrix of (A, b) and Q denotes any left inverse of Q, then the coefficient vector of the monic LCM p(s) = s r + c r s r 1 + +c 2 s + c 1,iec =[c 1,c 2,,c r ] t is defined by the first r-coordinates of the vector c = QA r b (39) 0 We consider now an alternative, direct approach for the characterisation and computation of the set of multiplies R ={r i (s), i k} This alternative approach is based on the factorization of g P (s) and is considered below Consider the right MFD of g P (s) ie g P (s) =[,p i (s) 1,]=[,r i (s), ] t p(s) 1 (310) It is clear from the coprimeness property that the numerator polynomials are the multipliers of P in p(s) Using the notation introduced above we have the following result: Corollary 32 For the set P with LCM p(s) and S(A,b,C) associated normal system, a right coprime MFD for g P (s) is given by g P (s) ={C Q adj(si Ā) Qb}{ si Ā } 1 = n r (s)p(s) 1 (311)

9 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) The vector n r (s) =[r 1 (s),, r k (s)] t defines the set of multipliers of P with respect to p(s) The results derived here provide a system theoretic characterisation of the LCM and corresponding multipliers and provide the basis for a numerical linear algebra procedure for their computation The general theoretical algorithm is summarised below: 31 Algorithm for LCM and multipliers Given the set P ={p i (s) R[s],i k} we construct by inspection the normal system S(A,b,C),whereA R σ σ, σ = k i=1 d i, d i = [p i (s)], i kthelcm p(s) and the corresponding set of multipliers R ={r i (s), i k : p(s) = r i (s)p i (s)} are defined as follows: Step 1: Compute the matrices Q i =[b,ab,,a i b] for i = 1, 2,,σ 1 and determine the smallest integer r such that Q r 1 has full rank, but Q r is rank deficient Step 2: Determine the dependency relationship amongst the vectors {b,ab,, A r 1 b,a r b} ie A r b = c 1 b c 2 Ab c r A r 1 b Step 3: Define the LCM p(s) as p(s) = s r + c r s r 1 + +c 2 s + c 1 Step 4: Define the reduced controllability matrix Q, Q =[b,ab,,a r 1 b] (312) and define a left inverse of Q, Q R r σ Step 5: From the computed p(s) = s r + c r s r 1 + +c 2 s + c 1 define the associated companion matrix c c 2 Ā = c c r and compute adj(si Ā) (313)

10 10 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 Step 6: The vector of multipliers is then defined as r 1 (s) r 2 (s) = C Q adj(si Ā) Qb (314) r k (s) The theoretical algorithm presented above involves a number of key computational issues, which will be considered in the following section We close this section by demonstrating the theoretical algorithm in terms of an example Example 34 Consider the set P ={p 1 (s) = s 2 + 3s + 2,p 2 (s) = s 2 + 4s + 3} The normal system is defined by inspection as 0 1 O A =, b = 0 1 0, C = O 3 4 Step 1: Compute the vectors b,ab,a 2 b since the LCM is of degree at least two ie b = 0, Ab = 3 1, 7 A2 b = The matrix [b,ab,a 2 b] has rank 3 and thus we compute next A 3 b and check the rank of [b,ab,a 2 ba 3 b] A 3 15 b = 13 and Q 3 = Since rank(q 3 ) = 3wehavethatr = 3 Step 2: Use elementary row operations to reduce Q 3 to a row echelon form Q 3 and then solve the equation Q 3 c = 0, ie Q 3 = Q 3 =

