Constructive Root Bound for k-ary Rational Input Numbers

Transcription

1 Constrctive Root Bond for k-ary Rational Inpt Nmbers Sylvain Pion, Chee Yap To cite this version: Sylvain Pion, Chee Yap. Constrctive Root Bond for k-ary Rational Inpt Nmbers. 19th Annal ACM Symposim on Comptational Geometry (SCG), Jn 003, San Diego, California, United States. pp.56-63, 003. <inria > HAL Id: inria Sbmitted on 0 Dec 008 HAL is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, whether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. L archive overte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 Constrctive Root Bond for k-ary Rational Inpt Nmbers Sylvain Pion Corant Institte of Mathematical Sciences New York University New York, NY 1001, USA Chee K. Yap Corant Institte of Mathematical Sciences New York University New York, NY 1001, USA ABSTRACT Constrctive root bonds is the fndamental techniqe needed to achieve garanteed accracy, the critical capability in Exact Geometric Comptation. Known bonds are overly pessimistic in the presense of general rational inpt nmbers. In this paper, we introdce a method which greatly improves the known bonds for k-ary rational inpt nmbers. Since majority of inpt nmbers in scientific and engineering applications are sch nmbers, this cold lead to a significant speedp for a large class of applications. We apply or method to the BFMSS Bond. Implementation and experimental reslts based on the Core Library are reported. Categories and Sbject Descriptors F.. [Nonnmerical Algorithms and Problems]: Geometrical problems and comptations; I.1.3 [Symbolic and Algebraic maniplation]: Langages and Systems Special-prpose algebraic systems General Terms Algorithms, Theory, Experimentation Keywords Constrctive root bonds, exact geometric comptation, robst nmerical algorithms, k-ary rational nmbers 1. INTRODUCTION The critical idea of the Exact Geometric Comptation (EGC) approach to robst algorithms is geometric exactness. This amonts to ensring that all comptational decisions in a program are error free. It translates to the ability to determine the sign of real nmerical qantities. The This research is spported by NSF/ITR Grant #CCR Sylvain s work is condcted nder a postdoc fellowship of this grant. SoCG 03, Jne 8 10, 003, San Diego, California, USA.. generalization of this is what we call garanteed accracy [13]. It is a generalization becase to garantee the exact sign determination of a nmber, it is eqivalent to garantee one relative bit of the nmber. Sch techniqes have been encoded into two general libraries LEDA real [5, 1] and Core Library [4, 6]. To ensre this form of nmerical control, the se of root bonds is central. Bt classical root bonds (e.g., [9]) may be highly non-constrctive. What we need are called constrctive root bonds in [8]. Sch bonds are defined relative to some set E of algebraic expressions. It is constrctive in two ways: (i) First, for each expression E E, we define a set of mtally recrsive parameters 1(E),..., m(e) (ii) Second, there is an explicit comptable root bond fnction β( 1,..., m) sch that if E is well-defined and E 0, then E β( 1(E),..., m(e)). (1) We will write β(e) instead of β( 1(E),..., m(e)). To be more precise, we may call β an exclsion root bond; if the ineqality in (1) were reversed, we wold have an inclsion root bond. The first examples of sch constrctive root bonds is Mignotte s constrctive Measre Bond [10], applied to the problem of identifying algebraic nmbers. In EGC, sch bonds were first introdced in the Real/Expr Package [15], where the degree-height bonds [15] and degree-length bonds [14, p. 177] were sed. Brnikel et al [] introdced the BFMS Bond that trns ot to be extremely effective for division-free expressions. Recently, this bond was improved to what we will call the BFMSS Bond [3]. In [8, 7], we introdced another constrctive root bond that overcomes some of the shortcomings of BFMS. If β, β are root bond fnctions, we can compare them in two ways: (i) efficiency and (ii) effectiveness. Efficiency refers to the complexity of compting the root bonds, and effectivity refers to the size of the bonds (a larger β(e) is more effective). Generally, the most interesting comparison is based on effectiveness (efficiency is less of an isse in most applications becase the rnning time is sally dominated by the mltiprecision arithmetic). If β (E) β(e) for all E E, we say β dominates β (over E). Among the crrent constrctive root bonds, there are three that are not dominated by any others over the class of constrctible expressions: degree-measre [10, ], BFMSS [3] and Li-Yap [8]. We give a comparison of the effectiveness of these three root bonds in Section 5. The starting point of this paper is the observation that (a) crrent constrctive bonds are qite effective for division-

