CC BY
40
UDC 512.7+519
A Variant of Gerdt's Algorithm for Computing Involutive
Bases
V. P. Gerdt*, Amir Hashemi^, Benyamin M.-Alizadeh
* Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow region, Russia
^ Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran
In [1], the first author presented an efficient algorithm for computing involutive (and reduced Grobner) bases. In this paper, we consider a modification of the algorithm which simplifies matters to understand it and to implement. We prove correctness and termination of the modified algorithm and also correctness of the used criteria. The proposed algorithm has been implemented in Maple. We present experimental comparison, via some examples, of performance of the modified algorithm with its original form described in [1] and implemented in Maple too. In doing so, we have taken care to provide uniform implementation details for the both algorithms.
Key words and phrases: Grobner bases, Buchberger's criteria, involutive bases, Gerdt's algorithm.
1. Introduction
The most important algorithmic object in computational algebraic geometry is Grobner basis. The notion of Grobner basis was introduced and an algorithm of its construction was designed in 1965 by Buchberger in his Ph.D. thesis [2]. Later on [3], he discovered two criteria for detecting some unnecessary, and thus useless, reductions that made the Grobner bases method a practical tool to solve a wide class of problems in polynomial ideals theory and in many other research areas of science and engineering [4].
In the 20s of the last century, French mathematician Janet [5] developed a constructive approach to analysis of certain systems of partial differential equations based on their completion to involution (cf. [6]). The Janet approach was generalized to arbitrary polynomially nonlinear systems by American mathematician Thomas [7]. Based on the related methods described in the book by Pommaret [8], Zharkov and Blinkov [9] introduced the notion of involutive bases in commutative algebra. The particular form of an involutive basis they used nowadays is called Pommaret basis [6,10].
Gerdt and Blinkov [10] have proposed a more general concept of involutive bases for polynomial ideals and designed algorithmic methods for their construction. The underlying idea of the involutive approach is to translate the methods originating from Janet's results into polynomial ideals theory in order to construct involutive bases by combining algorithmic ideas of the theory of Groobner bases and those in the theory of involutive differential systems. In doing so, Gerdt and Blinkov [10] introduced the concept of involutive division. Moreover, they have applied the involutive form of Buchberger's criteria. This led to a strong computational tool which was a serious alternative to the conventional Buchberger algorithm (note that any involutive basis is also a Grobner basis). Apel and Hemmecke in [11] proposed two more criteria for detecting unnecessary reductions in involutive basis computations (see also [12]) which, in the aggregate with the criteria by Gerdt and Blinkov [10], are equivalent to Buchberger's criteria. The first author in [1] described an efficient algorithm to compute involutive and Grobner bases using all these criteria. We refer to Seiler's
The contribution of the first author (V.P.G.) was partly supported by by the grant 01-01-00200 from the Russian Foundation for Basic Research and by the grant 3810.2010.2 from the Ministry of Education and Science of the Russian Federation.
book [6] for a comprehensive study and application of involution to commutative algebra and geometric theory of partial differential equations.
In this paper, we propose a variant of Gerdt's algorithm [1] for constructing involu-tive bases. This algorithm seems to be simpler both to understand and to implement. We prove correctness for the new version of algorithm which includes the criteria and its termination. We have implemented this algorithm in Maple and we compare here its performance with our implementation of the original Gerdt's algorithm via some examples.
The structure of the paper is as follows. Section 2 contains the basic definitions and notations related to theory of involutive bases, and a short description of the algorithm for their computing proposed in its initial form [10]. In Section 3, we briefly present Gerdt's algorithm from [1] to compute involutive bases. Section 4 is devoted to description of the modified algorithm together with a proof of its correction and termination. In Section 5, we compare, by using some benchmarking examples, performance of the last algorithm with our implementation of Gerdt's algorithm done also in Maple.
2. Preliminaries
In this section, we recall some basic definitions and notations from the theory of involutive bases which are used in the paper, and briefly describe the initial algorithm proposed in [10] for computing such bases.