11 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) from the essential part of Q 3 (nonzero part) we have that c c 2 c = 0 c = 11 6 c 4 1 and thus the LCM of P is defined by p(s) = s 3 + 6s s + 6 Step 3: Having found the LCM we may define Ā as well as Q 1 as: Ā = and Q = For this Ā we have s 2 + 6s s adj(si Ā) = s + 6 s 2 + 6s 11s 6 1 s s 2 and Thus C Q = [ ] 0 1 3, Q = C Q adj(si Ā) Qb = [ ] r1 (s) = r 2 (s) which verifies the fact that p(s) = p 1 (s)r 1 (s) = p 2 (s)r 2 (s) [ ] s + 3, s Numerical aspects of theoretical algorithm The numerical implementation of the theoretical algorithm involves a number of particular computational problems, which are considered below: 41 Computational problems (CP) (CPa) Computation of Q i (A, b) =[b,ab,,a i b], i = 1, 2,,r (CPb) Test of rank of Q i (A, b), i = 1, 2,,r, r σ (CPc) Computation of solution of Q r (A, b)x = 0 (CPd) Computation of left inverse of Q = Q r 1 (A, b) (CPe) Computation of adj(si Ā) and determination of the multipliers

12 12 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 Next we analyze these problems (CPa): Given the set P, the system S(A,b,C)is constructed by inspection and A contains the original data of P The numerical aspects of computation of a controllability matrix for general pair (A, B) has been considered in [14] and in certain cases may be an unstable process Here, however, the special structure of the pair (A, b) suggests that such computations are reduced to the simpler computation of the SISO phase variable form, where a µ a µ 1 A 0 =, b 0 =, a 0 = (41) a µ a µ 1 a µ 2 a 1 1 a 1 In fact, for i = 1, 2,,ρ, ρ µ, the block-diagonal structure of (27b), (28b) implies that if A i 0 b 0 = x i 1 x i 2 x i µ, then A i+1 0 b 0 = x i+1 1 x i+1 2 x i+1 µ, (42) where for all k>0wehave x i+1 1 = x2 i, xi+1 2 = x3 i,,xi+1 µ 1 = xi µ, xi+1 µ = a t 0 Ai 0 b 0 (42a) and thus every step involves the computation of one scalar product Error analysis: If we perform floating point arithmetic with unit round off u 1, then [16] fl(a0 t Ai 0 ) = at 0 Ai 0 b 0 + ɛ 0, where ɛ 0 (µ + 1)u 1 a t 0 Ai 0 b 0 The relative error (Rel) satisfies the relation a t 0 Rel (µ + 1)u Ai 0 b 0 1 a t 0 Ai 0 b 0 The relative error in the computed Q i (A, b) may not be small if a t 0 Ai 0 b 0 a t 0 Ai 0 b 0 [4] If we proceed in analyzing further the required inner product computation it holds that: 1 xµ i+1 = a i+1 a i xµ 1 a j xµ i+1 j, i = 0, 1, 2, (43) j=i 1

13 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) We see that For i = 0, xµ 1 = a 1, and thus no rounding error is introduced in this computation For i = 1, xµ 2 = a 2 + a1 2 There is danger for rounding errors if a 2 a1 2 since we will have the problem of subtracting approximate equal numbers In this case a t 0 A 0b 0 a t 0 A 0b 0 For i = 2, xµ 3 = a 3 + a 1 a 2 ( a 2 + a1 2)a 1Ifa1 2 a 2 and a 3 a1 3 we will have again the problem of subtracting approximate equal numbers Again for this case will hold a t 0 A2 0 b 0 at 0 A2 0 b 0 Starting from a set of data for which a t 0 Ai 0 b 0 at 0 Ai 0 b 0,fori = 1, 2the entries xµ 1,x2 µ will have a small value due to the subtraction of approximate equal numbers As proceeding further, since in the expression (43) the signs are interchanged, the difference between a t 0 Ai 0 b 0 and at 0 Ai 0 b 0 will be eliminated Thus, in the final computation of the LCM the introduced rounding error will be caused from the initial subtractions of approximate equal numbers, which will appear only if the coefficients of the given polynomials belong to special subvarieties, and will be characterised from the loss of accuracy that will appear in the final result, according of course to the available significant digits provided from the mantissa (see Example 44) Computational complexity: If we have k polynomials of degrees d 1,d 2,,d k the formulation of Q i (A, b) requires k j=1 (i 1)d j flops Thus we will have flops of order O((i 1)σ ) (CPb) (CPc): In the application of the algorithm we use the notion of numericalɛ rank of a matrix A [14] ie the number of singular values of matrix A that are greater than a specified accuracy ɛ According to this accuracy we will determine the smallest integer r such that Q r 1 has full numerical-ɛ rank and Q r is numerically-ɛ rank deficient The computation of the singular values of Q i (A, b) is a stable process When we have uncertainty in the coefficients we have to think how we select the accuracy ɛ, since this affects the estimated order r of the LCM Ifweplotthe numerical-ɛ ranks, ρ i,ɛ (A), as a function of i we would expect to get a nondecreasing function, which after some number of steps should reach a steady state value The minimum i required for this is the degree r i According to the used accuracy ɛ i the rank increases or remains the same Fig 1 demonstrates this situation Fig 1 Rank behaviour