3 free inpt expressions involving only integer inpts, and (b) the bonds become considerably worse in the presense of division. Even when the expression is division-free, the presence of rational inpt nmbers conts as introdcing division into the expression. Sch ineffective bonds can make some comptations impractical. We note that these ineffective bonds are sometimes intrinsic, becase it is easy to see that the worst case reqires exponential bit sizes. Fortnately, this is not the end of the story. The vast majority of nmerical inpt in scientific and engineering applications involves k-ary rationals for some integer k. Invariably k = (binary) or k = 10 (decimal). By a k-ary rational we mean a rational nmber whose denominator is a power of k. Ths k-ary rationals are generalizations of integers. We shall introdce a general techniqe that can take advantage of k-ary rationals. The techniqe seems orthogonal to previos techniqes in the sense that for any crrent constrctive root bond β, we can modify it to a k-ary version β k which is more effective. In this paper, we introdce the k-ary version of the BFMSS and Measre Bonds. These will be referred to as the BFMSS[k] and Measre[k] Bonds. In algorithms, especially in compter algebra, it is a well-known phenomenon that rational nmber arithmetic is mch slower than integer arithmetic. Frthermore, k-ary rational nmber arithmetic has a complexity that is intermediate between these two extremes. The techniqes of this paper will yield the same kind of intermediate complexity for root bonds of expressions with k-ary inpt nmbers. Some Examples. We briefly illstrate the possible improvements with or new techniqe. Instead of the root bond β(e), we normally consider the corresponding bitbond, defined as lg β(e). An example from [8] is the identically zero expression E 1(x, y) = x + y p x + y + xy. Sppose x, y are L-bit binary nmbers (i.e., nmerator are L-bit integers and denominator are L-bit powers of ). Table 1 compares some bit-bonds and timings (Cf. [3]). Line 1 gives the bit-bond as a fnction of L. Line gives the range of bit-bonds compted by or Core Library implementation when 10 random choices of machine dobles are sbstitted for x and y. Line 3 gives the time to evalate the 10 random examples of Line for 100 times each. When x, y are rational nmbers whose nmerator and denominator are L-bit integers, the Bit-Bond fnctions for BFMSS and Li-Yap are jst 96L + 30 and 8L + 60 (as in Line 1) while BFMSS[] drops to 8L On the other hand, when x, y are L-bit integers, the Bit-Bond fnction for all three methods is the same and eqal to 7.5L+30. This example illstrates or previos remark, that or new bitbonds for k-ary inpt nmbers lie between the bit-bonds for integers and for rational nmbers. Indeed, they are only slightly worse than the integer case. Next, consider the important and common sitation of evalating n n determinants where the inpt nmbers are L-bit binary nmbers. Sch nmbers have the form m k where m < L and 0 k L. Let E 0 be an expression for sch a determinant. First, assme E 0 is the co-factor expansion of the determinant (this is a polynomial with n! terms). Then the BFMSS Bond for E 0 gives a root-bit bond that is more than the BFMSS Bond, which gives a root-bit bond of nl. In or experiments (Section 6), we se a more efficient determinant expression: let E 1 be the determinant expression obtained by sing dynamic programming principles. Ths E 1 is a DAG while E 0 is a tree. E.g., when the inpt is a random 5 5 matrix and L = 100, or BFMSS implementation gives the bond lg E 10, 8, while or binary version of BFMSS gives lg E 36. Overview. Section gives a high-level view of what or k-ary transformation does to any constrctive root bond. Section 3 reviews the BFMSS Bond, while Section 4 gives the new BFMSS[k] Bond. We show that BFMSS[k] dominates BFMSS. In the fll version of this paper, we also give the new Measre[k] Bond, and again we show that Measre[k] dominates Measre. Experiments and comparisons are given in Section 5. We conclde in Section 6.. GENERIC K-ARY METHOD We propose a meta-method for exploiting k-ary inpt nmbers. The meta-method is applicable to any constrctive root bonding method. In particlar, we will apply it to the BFMSS Bond. The Measre Bond is similarly treated in the fll version of this paper. In general, if β is a root bond fnction as in (1), or k-ary transformation prodces a related root bond fnction β k. If β bfmss and β meas are the root bond fnctions corresponding the BFMSS and Measre Bonds, we will describe their binary version are β bfmss and β meas. As sal, we consider the class of expressions which are DAGs with rational nmbers at the leaves and whose internal nodes are algebraic operators. The typical class of algebraic operators are +,,, and algebraic root extraction, bt this may vary depending on context. Let val(e) be the algebraic nmber denoted by E. Since algebraic operators are partial fnctions, val(e) may be ndefined. In any ineqality involving val(e), it is nderstood that the ineqality is in effect only when both sides are defined. We sally write E instead of val(e) when this is clear from context. The basic idea of the k-ary transformation is to transform an expression E to another expression E k sch that E = k v(e) E k (3) and v(e) = v k (E) Z. What are the constraints on this transformation? If β(e) is the original root bond fnction, this transformation will lead natrally to a corresponding k-ary root bond β k (E). In this paper, or basic goal is to ensre that β k dominates β: β k (E) β(e) (4) for E E. Achieving this ineqality will depend on the natre of β. Assming both sides of (3) are well-defined, we have E 0 E k 0 E k > β(e k ) E > k v(e) β(e k ) (n!)nl. () This is exponentially worse in n than or binary version of Ths we define β k (E) := k v(e) β(e k ).