Let K be a field and R = K[x1,... ,xn] be the polynomial ring in the variables x1,... ,xn over K. Below, we denote a monomial x^1 ■ ■ ■ x£ R by xa where a = («i,...,an) £ N" is a sequence of non-negative integers. A monomial ordering on R used in theory of Grobner and involutive bases is a total order — on the set of all monomials satisfying the following properties:
— it is multiplicative; i.e., xa x^ implies xa+7 x^+7 for all a,/3,j £ N";
— the constant monomial is the smallest; i.e., 1 xa for all a £ N".
A typical example of a such monomial ordering is the pure lexicographical ordering, denoted by —iex. For two monomials xa, x^ £ R xa —¡ex if the left most nonzero entry of ft — a is positive.
Let I = {f1,..., fk) be the ideal of R generated by the polynomials f1,..., fk £ R. Also let f £ R and — be a monomial ordering on R. The leading monomial of f is the greatest monomial (with respect to -—) appearing in f, and we denote it by LM(/). If F c R is a set of polynomials, we denote by LM(F) the set {LM(/) | f £ F}. The leading coefficient of f, denoted by LC(/), is the coefficient of LM(/). The leading term of f is LT(/) = LC(/)LM(/). The leading term ideal of I is defined to be
LT(I) = {LT(/) | f £ I).
A finite set G = {g1,... ,gk} c I is called a Grobner basis of I w.r.t. — if LT(/) = {LT(^1),..., LT(^fc)). For more details, we refer to [13], pages 213-214.
We recall below the definition of a special kind of involutive bases, namely, Janet basis. For this purpose, we describe first the notion of an involutive division [10] which is a restricted monomial division [1] that, together with a monomial ordering, determines the properties and the structure of an involutive basis. This makes the main difference between involutive bases and Grobner bases. The idea is to partition the variables into two subsets of multiplicative and non-multiplicative variables, and only the multiplicative variables can be used in the divisibility relation.
Definition 1 (see [10]). An involutive division C on the set of monomials of R is given, if for any finite set U of monomials and any u £ U, the set of variables is partitioned into the subsets of multiplicative Mc(u, U) and non-multiplicative NMc(u, U) variables, and the following three conditions hold:
— u,v £ U, uC(u, U) n vC(v, U) = 0 u £ vC(v, U) or v £ uC(u, U)
- v G U, v G uC(u, U) C(v, U) с C(u, U)
- и G V and V с U C(u, U) с C(u, V)
where C(u,U) denotes the monoid generated by the variables in Mc(u,U). If v G uC(u,U) then и is called C— (involutive) divisor of v and the involutive divisibility relation is denoted by u|cv. If v has no involutive divisors in a set U, then it is called C—irreducible modulo U.
There are involutive divisions based on the classical partitions of variables suggested by Janet [5] and Thomas [7]. In this paper, we are interested in Janet division [10] only.
Example 1 (Janet division). Let U be a finite set of monomials and и G U. The variable x\ is multiplicative for и, if и has the maximum degree of x\ in U. For г > 1, the variable Xi is multiplicative for u, if it has the maximum degree of Xi in the set {v G U | degj(v) = deg^(и), 1 < j < i}, where deg^(v) denotes the degree of Xj in a monomial v.
Proposition 1 (see [10]). Janet division satisfies the conditions in Definition 1 and is noetherian.
Throughout this paper C is assumed to be the Janet division. Now, we define an involutive basis.
Definition 2 (see [1]). Let I с R be an ideal and ^ a monomial ordering on R. Let C be an involutive division. A finite set G с I is an (C-)involutive basis for I if for all f G I there exists g G G such that LM(#)|cLM(/).
From this definition and from that for Grobner basis [2, 13] it follows that an involutive basis for an ideal is its Grobner basis, but the converse is not always true.
Remark 1. By using an involutive division in the conventional division algorithm for polynomial rings, we obtain an involutive division algorithm. If F is a finite polynomial set and C is an involutive division, then we use NFc(f,F) to denote the remainder of f on involutive division by F.
