.

This work has been supported by the ANR-09-JCJC-0098-01 MaGiX project, as well as a Digiteo 2009-36HD grant and Région Ile-de-France.

On the complexity of skew arithmetic

Joris van der Hoeven

LIX, CNRS

École polytechnique

91128 Palaiseau Cedex

France

Email: vdhoeven@lix.polytechnique.fr

Web: http://lix.polytechnique.fr/~vdhoeven

December 27, 2015

In this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed almost linearly in terms of the cost of multiplication.

Keywords: Linear differential operators, algorithm, complexity, multiplication, local solution, division, gcrd, lclm

A.M.S. subject classification: 68W30, 68Q15, 34M03, 12E15

1.Introduction

Let be an effective field of constants of characteristic zero, so that all field operations can be carried out by algorithms. Given an indeterminate and the derivation , where , it is well known [26, 8, 12, 28, 29] that the skew polynomial ring behaves very much like an ordinary polynomial ring: there are skew analogues for each of the classical operations of division with remainder, greatest common divisors, least common multiples, etc. In this paper, we will study the complexity of these operations. For this purpose, it will be more appropriate to work in the ring instead of . In analogy with the commutative case, we will give bounds for the computational complexities of the various operations in terms of the complexity of operator multiplication.

For our complexity measures, we make the customary assumption that all field operations in can be carried out in constant time . We will try to express the complexities of our algorithms in terms of the following standard complexities:

We will denote by the subset of of polynomials of degree . Likewise, we denote by the set of operators of degree in and degree in .

Now consider two linear differential operators . We start with studying the following complexities:

The special case was first studied in [20], where it was shown that , using evaluation-interpolation techniques. The inverse bound has been proved in [5]; this paper also contains detailed information on the constant factors involved in these bounds. Recently (and after the writing of a first version of this paper), the quasi-optimal bound was proved in [2].

For fixed constants , one has , , , etc., by splitting the multiplicands in a finite number of pieces. In this paper, we will freely use this remark without further mention. In order to simplify our complexity estimates, it will be convenient to make a few additional assumptions. First of all, we will assume that , whence in particular . We will also assume that the function is increasing and that is increasing in both and . This will indeed be the case for the complexity bounds for that will be given in Section 3.

In Section 2, we will first prove (see Theorems 2 and 3) that the problems of multiplication and operator-vector application are essentially equivalent when . We also recall the best existing bounds for operator multiplication.

In Section 3, we show that the problems of computing fundamental systems of solutions and its inverse can be reduced to operator multiplication modulo a logarithmic overhead (see Theorems 7 and 8). This provides a dual way to perform operations on differential operators by working on their fundamental systems of solutions. In Section 3 and all subsequent sections, we always assume that . This is indeed required for the truncations of the fundamental systems of solutions at order to be linearly independent.

In Section 4, we start with the operations of exact right division and right division with remainder. In Section 5, we consider greatest common right divisors (gcrds) and least common left multiples (lclms). Again, we will show how to express the complexities of these operations essentially in terms of the complexity of multiplication (see Theorems 13, 15, 18 and 21).

For several of our algorithms, we need to work at a point where certain operators are non singular. If we only need the input operators to be non singular, then it is easy to find a point where this is the case. If we also need the output operators or certain auxiliary operators to be non singular (as in Section 5), then we resort to picking random points, which are non singular with probability 1. In Section 5.2 we present additional techniques for turning algorithms which rely on random point picking into randomized algorithms of Las Vegas type and into fully deterministic algorithms.

For technical reasons, we found it convenient to work with respect to the Euler derivation instead of . Nevertheless, operators in can be converted efficiently into operators in and vice versa, modulo an increase of the degree in with the degree in or (see Lemma 6). Using our assumption that , such increases of the degree by only gives rise to constant overheads in the complexity bounds. Hence, the complexity bounds for our main algorithms from Sections 3, 4 and 5 still hold when replacing by . In addition, some of the algorithms can be adapted to directly use instead of , without the need for any conversions (see Remark 11).

To the best of our knowledge, the idea to perform operations on linear differential operators via power series solutions was first proposed (but only partially worked out) in [4, Chapter 10]. In this paper, we use a slightly different technique: instead of a single power series solution, we prefer to consider a fundamental system of solutions. This has the advantage of forcing a clean bijection between operators and solution spaces, thereby avoiding part of the randomness in the proposals from [4, Chapter 10].

It is also possible to mimic classical divide and conquer algorithms for right division, greatest common right divisors and least common left multiples, while using adjoints in order to perform the recursive operations on the appropriate side. Such algorithms were partially implemented inside Mathemagix [24] and we plan to analyze this technique in more details in a forthcoming paper.