14 14 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 The determination of the dependency relationship amongst the vectors {b,ab,,a r 1 b,a r b} is performed in a numerically stable way if we compute, using the singular value decomposition, the one dimensional numerical-ɛ right null space (N r,ɛ (Q r )) of matrix Q r =[b,ab,,a r 1 b,a r b] For this index r we will have that N r,ɛ {Q i }=0,N r,ɛ {Q i+1 }=0,,N r,ɛ {Q r }=0,N r,ɛ {Q r+1 } /= 0 Extra care is needed for the appropriate estimation of the accuracy ɛ according to which N r,ɛ {Q r+1 } /= 0 The singular vector corresponding to the smallest singular value is used as the best approximation If there is uncertainty about the value of r, according to the specified accuracy, we can determine exact and approximate solutions (CPd): The computation of the left inverse Q of Q can be performed in a stable way using the singular value decomposition of Q If Q = U Σ V T is the SVD of Q, then Q can be expressed as: Q = V Σ 1 U T (CPe): The computation of multipliers is based on the following result Proposition 41 Consider the polynomial p(s) R[s], where p(s) = s r + c r s r 1 + +c 2 s + c 1 (44) with an associated companion matrix Ā and pencil si Ā s 0 0 c 1 1 s 0 c c 3 si Ā = (45) s + c r The adjoint of si Ā is a polynomial matrix P(s) R r r [s] with special structure The polynomial entries p ij (s) of P(s)are defined as follows: (a) The main diagonal elements are polynomials derived from p(s) and they all have degree r 1 This set is p ii (s) = i 1 j=r 1 c j+2 s j, i = 1, 2,,r (46) (b) For a given row i the p ii (s) element defines the rest of the elements of this row as follows: (i) The elements to the right of p ii (s) are 1 p i,i+1 (s) = c k s k 1, i < r, k=i p i,i+j (s) = s j 1 p i,i+1 (s), j = 2,,r i (47)

15 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) (ii) The elements to the left of p ii (s) are p i,i j (s) = i 1 k=r 1 c k+2 s k j, j = 1, 2,,i 1 (48) The above result is readily established by induction An interesting implication of formulas (45) (47) is that as long as we know the polynomial p(s), then adj (si Ā) is defined by using these formulas that does not involve any additional calculations and thus no additional errors This calculation can be directly done symbolically In the sequel, the computation of the vector of multipliers r k (s) R k [s] can be implemented symbolically as well by computing directly the product C Q P(s) Q b 42 Numerical results The proposed algorithm was programmed in the Matlab environment and tested on a Pentium machine over several sets of polynomials P characterised by various properties The symbolic math toolbox of Matlab was used for the symbolic computations of (CPe) The accuracy ɛ specifies the numerical ɛ rank used for the determination of the degree of the LCM In order to measure the accuracy of the computed results we estimate the relative error (rel) between the expected theoretical value of the LCM and the computed value of the LCM following the proposed algorithm Since the LCM is given from a polynomial of the form p(s) = s r + c r s r 1 + +c 2 s + c 1 we associate with it the vector p =[1 c r c r 1 c 1 ] t We say that the 2-norm of p(s) is equal to the 2-norm of its associated vector p 2 Thus, we will estimate the relative error using the 2-norm of the vector expressing the difference of the theoretical LCM and the computed LCM over the 2-norm of the vector expressing the theoretical LCM Throughout the following examples, we will always use three significant digits for the estimation of rel Example 41 The following polynomial set contains real polynomials p 1 (s) = s s s + 78 = (s + 15)(s s + 52), p 2 (s) = s s = (s + 15)(s + 37) For ɛ = we get the following results: computed LCM = s s s s The relative error concerning the LCM computation is: rel =