The following theorem provides an algorithmic characterization of involutive basis for a given ideal which is an involutive analogue of the Buchberger characterization of Grobner basis.
Theorem 1 (see [10]). Let I с R be an ideal, ^ a monomial ordering on R, and C an involutive division. Then a finite subset G с I is an involutive basis for I if for each f G G and each x G NMc(LM(f ), LM(G)), we have NF c(xf,G) = 0.
Based on this theorem, one can design an algorithm to compute involutive bases. We recall here the InvBasis algorithm from [10] (see also [1]) which computes a minimal involutive basis for an ideal. An involutive basis G is called minimal if for any other involutive basis G the inclusion LM(G) Ç LM(G) holds. A minimal involutive basis exists and being monic and involutively autoreduced, i.e. satisfying
(Уд G G) [g = NFc(g,G\Ы)]
is unique for a given ideal, a monomial ordering and a constructive involutive division (see [10] for the definition of constructivity).
Here, as it has been mentioned in [1], in comparison to the algorithm in [10] another selection strategy is used. By this strategy, a polynomial whose leading monomial has no proper divisor is chosen. However, this algorithm is not efficient in practice, it processes the repeated prolongations and does not use any criterion to avoid the unnecessary reductions. For this purpose, we consider in the next section, the Gerdt's algorithm from [1] as an improvement of InvBasis.
Algorithm 1 InvBasis
inputs
F, a set of polynomials; £, an involutive division;, a monomial ordering. outputs
a minimal involutive basis for (F} w.r.t £ and Select / G F with no proper divisor of LM(/) in LM(f)
G := {/}; Q := F \ G; while Q = 0 do
Select and remove p G Q with no proper divisor of LM(p) in LM(Q); h := NFc(p, G); if h = 0 then
for g G G which LM(g) is properly divisible by LM(h) do
Q := Q u{9}; G := G \{9};
end for
G := G U{h};
Q := Q U{xg | g G G,x G NMc(LM(g), LM(G))}; end if end while Return (G)
3. Gerdt's algorithm
The Gerdt's algorithm [1] improves the above algorithm InvBasis. It uses the involutive form of Buchberger's criteria to avoid unnecessary reductions and avoids the repeated processing of non-multiplicative prolongations. In order to explain the structure of this algorithm, we recall some more definitions and notations.
Definition 3 (see [1]). An ancestor of a polynomial f £ F c R \ {0}, denoted by anc(/), is a polynomial g £ F of the smallest deg(LM(g)) among those satisfying LM(/) = uLM(g) where u is either the unit monomial or a power product of non-multiplicative variables for LM(g) and such that NF^(/ — ug, F \ {f}) = 0 if f = ug.
This additional information for an element in a polynomial set is useful to avoid unnecessary reductions specially by applying the adapted Buchberger's criteria (see below).
Proposition 2 (see [11]). Let I c R be an ideal and G c I be a finite set. Let also — be a monomial ordering on R and C an involutive division. Then G is an C—involutive basis for I if for all f £ G one of the following conditions holds (we use "IZ" to denote proper conventional division):
1) for all x £ NMc(LM(f), LM(G)) the equality NFc(xf,G) = 0 holds;
2) there exists g £ G with LM(g)I^LM(f) satisfying one of the following:
(Ci) LM(anc(/))LM(anc(9)) = LM(/);
(C2) lcm(LM(anc(/)),LM(anc(9))) z LM(/);
(C3) there exists t £ G such that:
- lcm(LM(i),LM(/)) Z lcm(LM(anc(/)),LM(anc(^)));
- lcm(LM(i), Lm(#)) z lcm(LM(anc(/)), LM(anc(^)));
(C4) there exists t £ G computed before anc(/) and y £ NMc(LM(t), LM(G)) s.t.
- yLM(i) = LM(/);
- lcm(LM(anc(/)), LM(anc(i))) z LM(f).