Various complexity results for computations with linear differential operators and other skew polynomials were previously obtained [17, 14, 15, 25, 16, 4]. Especially the computation of greatest common right divisors and least common left multiples of two or more operators has received particular attention. After the publication of a first version of this paper [23], the complexities of several classical algorithms [19, 33, 25] for the computation of least common right multiples were studied in great detail in [6], and new improvements were proposed there.

The complexities of most of the algorithms in this paper are stated in terms of the input and output sizes. The uncertified randomized algorithms for gcrds and lclms are optimal up to logarithmic factors from this perspective, which yields an improvement with respect to the previously known complexity bounds. In the context of certified randomized algorithms (i.e. of Las Vegas type), the complexity bounds remain quasi-optimal in terms of the size of a suitable certificate. From the deterministic point of view, the new algorithms for gcrds and lclms are suboptimal.

Acknowledgment. We are grateful to the second referee whose questions and remarks have lead to several improvements with respect to the first version of this paper.

2.Evaluation and interpolation

The key argument behind the proof from [20] that is the observation that an operator is uniquely determined by its images on the vector . This makes it possible to use a similar evaluation-interpolation strategy for the multiplication of differential operators as in the case of FFT-multiplication of commutative polynomials. More precisely, given , let be the matrix of the mapping with respect to the bases and :

The evaluation and interpolation steps can be done efficiently using the following lemma, which is essentially contained in [5]:

Lemma 1. Let . Then

  1. We may compute as a function of in time .

  2. We may recover from in time .

Proof. Consider the expansion of with respect to

For all , we have

In other words, is a lower triangular band matrix

of bandwidth . The coefficients on the -th subdiagonal of are exactly the result of a multipoint evaluation of at . It is classical [27, 35, 3] that both multipoint evaluation and the inverse operation of interpolation can be performed in time . Doing this for each of the polynomials yields the result.

Theorem 2. If , then

Proof. Let and assume that we want to compute . We may evaluate in time . We may also evaluate in time . Using Lemma 1, we may recover from in time . This completes the proof.

Theorem 3. If , then

Proof. Assume now that we are given , as well as a vector and that we want to evaluate . This is equivalent to evaluating the operator at the vector . It is classical [1] that can be computed in time . Using Lemma 1, we may compute the unique operator with in time . We may next compute the product in time . Lemma 1 finally allows us to evaluate at in time , thereby yielding .

The above results immediately imply the bound from [20] by the computation of a product to the computation of a matrix product

After the publication of a first version of this paper, the following quasi-optimal bound for was established in [2, Theorems 3 and 5].

Theorem 4.

  1. For , we have .

  2. For , we have .

The inverse bound from [5] can also be generalized:

Theorem 5. If , then the product of an matrix and an matrix with coefficients in can be computed in time .

Proof. By the result from [5], the problem is equivalent to the computation of operators in with a fixed operator . Setting , we may compute in time . We may directly read off the products from the result.

In this paper, we have chosen to work with respect to the derivation instead of . The following result from [5, Section 3.3] can be used to efficiently convert between operators in and (in [20], we proved a somewhat weaker result which would also suffice for the purposes of this paper). We have written for the set of operators of degree in and degree in .

Lemma 6.

  1. Any operator in can be converted into an operator in in time .

  2. Any operator in can be converted into an operator in in time .

3.Local solutions

From now on, we will assume that . We recall that an operator of order is said to be non singular at , if its leading coefficient does not vanish at . We will say that an operator of order is non singular (at the origin) if and is non singular as an operator in .

Given a non singular differential operator of order , the equation admits a canonical fundamental system of solutions in , with the property that and for all with . Conversely, given a -linearly independent vector of power series , there exists a unique monic operator of order with . Let us show how to convert efficiently between these two representations.

Theorem 7. Let be a differential operator of order , which is non singular at the origin, and let be its canonical fundamental system of solutions. Then we may compute up to order in time . In other words,

Proof. Modulo multiplying on the left by , we may assume without loss of generality that is monic. Since is non singular at the origin, we have . Rewritten in terms of , this means that is of the form

for certain . Setting , we observe that maps into . We now compute using the “recursive” formula

where

The equation (4) is a schoolbook example for applying the strategy of relaxed resolution of power series equations [21, 22]. Since operates coefficientwise, it can be computed in linear time. The main cost of the computation therefore reduces to the relaxed evaluation of . Using fast relaxed multiplication, this amounts to a cost

Using the monotonicity assumption and Theorem 3, the result follows.

In what follows, given a non zero series in , we denote by its valuation. Given a vector of elements in a -vector space, we will also denote by the subvector space generated by the entries of , and

Notice that .