16 16 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 Using seven significant digits, we get the following set of multipliers: [ ] [ ] r1 (s) s = r 2 (s) s s The relative error concerning the computation of multipliers r 1 (s) and r 2 (s) is: rel(r 1 (s)) = , rel(r 2 (s)) = The relative error can be further reduced if we perform the symbolic computation using more significant digits Example 42 The following polynomial set contains integer polynomials of rather high degree p 1 (s) = s s 9 + 3s 8 + 4s 7 + 6s s 5 + 8s 4 + 4s 3 + 9s 2 + 6s + 1, = (s 6 + 2s 5 + 3s 4 + 2s 3 + s 2 + 4s + 1)(s 4 + 2s + 1) p 2 (s) = s 8 + 2s 7 + 4s 6 + 4s 5 + 4s 4 + 6s 3 + 2s 2 + 4s + 1, = (s 6 + 2s 5 + 3s 4 + 2s 3 + s 2 + 4s + 1)(s 2 + 1) p 3 (s) = s 8 + 2s 7 + 5s 6 + 6s 5 + 7s 4 + 8s 3 + 3s 2 + 8s + 2, = (s 6 + 2s 5 + 3s 4 + 2s 3 + s 2 + 4s + 1)(s 2 + 2) p 4 (s) = s 7 + 5s 6 + 9s s 4 + 7s 3 + 7s s + 3, = (s 6 + 2s 5 + 3s 4 + 2s 3 + s 2 + 4s + 1)(s + 3) For ɛ = and using 10 significant digits we get the following results: computed LCM = s s s s s s s s s s s s s s s The relative error concerning the LCM computation is: rel = Using eight significant digits, we get the following set of multipliers: s 5 + 3s 4 + 3s 3 + 9s 2 + 2s + 6, r 1 (s) r 2 (s) r 3 (s) = s 7 + 3s 6 + 2s 5 + 8s 4 + 7s 3 + 7s s + 6, s 7 + 3s 6 + s 5 + 5s 4 + 7s 3 + 5s 2 + 7s + 3, s r 4 (s) s s s 4 + 6s s s The relative error concerning the computation of the multiplier r 4 (s) is: rel =

17 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) Example 43 coefficients The following polynomial set contains polynomials with varying t 1 (s) = s 2 3s + 2 = (s 1)(s 2), t 2 (s) = s 2 (3 ε 1 )s + (2 ε 2 ) For the case of exact coefficients (ε 1 = ε 2 = 0), we have t 1 (s) = t 2 (s) Thus the exact GCD of the polynomials and the LCM is s 2 3s + 2 However if the coefficients of the polynomials become inexact (ε 1,ε 2 /= 0) we have a lot of LCMs whose degrees are 3 or 2 These depend strongly on the selection of the accuracy ɛ that will define the numerical-ɛ rank More precisely, if (i) ε 1 = ε 2 = For ɛ = : approximate LCM = s s 2 + s 3 Thus we compute an approximate LCM that equals to s s s (keeping five significant digits) Almost the same approximate LCM was computed in [7] (ii) ε 1 = ε 2 = ɛ = : approximate LCM = s + s 2 Thus we compute an approximate LCM that equals to s s (keeping six significant digits) Example 44 subvarieties The coefficients of the following polynomial set belong to special p 1 (s) = s s s , p 2 (s) = s + 1 For this set we have that a 2 a1 2 and a 3 a1 3 The existence of this property in the coefficients of the polynomials is not a generic phenomenon The theoretical LCM is: s s s s We start the computation using a mantissa that can contain five significant decimal digits We notice that: For i = 1, a t 0 Ai 0 b 0 = at 0 Ai 0 b 0 = For i = 2, a t 0 Ai 0 b 0 = at 0 Ai 0 b 0 =