Now we can present Gerdt's algorithm for computing involutive bases in which we associate to each polynomial f, the triple p = {f,g,V} where f = poly(p) is the polynomial itself, g = anc(p) is its ancestor and V = NM(p) is the list of non-
multiplicative variables of f which have been already processed in the algorithm. If P is a set of triples, we denote by poly(P) the set {poly(p) | p £ P}. If no confusion arises, we may refer to a triple p instead of poly(p), and vice versa.
Algorithm 2 Gerdt inputs
F, a set of polynomials; £, an involutive division; a monomial ordering outputs
a minimal involutive basis for (F} w.r.t £ and ^ Select f G F with no proper divisor of LM(/) in LM(f) T := {{f, f, 0}}; Q : = {{q,q, 0} | q G F \ {f}}; Q :=HeadRedüce(Q, T, £, while Q = 0 do
Select and remove p G Q with no proper divisor of LM(poly(p)) in LM(poly(Q));
if poly(p) = anc(p) then
for q G T with LM(poly(p)) С LM(poly(g)) do
Q := Q u {q};
T := T \{q}; end for
end if
h :=TailNormalForm(p, T, £,
T := T U {{h, anc(p), NM(p)}};
for q G T and x G NMc(LM(poly(q)), LM(poly(T))) \ NM(q) do Q := Q U {{x poly(g), anc(g), 0}};
NM(q) := NM(q) П NMC(LM(poly(q)), LM(poly(T))) U {x};
end for
Q :=HeadRedüce(Q, T, £, end while Return (poly(T))
This algorithm includes three subalgorithms HeadReduoe, HeadNormalForm and TailNormalForm which we present below.
Algorithm 3 HeadReduoe
inputs
Q and T, sets of triples; £, an involutive division; a monomial ordering outputs
The Q of triples whose polynomials are £-head reduced modulo T
S := Q Q := 0;
while S = 0 do
Select and remove p G S h :=HeadNormalForm(p, T, £); if h = 0 then
if LM(poly(p)) = LM(h) then Q := Q U{{h,h, 0}};
else
Q := Q U {P};
end if else
if LM(poly(p)) = LM(anc(p)) then
S := S \{q G S | anc(g) = poly(p)}; end if end if end while Return (Q)
In the below subalgorithm HeadNormalForm the Boolean expression Criteria(p, g) is true if at least one of the four criteria in Proposition 2 holds for p and g.
Algorithm 4 HeadNormalForm
inputs
T, a set of triples; p, a triple; £, an involutive division; a monomial ordering outputs
£-head normal form of poly(p) modulo T
h := poly(p);
G := poly(T);
if LM(h) is £-irreducible modulo G then
Return (h) else
Select g G G with LM(poly(g))|£LM(h); if LM(h) = LM(anc(p)) then if Criteria(p, g) then
Return (0) end if else
while h = 0 and LM(h) is £-reducible modulo G do Select g G G with LM(g)|£LM(h);
h h LT(fe) h := h - ;
end while end if end if Return (h)
Algorithm 5 TailNormalForm
inputs
T, a set of triples; p, a triple, which is £-head reduced modulo T; £, an involutive division; a monomial ordering outputs
£-Normal form of poly(p) modulo T
h := poly(p);
G := poly(T);
while h has a term t which is £—reducible modulo G do Select g G G with LM(g)|£i;
h := h — g-
LT(fl)' end while Return (h)
4. Modified algorithm
In this section, we present the following variant of Gerdt's algorithm for computing involutive bases. This variant seems to be simpler both to understand and to implement. We also present here a proof of its correctness and termination including the correctness proof for the used criteria.