4 Table 1: Comparison of BFMSS, Li-Yap and BFMSS[] Method BFMSS Li-Yap BFMSS[] (new) 1 Bit-Bond fnction 96L L L + 30 Bit-Bond Range (L = 53) Timing (L = 53, times) 46.7 s 8.35 s 3.58 s and so the ineqality (4) amonts to β(k v(e) E k ) k v(e) β(e k ). To simplify 1 the presentation below, we will choose k =. Also, we will simply write v(e) instead of v (E). Generalizing this to a general k > is mostly straightforward. A frther generalization is to maintain the powers of two or more k s simltaneosly. It seems that (k, k ) = (, 5) will yield most of the benefits of the method, since actal inpt nmbers in comptation are overwhelmingly decimal or binary. This amonts to the following transformation (cf. (3)): where v k (E) Z (for k =, 5). E = v (E) 5 v 5(E) E,5 (5) 3. THE BFMSS BOUND We first review the BFMSS Bond [, 3] for algebraic expressions. Let E be an expression as represented by a DAG, with integers at its leaves, and whose internal nodes correspond to the operators in colmn 1 of Table. The diamond operator in the last row of the Table extracts the jth largest real root of the polynomial P n i=0 FiXi where F i are expressions. With the diamond operators of a degree n, we associate an inclsion root bond fnction (in the sense of [3]) Φ(a n 1,..., a i,..., a 0) = Φ(..., a i,...) (6) where it is nderstood that the index i decreases from n 1 to 0. Since there are several possible choices Φ 1, Φ, etc for Φ, we may jst compte the bond given by each Φ i and take the best. This procedre amonts to the observation that if Φ 1 and Φ are inclsion root bond fnctions, then max{φ 1, Φ } is also an inclsion root bond. The BFMSS bond constrctively maintains two real parameters (E) and l(e) as shown in Table. To avoid cltter in the table, we write,, for (E ) and (E ); similarly for l, l. Frthermore, the diamond operator involves sbexpressions F 0, F 1,..., F n; in this case, we write D i := (Fi) l(f i) ny l(f j). (7) j=0 The degree of a node in E is p if the node is the operator p, and n if the node is the diamond operator of degree n. Otherwise the degree is 1. Moreover, let D(E) be the prodct of all the degrees of the distinct nodes in the dag of E. The degree of val(e) is bonded by D(E). The BFMSS bond says that if val(e) 0 then val(e) 1 (E) D(E) 1 l(e). (8) 1 The case k = is the most important case. Also, the reslting formlas are easier to read as we avoid the se of the variable k. Hence we may define the BFMSS root bond fnction as β bfmss (, l, D) := 1 D 1 l, (9) with the sal convention we write β bfmss (E) for β((e), l(e), D(E)). The BFMSS Rles are given in Table. Or rle for p E in this table is a nification of the two cases in the BFMSS presentation (an improvement noted by Yap). 4. GENERALIZATION OF BFMSS Let α be an algebraic nmber. As in [8], let µ(α) = max{ α i : i = 1,..., n} where α = α 1,..., α n are all conjgates of α. We call a triple (, l, v) a set of l[]- parameters for α if, l R 0 and v Z and there exist algebraic integers α 1, α sch that α = v α 1 α, (10) µ(α 1) and µ(α ) l. If is replaced by an integer k >, we have the analogos set of l[k]-parameters. When α is non-zero with degree D, we have α β (, l, v, D) := v 1 D 1 l (11) where β (, l, v, D) = β bfmss (, l, v, D) is the binary version of the BFMSS root bond fnction. The expression (10) is non-niqe. Indeed, there is some leeway for designing a sitable set of l[]-parameters for α becase in general the best choice is not easily given by a fixed rle. Ths, if (, l, v) is a set of l[]-parameters for α, then either ( v, l, 0) or (, l v, 0), depending on whether v 0 or