Theorem 8. Let be -linearly independent. Then there exists a unique monic operator with . Moreover, given the truncation of at order , we may compute at order in time . In other words,

Proof. Modulo a triangularization of , we may assume without loss of generality that . We define operators by

Then annihilates and for any other operator with , we would have , which is in contradiction with the fact that . Moreover, by induction over , we observe that the coefficient of in is given by and the coefficients of in can be expressed in terms of the coefficients of in . In particular, the truncation of at order is uniquely determined by the truncation of at order .

In order to explicitly compute up to a given order, it is more efficient to use a divide and conquer approach. More precisely, given we compute using the following method:

If , then it is easy to check that . For a fixed constant , we thus have

The result now follows from the monotonicity assumption.

Remark 9. If is increasing in for some , then the bound further simplifies to .

Remark 10. We notice that the operator in Theorem 8 is singular if and only if , and if and only if .

Remark 11. The algorithm from the proof can be adapted so as produce a vanishing operator in instead of . Indeed, for this, it suffices to take

and carefully adapt the truncation orders.

Although a general operator can be singular at the origin, many operations on operators (such as right division and greatest common right divisors) commute with translations , and Lemma 6 may be used in conjunction with the following lemma in order to reduce to the case when is non singular at the origin.

Lemma 12. Given a non zero operator , we may find a point where is non singular in time .

Proof. Let be the leading coefficient of . Since , we have for some . Using fast multipoint evaluation [7], we may find such a point in time .

4.Right division

From the formula (3) it is clear that both the degrees in and are additive for the multiplication of operators . In particular, if and is left or right divisible by , then the quotient is again in .

Theorem 13. Let be such that for some and . Then we may compute in time .

Proof. By Lemmas 12 and 12, and modulo a shift , we may assume without loss of generality that and are non singular at the origin. We now use the following algorithm:

Since each of the steps can be carried out in time , the result follows.

It is classical that euclidean division generalizes to the skew polynomial ring . In other words, given operators where , there exist unique operators and in with

and . If and is the leading term of with respect to , then left multiplication of by allows us to remain in the domain : there exist unique and in with

and . The operators and are usually called pseudo-quotients and pseudo-remainders. In some cases, a non trivial polynomial can be factored out in the relation (6). Let be monic, of maximal degree, such that . Then we call and the “simplified” pseudo-quotient and pseudo-remainder of and .

Lemma 14. Let be -linearly independent and define . Given , there exists a unique operator of order with and we may compute its first terms with respect to in time .

Proof. Let for each . Modulo a base change, we may assume without loss of generality that . Let be the operator with

and let denote the inverse operator. Let be the operator with

Writing and , we have

In other words, and its inverse operate coefficientwise and coefficients can be computed in time .

Putting with for each , we may rewrite the equation as

and we observe that the coefficient of in the righthand side of (8) only depends on earlier coefficients of in . In particular, we may solve the equation using a relaxed algorithm. Then the main cost is concentrated in the relaxed evaluation of . As in the proof of Theorem 7, this evaluation can be done in time .

Theorem 15. Let with and . Right pseudo-division of by and simplification yields a relation

where . If is such that , then and can be computed in time .

Proof. Modulo a shift , we may assume without loss of generality that and are non singular at the origin. We now use the following algorithm:

The total complexity of this algorithm is bounded by .

Remark 16. In the above proof, we have assumed that is known beforehand. In general, we may still apply the above algorithm for a trial value . Then the algorithm may either fail (for instance, if ), or return the triple under the assumption that . We may then check whether the triple is correct in time . Applying this procedure for successive guesses , the algorithm ultimately succeeds for an with . Using the monotonicity hypothesis, the total running time thus remains bounded by .

5.Euclidean operations

5.1.Randomized algorithms

It is classical that greatest common right divisors and least common left multiples exist in the skew euclidean domain : given two operators , the greatest common right divisor and the least common left multiple are the unique monic operators with

Assume now that and let and be monic polynomials of minimal degrees, such that and are in . Then we call and the (simplified) pseudo-gcrd and pseudo-lclm of and .

Lemma 17. Let be such that and are non singular at the origin, as well as or . Let and be the canonical fundamental systems of solutions to and . Then

Proof. Let , , and . If is non singular, then it admits a canonical fundamental system of solutions with and for all with . In particular, . Since is the least common left multiple of and , we also have , which completes the proof of the first equality. If is non singular, then we obtain the second equality in a similar way.

If is non singular, then we also have and , since and are non singular. Now , whence . If is non singular, then we obtain the first equality in a similar way.

Theorem 18. Let and be such that and . Assume that , and (or ) are non singular at the origin. Then we may compute in time .