Algorithm 6 VarGerdt inputs
F, a set of polynomials; £, an involutive division; a monomial ordering outputs
a minimal £—involutive basis for (F} Select f G F with no proper divisor of LM(/) in LM(f) T := {{/, f, 0}}; Q : = {{q,q, 0} | q G F \ {/}}; while Q = 0 do
Select and remove p G Q with no proper divisor of LM(poly(p)) in LM(poly(Q));
h :=NormalForm(p, T, £,
if h = 0 and LM(poly(p)) = LM(anc(p)) then
Q := {ч G Q 1 anc(g) = poly(p)};
else
if LM(poly(p)) = LM(h) then
for q G T with LM(poly(h)) С LM(poly(q)) do
Q := Q u{^};
T := T \{g}; end for
Г := Г U {{h,h, 0}}; else
T := T U {{h, anc(p), (p)}}; end if
for q G T and x G NML(LM(poly(g)), LM(poly(T))) \ NM(q) do Q := Q U {{x poly(g), anc(g), 0}};
NM(q) := NM(q) П NMC(LM(poly(g)), LM(poly(T))) U {x}; end for end if end while Return (poly(T))
Algorithm 7 NormalForm inputs
p, a triple; T, a set of triples; an involutive division; a monomial ordering outputs
^-Normal form of p modulo T h := poly(p); G := poly(T);
while h has a term t which is C—reducible modulo G do Select g e G with LM(5)|£i; if t = LT(poly(p)) and Criteria(h, g) then
Return (0) else
h := h — 9 LT(^); end if end while Return (h)
The subalgorithm NormalForm returns the L—normal form of a polynomial w.r.t. a given set of polynomials (see below). Moreover, this subalgorithm detects some unnecessary reductions using the involutive form of Buchberger's criteria. It is worth noting that Gerdt and Yanovich [14] have shown that the use of C3 and C4 criteria (see Proposition 2) may notably slow down the computation of involutive bases. That is why, we use only C\ and C2 criteria. Below, we give a simple proof of correctness for these criteria.
Lemma 1. Let I C R be an ideal, -< a monomial ordering on R and L an in-volutive division. Let T c I be the last computed basis during a computation of an involutive basis for I, and f £ I. Then NFc(/, T) = 0 if there exists q £ T with LM(q)lcLM(f) satisfying one of the following conditions:
(Ci) LM(anc(/))LM(anc(g)) = LM(/) (c2) lcm(LM(anc(/)),LM(anc(g))) C LM(/).
Proof. Suppose that (f,q) satisfies Ci. Then, for lcm(LM(anc(/)),LM(anc(g))) two cases are possible: If it is a proper divisor of LM(anc(/))LM(anc(g)) = LM(/), there exists a monomial s = 1 such that LM(/) = slcm(LM(anc(/)),LM(anc(g))). Let f = u ■ anc(/), q = v ■ anc(g) and LT(/) = t ■ LT(g) for some monomials u and v and some term t. Thus, we can write f — tq = u ■ anc(/) — tv ■ anc(g) = s(u' ■ anc(/) — v' ■ anc(g)) where u = su' and vt = sv' for some monomials u', v'. Since, lcm(LM(anc(/)), LM(anc(g))) is a proper divisor of LM(/), then u' ■ anc(/) — v' ■ anc(^) has been computed before f (see the selection strategy for choosing polynomials in the algorithm), and therefore, it has a standard representation w.r.t. T. This implies that f — tq has also a standard representation w.r.t. T, and NFc(f,T) = 0. As the second case, if lcm(LM(anc(/)), LM(anc(g))) is equal to LM(anc(/))LM(anc(g)), then by the Buchberger's first criterion, we can conclude that u' ■ anc(/) — v' ■ anc(^) has a standard representation w.r.t. T, and this proves the assertion like the first case.
Now, if (f,q) satisfies C2, the proof is similar to that for the above first case. □
Just as above, for two polynomials p and q, the Boolean expression Criteria(p, q) is true if they satisfy at least one of the criteria Ci or C2, and false otherwise.
Theorem 2. The VarGerdt algorithm computes a minimal involutive basis for the ideal generated by the input polynomial set.