not. More generally, it is always possible to redce v towards 0 in any set of parameters (, l, v). A set of l[]- parameters where v = 0 can be regarded as a generalization of the BFMSS parameters. The BFMSS[] Rles. The binary transformation of BFMSS is given in Table 3. The table incorporates a refinement of the l[]-parameters, whereby v(e) is represented by two nmbers v + (E) 0 and v (E) 0 satisfying the relation v(e) = v + (E) v (E). This refinement will better qantify or gain over the original BFMSS bond (see Lemma 1 below). In actal implementation, it is sfficient to jst maintain v(e). In this case, to apply the rles, we will define v + (E) to be v(e) if v(e) 0 and otherwise let v + (E) = 0. Similarly, v (E) is defined to be v(e) if v(e) < 0 and otherwise v (E) = 0. Call this variation the redced version of the BFMSS[] Rles (in contrast to the refined version where v +, v are independent). When α is represented by an expression E (in the dag form), this table defines a niqe set of l[]-parameters for

5 Table : BFMSS Rles E (E) l(e) integer n n 1 E ± E l + l l l E E l l E E l l p E min( p l p 1, ) min(l, p p 1 l ) (j, F n, F n 1,..., F 0) Φ(..., (D n) i D n i,...) where D i is given in (7) D n E, ( (E), l (E), v(e)). The BFMSS[] root bond for E is E 0 E v(e) (E) D(E) 1 l (E) (1) In the table, (, l, v ) denotes the l[]-parameters of sbexpression E ; similarly (, l, v ) is for E. Most of the rles in Table 3 can be read off the table; bt the more complex diamond operator will be explained here. We want a set of l[]-parameters for (j; F n, F n 1,..., F 0). Sppose Φ(a n 1, a n,..., a 0) is a root bond fnction in the sense of [3]. Write v i for v + i v i = v + (F i) v i (Fi). Define! nx w i := v i + and j=0 v j = v v i 1 + v + i + v i v n C i = w i (Fi) l (F i) (13) ny l (F j). (14) j=0 Jst as in BFMSS, the diamond operator (if well-defined) (j; F n, F n 1,..., F 0) specifies an algebraic nmber α where α = U/L and U, L are algebraic integers satisfying µ(u) Φ(..., (C n) i C n i,...), µ(l) C n. Also, a set of l[]-parameters for α is (Φ(..., (C n) i C n i,...), wn C n, w n). (15) This jstifies the rle for diamond operator in Table 3 (Other rles will be jstified below). If we know more abot the natre of Φ, improved bonds may be possible. E.g., sing the Lagrange-Zassenhas bond [14], we get the simpler set of l[]-parameters, (Φ(..., D n i,...), 1, 0). BFMSS[] dominates BFMSS We first prove a key relationship between the BFMSS Rles and the new BFMSS[] Rles. Lemma 1. Let (, l), (, l, v +, v ) be the parameters for an expression E given by Table and Table 3, respectively. Then = v+, l =. Proof. We se indction on the strctre of E. The base case is obvios. CASE E = E ± E : = l +l l l l = v + +v l +v +v + l v +v l l = v+ ( v + +v v + l +v +v + v + l ) v +v l l = v+ where v + = min(v + + v, v + v ) and v = v + v. We want to conclde from this derivation that = v+, l =. This is only valid if, in the above derivation, we never apply any cancellation of terms between the nmerator and denominator. The reader may verify this is the case. In other words, althogh we presented the argment as a seqence of eqations involving ratios, it shold be read as a pair of parallel transformations involving the nmerator and denominator separately. This will also be tre in all the other derivations in this proof. CASE E = E E : = l l l = v + +v + v +v l l (BFMSS) (indction) = v+ (BFMSS[]) where v + = v + + v + and v = v + v. The division case is similar. CASE E = E E : = l l l = v + +v l v +v + l (BFMSS) (indction) = v+ (BFMSS[]) where v + = v + + v and v = v + v +. CASE E = p E : The rles here split into two cases, depending on whether v l. The critical observation is that v l is eqivalent to l (the corresponding criteria for choosing the two cases in the BFMSS Rle). First assme v l. Let ev = v + + (p 1)v, v + =