18 18 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 For values of i/geq3 the above quantities are approximate equal The computed LCM equals to: s s s s The relative error is equal to Remark 45 If we compare the proposed method with the balanced minimal realization method we notice that the balanced minimal realization method requires much more flops since many transformations of the original data are needed More specifically in the Example (41) the computation of the LCM with the proposed method requires 5052 flops whereas the same computation with the balanced minimal realization method requires 48,444 flops Furthermore the balanced minimal realization method cannot define various approximate LCM s according to the specified accuracy Thus in the Example (43) the proposed method, according to the selected accuracy, computed two approximate LCM s of degrees 3 and 2 whereas the balanced minimal realization method computed only the LCM of degree 3 with of course much more flops (7059 flops were needed instead of 2703 that were required using the proposed method) 5 Conclusions A new characterisation of the LCM of a set of polynomials has been given based on the properties of the controllable space of a linear system that is associated with the given set of polynomials The distinguishing property of the new procedure is that the system is defined from the original data, without involving transformations and that it also provides the set of associated multipliers The essential part of the numerical procedure is the determination of the successive ranks of parts of the controllability matrix, which due to the special structure of the system may be computed involving numerically stable procedures The procedure has the advantage that it provides a meaningful way for determining the LCM in a numerically robust way and also indicates that approximate solutions to LCM are equivalent to determining the controllable space, with given accuracy of the associated natural realization Appendix A Proof of results Proof of Proposition 21 (i) If s = z is a zero of g P (s), theng P (z) = 0 and thus p i (z) 1 = 0, i k, which contradicts the fact that p 1 i (s) has no finite zero

19 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) (ii) If ν is defined as above, then we can use the valuation representation [15] ie 1 p i (s) = 1 s d u i(s), i k, i where u i (s) is an appropriate unit of R pr (s) Clearly, then from the definition of valuations and infinite zeros [15] we have that ν = min{d i, i k} (iii), (iv) These parts are standard from linear systems [3,6] Proof of Proposition 22 By defining the realization of p i (s) 1 = c t i (si A i) 1 b i then c t 1 (si A 1) 1 b 1 g P (s) = c t i (si A i) 1 b i = bl-diag{,c t i,}bl-diag{,si A i,} 1 b i c t k (si A n) 1 b k and this completes the proof of the result Proof of Proposition 23 The block-diagonal structure of (A P,C P ) pair suggests that observability is equivalent to the fact that each pair (A i,c t i ) is observable However, for SISO systems the phase variable realization is both observable and controllable and this establishes the first part of the result Note that if the polynomials of P are pairwise coprime, then LCM is defined as p(s) = k i=1 p i (s) and {p(s)} = σ = k i=1 σ i is the McMillan degree of g P (s) which coincides with the dimension of the realization This completes the proof Proof of Theorem 31 We note that S P (A P,b P,C P ) is objervable but not necessarily controllable Let x = Px be the coordinate transformation that reduces to the controllable normal form, then r is the dimension of the controllable space of (A P,b P ) For the new description we have: g P = C P (si A P ) 1 b P = C P (si à P ) 1 b P =[ C P, C P ] (si Ā P ) 1 X b P O (si Ā P ) 1 0 = C P (si Ā P ) 1 b P Clearly, the triple (Ā P, b P, C P ) is controllable and observable (the original system is observable) and thus it is a minimal realization of g P (s) Condition (210) is then readily established