Proof. Let F C R be a finite set of polynomials taken as the input of VarGerdt algorithm. Let ^ be a monomial ordering on R and L be a constructive noetherian involutive division [10]. Let G be a a polynomial set that is the output of the algorithm for F. Since (by the structure of the algorithm) for each f £ G and each x £ NMc(LM(f), LM(G)), we examine xf (we either compute its L—normal form w.r.t. G or we discard it by the criteria), it follows that NFc(xf,G) = 0. Therefore, G is an involutive basis for {F) by Theorem 1. Moreover, this basis is minimal by the definition of this concept due to the structure of the VarGerdt algorithm and constructivity of L.
The termination of VarGerdt algorithm follows from the noetherianity of L, see Proposition 1. □
Remark 2. The algorithm VarGerdt as well as Gerdt's algorithm not necessarily outputs an involutively tail autoreduced basis. If the output is G, then, in accordance to the definition that follows Theorem 6 of Section 2, the tail autoreduc-tion is provided by the command
(Vg £ G) [g := NFc(g,G\M)].
5. Experiments and Results
We have implemented both algorithms VarGerdt and Gerdt in Maple 141. For an efficient implementation of Gerdt algorithm, we refer to [15]. It is worth noting that, in this paper, we are willing to compare the structure and behavior of VarGerdt and Gerdt algorithms on the same platform. Therefore, we do not compare our implementations with [15].
1 The Maple code of our programs and examples is available on the Web page http://invo. jinr.ru/
To compare the behavior of these algorithms, we used some well-known examples from the the collection [16]. These examples have been widely used for verification and comparison of different software packages for construction of Grobner bases. In our benchmarking we applied the following selection strategy for the polynomials in Q. An element at the initialization step to be inserted in set Q is that with has the lowest leading monomial with respect to the degree-reverse-lexicographic ordering. The same strategy is used for selected elements from Q in the main loop. In the case of several such polynomials, that one is selected whose ancestor has been examined earliest.
The results are shown in the following tables where computation was performed on a personal computer with 2.33 GHz, 2xIntel(R) Xeon(R) Quad core, 16 GB RAM and 64 bites under the Linux operating system. The computation was done over Q and the monomial ordering is always the degree-reverse-lexicographical one.
The time (resp. memo., reds., C\ and C2) column shows the consumed CPU time in seconds (resp. amount of megabytes of memory used, number of reductions to zero, the number of polynomials removed by C\ and C2 criteria) by the corresponding algorithm. The seventh column indicates the number of polynomials eliminated by Rewritten criterion: in VarGerdt algorithm, as well as in Gerdt algorithm, when the polynomial part of a triple {f, g, V} reduces to zero, and f = g, then we can discard any polynomial whose ancestor is f. We call this criterion the Rewritten criterion. The last column shows the largest degree of intermediate polynomials treated during the computation of involutive bases.
A comparison of the timing columns in the above tables and our test for some other examples shows that VarGerdt is more efficient and stable than Gerdt. Indeed, the main difference between these algorithms is that in Gerdt algorithm, we do first the head reduction of all non-multiplicative prolongations (which have not been examined yet) and then we choose one of them to complete its involutive reduction to the full normal form, while in VarGerdt we choose a polynomial (among the non-multiplicative prolongations), and we compute its full normal form.
In Gerdt's algorithm the head involutive reduction of the whole set of non-multiplicative prolongation is done in order to provide a platform for matching a good selection strategy. We refer to paper [17] for details on heuristically good selection strategies for the Janet division. The choice of such selection strategy as well as the use of proper data structures for fast search of involutive divisor [1] plays a key role in providing a high computational efficiency of the involutive algorithms. In this paper we do not consider these important aspects of the practical construction of involutive bases and compare two algorithms experimentally for their simplest and identical for the both algorithms implementation.
Acknowledgements.
The research presented in the paper was completed during the stay of the second author (A. Hashemi) at the Joint Institute for Nuclear Research in Dubna, Russia. He would like to thank Professor V. P. Gerdt for the invitation, hospitality and support. The contribution of the first author (V. P. Gerdt) was partially supported by the grant 01-01-00200 from the Russian Foundation for Basic Research and by the grant 3810.2010.2 from the Ministry of Education and Science of the Russian Federation.