6 Table 3: The Refined BFMSS[] Rles E = (E) l = l (E) v + = v + (E) v = v (E) binary rational n m n 1 max(0, m) max(0, m) E ± E v + +v v + l + v +v + v + l l l min(v + + v, v + v + ) v + v E E l l v + + v + v + v E E l l v + + v v + v + q p E, v l p ev pv+ l p 1 l ev/p where ev = v + + (p 1)v v p E, v < l (j; F d,,..., F 0) Φ(..., C i nc n i,...) (see (14)) q p ev pv p 1 l v + ev/p where ev = (p 1)v + + v wn C n 0 w n (see (13)) ev/p and v = v. We have = p l p 1 l q l = p v + +(p 1)v l p 1 v l (BFMSS) (indction) = v+ (BFMSS[]). The other case, when v < l is similarly shown. CASE E = (F n,..., F 0): For i = 0,..., n, we have Ths D i = (F Q i) n l(f i ) j=0 l(fj) (BFMSS) = w i (F i ) Q n l (F i ) j=0 l(fj) (indction) = C i (BFMSS[]) (E) = Φ(..., D i nd n i,...) = Φ(..., C i nc n i,...) = (E) = v+ (E). Similarly, l(e) = D n = C n = wn l (E) = (E). Q.E.D. Or main reslt concerning the BFMSS and BFMSS[] Rles is the following domination relation: Theorem. For expression E spported by Table, we have β bfmss (E) β bfmss (E) Proof. Let β(e) = 1 and D 1 β = v l D 1 l be (respectively) the BFMSS and BFMSS[] bonds for expression E. From Lemma 1, we conclde β β = v ( v + ) D 1 () = v + D 1. D 1 l Q.E.D. Correctness and the Umbral Convention. We now jstify the BFMSS[] Rles in Table 3. The correctness of a set (, l, v) of l[]-parameters for an expression E depends on the existence of algebraic integers U, L sch that E = v U L. (16) with µ(u ), l µ(l ). We have not given explicit rles for maintaining U, L, bt these are easily dedced from Table 3. That is becase the rles for maintaining, l is a shadow of the corresponding rles for U, L. Let s illstrate this: when E = E ± E, we have the rle = v + +v v + l + v +v + v + l (17) This is a shadow of the corresponding rle for U : U = v + +v v + U L ± v +v + v + L U (18) REMARKS: The original BFMSS rles also have sch an mbral connection between (, l) and the pair of expressions (U, L), althogh this was only implicit. Sch a shadowing techniqe is similar to the mnemonic device called symbolic or mbral calcls from the invariant theorists, and developed by Rota and his collaborators [11] as a form of linear operator. The mbral relation between (, l ) and (U, L ) is jstified by the following: Lemma 3. For any expression E, (i) The expressions U (E) and L (E) are algebraic integers. (ii) The following ineqalities hold: µ(u ), l µ(l ). (19) Proof. (i) We sketch the jstification of the rles for U (E); the jstification of L (E) is analogos. Consider the case when E = E ± E. Then U (E) is given by (18), and this is an algebraic integer becase v + + v v + 0 and v + v + v + 0 (also, indctively, the sbexpressions U, U are algebraic integers). In the case of radicals, we se the fact that p E is an algebraic integer when E is an algebraic integer. The remaining cases are jst as easily shown. (ii) We sketch the argment for part (ii). The relationship (19) holds becase for algebraic integers A, B, if a µ(a) and b µ(b) then a + b µ(a ± B), ab µ(ab), p a µ( p A). In particlar, this jstifies why (17) is an pper bond on the algebraic integer (18). Q.E.D. We are ready to prove the correctness of or rles. Note that the rles for, l shadow the rles for U, L, bt not vice-versa, becase ± for U, L becomes a + for, l. This can be seen by comparing (17) and (18).