20 20 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 Proof of Lemma 31 Assume that r = rank Q(A P,b P ) and let us denote by q i = A i P b P Assume now that q r = c r 1 q r 1 + +c 1 q 1 + c 0 q 0, then q r+1 = Aq r = c r 1 Aq r 1 + +c 1 Aq 1 + c 0 Aq 0 = c r 1 q r + +c 1 q 2 + c 0 q 1 r 1 = c r 1 c j q j + c r 2q r 1 + +c 1 q 2 + c 0 q 1 j=0 = c r 1 q r 1 + +c 1 q 1 + c 0 q 0 and thus q r is dependent on {Q r }; by induction we may prove that for all j>r+ 1 this property also holds true The above establishes the dependence of all q j,for j r on {Q r } Note now that the set {Q r } has to be independent, since otherwise Q(A P,b P ) will have rank less than r and this completes the proof Proof of Theorem 32 It is known that the controllable subspace Q P is A P -invariant [17] The construction of the coordinate transformation that leads to the derivation of the controllable form in (8) is x = Px, where [3] P 1 Q =[q 1,q 2,,q r ; q r+1,,q σ ] (A1) with the first columns [q 1,q 2,,q r ] defined by the vectors in Q r 1 (A P,b P ) and the remaining σ r vectors entirely arbitrary, so long as the matrix Q is nonsingular The transformation defined above reduces the (A P,b P ) pair to the (Ã P, b P ) description in (31) and clearly, Ā P is the restriction of A P on R P To compute the restriction map Ā P we consider the basis [q 1,q 2,,q r ] for R P Then A P [q 1,q 2,,q r ]=[Aq 1,Aq 2,,Aq r 1,Aq r ] =[q 2,q 3,,q r,a r b P ] If A r P b P = q is expressed as in (11b) then (A2) leads to r+1 [ ] r A P [q 1,q 2,,q r ]= q 2,q 3,,q r, c i q i i= c c 2 =[q 1,q 2,,q r ] c c r (A2) (A3)

21 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) and thus with respect to the basis [q 1,,q r ], the restriction map is c c 2 Ā P = c c r and this completes the proof (A4) Proof of Corollary 31 By (36) the coefficients of LCM are defined by c 1 c 2 [b,ab,,a r 1 b,a r b] = 0 (A5a) c r 1 Given that [b,ab,,a r 1 b] has rank r, there exists T R σ σ, T /= 0suchthat I r T [b,ab,,a r 1 b]= (A5b) 0 Given that A r b is dependent on {b,ab,,a r 1 b}, it follows that for the transformation T defined above we also have T [b,ab,,a r 1 b,a r I r d b]= (A5c) 0 d From the dependence of A r b on {b,ab,,a r 1 b}, it follows that d = 0, since otherwise rank{[b,ab,,a r 1 b,a r b]} >r If we now denote by d =[d 1,d 2,, d r ] t, then by multiplying (A5a) on the left by T and using (A5c), it is readily shown that d i = c i, i r (A5d) and this completes the proof Proof of Corollary 32 Consider the normal system S(A,b,C) of P :ẋ = Ax + bu, y = Cx and carry out the coordinate transformation x = Px, x = P 1 Q such that P 1 = Q =[b,ab,,a r 1 b; q r+1,,q σ ] =[q 1,q 2,,q r ; q r+1,,q σ ]=[Q 1,Q 2 ], (A6a)