References
1. Gerdt V. P. Involutive Algorithms for Computing Gröbner Bases // Computational Commutative and Non-Commutative Algebraic Geometry. — Amsterdam: IOS, 2005. — Vol. 196 of NATO Sci. Ser. III Comput. Syst. Sci. — Pp. 199-225.
2. Buchberger B. Ein Algorithms zum Auffinden der Basiselemente des Restklassenrings nach einem nuildimensionalen Polynomideal: Ph.D. thesis / Universitöt Innsbruck. — 1965.
Table 1
Noon4 time memo. reds. Ci C2 R deg.
VarGerdt 14.29 819.2 56 6 19 0 10
Gerdt 14.92 821.4 50 6 19 0 9
Cyclic5 time memo. reds. Ci C2 R deg.
VarGerdt 12.15 701.8 77 36 9 18 9
Gerdt 12.80 686.6 83 40 5 0 8
Sturmfels time memo. reds. Ci C2 R deg.
VarGerdt 64.29 4152.6 91 43 212 0 6
Gerdt 80.72 5103.7 80 42 218 0 5
Katsura5 time memo. reds. Ci C2 R deg.
VarGerdt 97.18 9636.8 47 22 1 0 12
Gerdt 54.50 4731.8 47 22 1 0 10
Eco7 time memo. reds. Ci C2 R deg.
VarGerdt 102.62 6066.8 132 51 48 20 6
Gerdt 80.21 4249.7 124 46 40 0 4
Liu time memo. reds. Ci C2 R deg.
VarGerdt 2.65 155.1 18 6 3 0 0
Gerdt 3.81 236.0 18 6 3 0 0
Lichtblau time memo. reds. Ci C2 R deg.
VarGerdt 111.21 22302.9 25 0 9 1 11
Gerdt > 8 hours ? ? ? ? ? ?
Katsura6 time memo. reds. Ci C2 R deg.
VarGerdt 213.59 16124.3 128 44 3 0 8
Gerdt 4227.32 1632783.6 128 44 3 0 7
Cyclic6 time memo. reds. Ci C2 R deg.
VarGerdt 1313.88 222625.1 542 169 7 47 11
Gerdt 2571.84 655916.5 476 152 18 0 10
3. Buchberger B. A Criterion for Detecting Unnecessary Reductions in the Construction of Grobner-bases // Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille, 1979). — Berlin: Springer, 1979. — Vol. 72 of Lecture Notes in Comput. Sci. — Pp. 3-21.
4. Grobner Bases and Applications / Ed. by B. Buchberger, F. Winkler. — Cambridge: Cambridge University Press, 1998. — Vol. 251 of London Mathematical Society Lecture Note Series. — ISBN 0-521-63298-6, Pp. viii+552. — Papers from the Conference on 33 Years of Grobner Bases held at the University of Linz, Linz, February 2-4, 1998.
5. Janet M. Les systemes d'equations aux derivees partielles // J. Math. Pures Appl. — 1920. — Vol. 3. — Pp. 65-151. — ISSN 0021-7824.
6. Seiler W. M. Involution. — Berlin: Springer-Verlag, 2010. — Vol. 24 of Algorithms and Computation in Mathematics. — ISBN 978-3-642-01286-0, Pp. xxii+650. — http://dx.doi.org/10.1007/978-3-642-01287-7. — The formal theory of differential equations and its applications in computer algebra.
7. Thomas J. Differential Systems. — New York: American Mathematical Society, 1937. — Pp. xx+363.
8. Pommaret J.-F. Systems of Partial Differential Equations and Lie Pseu-dogroups. — New York: Gordon & Breach Science Publishers, 1978. — Vol. 14 of Mathematics and its Applications. — ISBN 0-677-00270-*, Pp. xiv+411. — With a preface by Andre Lichnerowicz.