7 Theorem 4. Table 3 is correct: for each expression E, the triple ( (E), l (E), v(e)) is a set of l[]-parameters for E. Proof. Since we already know Lemma 3, it remains to show the relation (16). The BFMSS rles prodce a pair of algebraic integer expressions U(E), L(E) sch that E = U(E)/L(E). Lemma 1 shows that + l = v. From the mbral relation between (, l) and (U, L), and also between (, l ) and (U, L ), we conclde that + U U L = v = v U. v L L Q.E.D. Generalization. We can generalize the l[]-parameters to l[k]-parameters for any integer k >. Since the majority of inpt constants in scientific and engineering comptations is covered by the l[] or l[10], the following generalization will be sefl: if q 1,..., q n are relatively prime, it is easy to define a set ((E), l(e), v q1 (E),..., v qn (E)) of l[q 1,..., q n]-parameters for E, so that E = (E) ny q vq i i l(e) Special Cases. The binary BFMSS Rles allow the root bonds of a floating point constant to behave like an integer (i.e., l(e) = 1). As long as there is no explicit division in or expression, the expression contines to behave like an integer. This is a very important case in practice. Let s consider some specialization of or rles. Sppose E and E are almost division-free in the sense that l = l = 1 (they may not be algebraic integers since v 1, v can be negative). Then the rle for E = E ± E in Table 3 gives i=1 = v + +v v + + v +v + v +. (0) When v = v, this frther simplifies to = +. Similarly l = 1 and v = v. Sppose x, y are two L-bit binary nmbers. Sch nmbers can be represented by a binary string of length L with a binary point somewhere in the string. So the triple (4 L, 1, L) is a set of l[]- parameters for x and for y. From the preceding, x + y has l[]-parameters ( 4 L, 1, L). Similarly, xy has the l[]- parameters (4 L, 1, L). Now sppose E is the determinant of an n n matrix with entries which are L-bit binary nmbers. Viewing E as the standard sm of n! terms, we easily see that E has (4 nl n!, 1, nl) (1) as a set of l[]-parameters. Frthermore, since D(E) = 1, β bfmss (E) = nl. This jstifies the root bit bond given in (). 5. EXPERIMENTAL RESULTS The timings in this paper are based on rns on an Ultrasparc 10 machine with a 440 MHz CPU. The software is Core Library Version 1.5+, which implements 3 the Measre Bond, the Li-Yap Bond and a choice between the original BFMSS, the BFMSS[], or the BFMSS[,5] Bond. To give empirical data on the relative effectiveness of these three bonds families, we rn the Core Library Test Site and conted the nmber of times that each bond is the best one. The reslts are shown in Table 4. Note that more than one bond may be the best for any given expression, so for each, we give a pair of nmbers, the first one is the nmber of times the given bond is eqal to the best one, and the second one is the nmber of times it is the only one which is eqal to the best one. The first colmn gives the reslt of a rn with the original BFMSS Bond sed, the second colmn gives the reslt of a rn with the BFMSS[] Bond sed, and the third colmn gives the reslt of a rn with the BFMSS[,5] Bond sed. Experiment 1 involves the expression E 1(x, y) given in the introdction. We assme that E 1(x, y) does not share sbexpressions. For example, we can redce the degree from 16 to 8 by sharing, and the bit-bond fnction for BFMSS improves to 48L +. Experiment involves the expression E (x, y) = x y x y x y x y, an example from [3]. When x, y are integers, the bit-bond from BFMSS and Yap are 6L + 64 and 65L + 91, respectively. Bt when x, y are L-bit binary nmbers, the bit-bond of BFMSS[] is 7.5L When we sbstitte varios machine doble vales, we obtain bit-bonds whose ranges are: (BFMSS), (BFMSS[]). Rnning these 1000 times gives timings of 36 seconds (BFMSS) and.8 seconds (BFMSS[]). Althogh there is an improvement, it is not of the order of magnitde one might expect from bit-bond ranges, Determinants. Experiment 3 involves the determinant example in the introdction. Let A be a n n matrix whose entries are L-bit binary rationals. By definition, the entries has the form n k where 0 n < L and 0 k L. There are two special cases that we consider: (1) If n L 1, we say the L-bit binary rational is strict. All the nmbers in A are strict in or experiment. () If k = L, then we say the L-bit binary rational is normal. We noted that if E 0 is the co-factor expansion of matrix A, then the BFMSS bond gives lg β bfmss (E 0) (n!)nl, while the binary BFMSS bond gives lg β bfmss (E) nl. If E is the dynamic programming implementation of the determinant of A, then β bfmss (E ) may be strictly greater than β(e). For instance, if a, b, c are L-bit binary nmbers then β bfmss (a(b + c)) = 3L while β bfmss (ab + ac) = 4L. On the other hand, β bfmss (a(b + c)) = β bfmss (ab + ac). Table 5 compares the root bit bonds of BFMSS and the binary version on random matrices whose entries are 100 bit binary rationals. These empirical bonds are (as expected) better than the worst case estimate. If we se normal 100-bit binary rationals, the Table 6 gives the same comparison when those entries are normalized 100 bit binary rationals. Or 3 Version 1.5+ refers to the modifications of the released Version 1.5 necessary to spport the experiments of this paper. Or implementation of these bonds will generally be slightly worse bond than the theory predicts becase we maintain pper bonds on lg M(E), lg (E), etc, instead M(E), (E), etc. The Core Library Test Site is a set of abot 30 sample programs that is distribted with the library.