22 22 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) 1 23 where Q 1 is the reduced controllability matrix and Q 2 provides a completion of a basis to R σ The tranformed system is {à = PAP S(Ã, b, C) : 1 = Q 1 AQ, b = Pb = Q 1 b, C = CP 1 (A6b) = CQ, where also à = Q 1 Ā Ā 12 AQ = (A6c) O Ā The resolvent (si Ã) 1 may be computed by using si Ā Ā 12 X 1 (s) X 2 (s) = I r O O si Ā X 3 (s) X 4 (s) O I σ r from which it follows that (si Ā)X 1 (s) Ā 12 X 3 (s) = I, (si Ā )X 3 (s) = 0, (si Ā)X 2 (s) Ā 12 X 4 (s) = 0, (si Ā )X 4 (s) = I and from the above we readily have that si Ā 1 Ā 12 (si Ā) 1 (si Ā) 1 Ā 12 (si Ā ) 1 = O si Ā O (si Ā ) 1 (A6d) If we set Q 1 = Q, then Q 1 r Q = Q 2 σ r b Q 1 b and b = Pb = Q 1 b = = 0 0 (A6e) Note that Q 2 b = 0, since the coordinate transformation reduces (A, b) to the controllable form (see Theorem (31)) Similarly, if r σ r [ Q 1, Q 2 ] then C = C[Q 1,Q 2 ]=[CQ 1,CQ 2 ] (A6f) Q = By the definition Q we have [ ] Q 1 [Q Q 1,Q 2 ]= 2 [ ] Ir O Q O I 1 Q 1 = I r σ r (A6g)

23 N Karcanias, M Mitrouli / Linear Algebra and its Applications 381 (2004) and thus Q 1 is a left inverse of Q 1 The overall transfer function may thus be expressed as: (si Ā) 1 (si Ā) 1 Ā 12 (si Ā ) 1 Q 1 b g P (s) =[CQ 1 ; CQ 2 ] O (si Ā ) 1 0 = CQ 1 (si Ā) 1 Q 1 b ={CQ 1 adj(si Ā) Q 1 b}{ si Ā } 1 (A6h) Clearly, since the pole polynomial si Ā is the LCM, the factorisation is minimal and thus the multipliers may be readily found from the numerator [r 1 (s),, r k (s)] t = CQ 1 adj(si Ā) Q 1 b (A6i) References [1] S Barnett, Greatest common divisor of several polynomials, Proc Camb Philos Soc 70 (1971) [2] T Beelen, P Van Dooren, A pencil approach for embedding a polynomial matrix into a unimodular matrix, SIAM J Matrix Anal Appl 9 (1988) [3] CT Chen, Linear Systems Theory and Design, Holt-Saunders International Edit, 1984 [4] GH Golub, CF Van Loan, Matrix Computations, Johns Hopkins Univ Press, Baltimore, 1989 [5] MW Hirsch, S Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974 [6] T Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1980 [7] N Karcanias, M Mitrouli, Numerical computation of the least common multiple of a set of polynomials, Reliab Comput 6 (4) (2000) [8] N Karcanias, M Mitrouli, A matrix pencil based numerical method for the computation of GCD of polynomials, IEEE Trans Automat Control 39 (1994) [9] N Karcanias, M Mitrouli, Approximate algebraic computations of algebraic invariants, in: Symbolic Methods in Control Systems Analysis and Design, IEE Control Eng Ser, vol 56, 1999, pp [10] V Kucera, Discrete Linear Control, The Polynomial Equation Approach, John Wiley, New York, 1979 [11] A Laub, Computation of balancing transformations, in: Proc JACC, vol 1, 1980, paper FA8-E [12] M Mitrouli, N Karcanias, Computation of the GCD of polynomials using Gaussian transformation and shifting, Internat J Control 58 (1993) [13] M Mitrouli, N Karcanias, C Koukouvinos, Numerical aspects for nongeneric computations in control problems and related applications, Congr Numer 126 (1997) 5 19 [14] P Petkov, N Christov, M Konstantinov, Computational Methods for Linear Control Systems, Prentice-Hall International, UK, 1991 [15] AIG Vardulakis, DJN Limebeer, N Karcanias, Structure and Smith McMillan form of rational matrix at infinity, Internat J Control 35 (1982) [16] J Wilkinson, Rounding errors in algebraic processes, Her Majesty s Stationery Office, London, 1963 [17] WM Wonham, Linear Multivariable Control: A geometric Approach, second ed, Springer-Verlag, New York, 1984