9. Zharkov A. Y., Blinkov Y. A. Involution Approach to Investigating Polynomial Systems // Math. Comput. Simulation. — 1996. — Vol. 42, No 4-6. — Pp. 323332. — ISSN 0378-4754. — http://dx.doi.org/10.1016/S0378-4754C96) 00006-7. — Symbolic computation, new trends and developments (Lille, 1993).
10. Gerdt V. P., Blinkov Y. A. Involutive Bases of Polynomial Ideals // Math. Comput. Simulation. — 1998. — Vol. 45, No 5-6. — Pp. 519-541. — ISSN 0378-4754. — http://dx.doi.org/10.1016/S0378-4754(97)00127-4. — Simplification of systems of algebraic and differential equations with applications.
11. Apel J., Hemmecke R. Detecting Unnecessary Reductions in an Involutive Basis Computation // J. Symbolic Comput. — 2005. — Vol. 40, No 4-5. — Pp. 11311149. — ISSN 0747-7171. — http://dx.doi.org/10.1016/j.jsc.2004.04.004.
12. Gerdt V. P. On an Algorithmic Optimization in the Computation of Involutive Bases // Programming and Computer Software. — 2002. — Vol. 28, No 2. — Pp. 62-65. — ISSN 0132-3474. — http://dx.doi.org/10. 1023/A:1014816631983.
13. Becker T., Weispfenning V. Grobner Bases. — New York: Springer-Verlag, 1993. — Vol. 141 of Graduate Texts in Mathematics. — ISBN 0-387-97971-9, Pp. xxii+574. — A computational approach to commutative algebra, In cooperation with Heinz Kredel.
14. Gerdt V. P., Yanovich D. A. Effectiveness of Involutive Criteria in Computation of Polynomial Janet Bases // Programming and Computer Software. — 2006. — Vol. 32, No 3. — Pp. 134-138. — ISSN 0132-3474. — http://dx.doi.org/10. 1134/S0361768806030030.
15. The Maple package "Janet": I. Polynomial Systems and II. Linear Partial Differential Equations / Y. A. Blinkov, V. P. Gerdt, C. F. Cid et al. // Computer Algebra in Scientific Computing (CASC 2003), V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, eds. — Institute of Informatics, Technical University of Munich, Garching, 2003. — Pp. 31-54.
16. Bini D., Mourrain B. Polynomial Test Suite. — http://www-sop.inria.fr/ saga/POL/.
17. Gerdt V. P., Blinkov Y. A. On Selection of Nonmultiplicative Prolongations in Computation of Janet Bases // Programming and Computer Software. — 2007. — Vol. 33, No 3. — Pp. 147-153. — ISSN 0132-3474. — http://dx.doi.org/10. 1134/S0361768807030048.
УДК 512.7+519
Вариант алгоритма Гердта для вычисления инволютивных
базисов
В. П. Гердт*, А. Хашеми^, Б. М.-Ализадех^
* Лаборатория информационных технологий Объединённый институт ядерных исследований ул. Жолио-Кюри д.6, Дубна, Московская область, 141980, Россия
^ Отделение математических наук Исфаханский технологический университет 84156-83111, Исфахан, Иран
В работе [1] первого автора был представлен эффективный алгоритм вычисления инволютивных базисов и приведенных базисов Гребнера. В данной работе мы рассмотрим модификацию указанного алгоритма, которая упрощает его понимание и облегчает программную реализацию. Мы докажем корректность модифицированного алгоритма и используемых вместе с использованными в нем критериями и его оканчиваемость. Предложенный алгоритм реализован на языке Maple. Путем вычислительных экспериментов с рядом полиномиальных систем мы сопоставим времена счета модифицированного алгоритма с его оригинальной формой, описанной в [1] и также реализованной на языке Maple. При этом детали оба алгоритма были реализованы схожим образом, чтобы обеспечить корректность сравнения их вычислительной эффективности.
Ключевые слова: базисы Гребнера, критерии Бухбергера, инволютивные базисы, алгоритм Гердта.