8 Table 4: Relative effectiveness of 3 Root Bonds on CORE Test Site original BFMSS BFMSS[] BFMSS[,5] BFMSS family 5571/ / /1577 Li-Yap 51669/ / /3 degree-measre 4/4 4/4 0/0 Total nmber of expressions Table 5: Bitbond for dynamic programming determinant for random binary entries (L = 100) n (n!)nl BFMSS nl BFMSS[] , , Table 6: Bitbond for dynamic programming determinant for random normal binary entries (L = 100) n (n!)nl BFMSS nl BFMSS[] , 000 3, implementation of the β bfmss bond practically matches the theoretical pper bond of nl. We next compare timing for BFMSS, BFMSS[] and BFMSS[,5]. Despite the wide gap in the root bonds, the timings is not expected to be different for random matrices. That is becase a random determinant is nlikely to be zero and so the floating point filter will be in effect. Instead, we convert the above data into degenerate matrices, jst by making the last row a dplicate of the previos row. Srprisingly, there was no detectable difference in timing between BFMSS and BFMSS[]. This cold be explained as follows: the internal representation of the nmbers was in binary, and even when the root bond asks for many bits of precision, or implementation of BigFloat ensres that no redndant bits are transmitted (i.e., trailing zeroes are omitted). Hence the speedp cold only be observed if we se inpts that are not prely binary. Therefore, in or next set of experiments for timing, we se decimal rationals. We se random matrices whose entries are strict 50-digit decimal rationals. Table 7 compares the speed of BFMSS, BFMSS[] and BFMSS[,5]. The timing are for 10, 000 evalations of each determinant. 6. OPEN PROBLEMS AND FUTURE WORK This paper introdce the factoring techniqe into constrctive root bonds, and demonstrated its effectiveness. In general, the problem of constrctive root bonds will become more important as EGC techniqes and sch algorithms become more widely sed. The trade-offs between Table 7: Dynamic programming determinant for degenerate strict matrices with 50-digit decimal rationals n BFMSS BFMSS[] BFMSS[,5] time bitbd time bitbd time bitbd , , effectiveness (i.e., small root-bit bonds) and efficiency (i.e., low comptational complexity) is not nderstood. Between the extremes of simple recrsive rles (as constitte the blk of crrent bonds) and (say) compting minimal polynomials, we wold like to see methods with intermediate comptational complexity. Or factoring method can be seen as one step in this direction. We list some open problems and ftre work: Or k-ary method can be generalized to maintain arbitrary rational factors, in addition to k-ary factors. (e.g., transform E to q v E where v Z, q Q). The benefit of the rational factors is less predictable, and hence experimentation is called for. Crrent constrctive root bond techniqes are mostly static in natre. More dynamic root bond techniqes shold be exploited. An idea of Sekigawa[1] can be prsed. Sekigawa proposed some methods in the case of the measre bond, bt they do not seem to have been implemented. We cold combine with the most significant bit (MSB) bond that is maintained in the Core Library [7]. It is clear that the k-ary method can also be applied to the Li-Yap Bond. The general treatment of the diamond operators nder the Measre Bond is sbject for frther research. The incorporation of the Sekigawa improvements into the crrent Measre[] Rles is immediate if there is no division. It is possible to give rles that incorporate these improvements for division, bt it is nclear how to ensre that the binary bond dominates the original bond.

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