Proof. We compute using the following algorithm:

This algorithm requires a total running time .

Remark 19. In the above proof, we have again assumed that is known beforehand. Below, we will discuss ways to check the correctness of the computed result for a trial value , after which a similar strategy as in remark 16 can be applied. During the relaxed computation of and , we may also check whether at each next coefficient. In the particular case when , the running time of the algorithm will then be bounded by , where is the smallest order at which common solutions no longer exist. This kind of early termination only works for this very special case.

Remark 20. Notice that might be singular at the origin, even if , and are not. This happens for instance when is the minimal annihilator of the vector and the minimal annihilator of the vector , so that .

Theorem 21. Let and be such that and . If , and (or ) are non singular at the origin, then we may compute in time .

Proof. Similar to the proof of Theorem 18, by taking instead of .

5.2.Certifying correctness

The assumption that should be non singular is still a bit unsatisfactory in Theorems 18 and 21, even though the probability that a randomly chosen point is singular is infinitesimal. If we drop this assumption, then we still have in the proof of Theorem 18. Consequently, “candidate” pseudo-gcrds found by the algorithm are genuine pseudo-gcrds whenever pseudo-divides both and . Using the right division algorithms from the previous section, this can be checked in time in the case of gcrds and in the case of lclms.

Remark 22. Using the polynomial linear algebra techniques from [17, 6], it is likely that one may prove that for some and . If this is indeed the case, then the trial divisions of and by can actually be carried out in time .

An alternative way to check whether candidate gcrds and lclms are correct is to compute Bezout and Ore relations. More precisely, given with , there exist operators with

and . The matrix at the righthand side will be called the Euclidean matrix of and . In a similar way as above, we may define a (simplified) pseudo-Euclidean matrix with entries in , whenever . We will say that is non singular at , if the denominators of and do not vanish at .

Theorem 23. Let and be such that and . If , , and are non singular at the origin, then we may compute in time .

Proof. Assuming known, we compute at order as follows:

We finally compute from and using rational function reconstruction. The complexity analysis and the remainder of the proof is done in a similar way as in the proofs of Theorems 15 and 18.

With the above techniques, we may at least verify whether computed pseudo-gcrds or pseudo-lclms are correct. For a fully deterministic algorithm, we still need a way to find a point where is non singular. This can be done by brute force. Let us state the result in the most general setting of pseudo-Euclidean matrices.

Theorem 24. Let and be such that and . Then we may compute in time .

Proof. Let , , and assume first that we know . Let be be pairwise distinct, randomly chosen points in at which and are non singular. At each , we compute canonical fundamental systems of solutions and for and at order . We claim that this can be done in time .

Indeed, it requires a time to rewrite each operator with respect to . We next perform a multipoint evaluation of the coefficients of these operators to obtain the shifted operators at (this requires a time ). The truncations of these operators at order are then converted back to the respresentation with respect to . This can be done in time . Using Theorem 7, we finally compute the required fundamental systems of solutions in time .

From , we get . Since we assumed to be sufficiently large, it follows that is non singular at one of the points . At such a point , the canonical fundamental systems of solutions and generate a vector space of maximal dimension , and with a basis such that for all . We finally apply Theorem 23 in order to obtain . If is unknown, then we use a sequence of guesses , as in the previous proofs.

Remark 25. In the case of least common left multiples, we may directly compute using Theorem 21 and certify the result using trial division by and . This allows us to use the weaker assumption instead of , whereas the complexity bound becomes .

5.3.Summary of the complexity bounds for Euclidean operations

We have summarized our complexity bounds for Euclidean operations on two operators in Table 1. We systematically write for the degree in of the result. We also write for the degree of the Euclidean matrix in .

The algorithms in the first line correspond to applying Theorems 18, 21 and 23 at a randomly chosen point, without checking the result. The second line corresponds to the Las Vegas randomized algorithm for which the answers are certified through trial division (the bound for gcrds might further drop to in view of Remark 22; more generally, the bounds can be restated in terms of sizes of certificates). In the third line, we rather use Euclidean matrices for the certification. The fourth line shows complexity bounds for the deterministic versions of our algorithms.

In comparison, several randomized Las Vegas algorithms were given in [6] that achieve the complexity bound for lclms. This is in particular the case for Heffter's algorithm [19], when using Theorem 4. The non determinism is due to the use of a fast Las Vegas randomized algorithm for the computation of kernels of matrices with polynomial entries [6, Theorem 2]. Grigoriev established complexity bounds for gcrds which rely on a similar reduction to polynomial linear algebra. Along the same lines as in [6], this should lead to a Las Vegas randomized algorithm of complexity , although we did not check this in detail.

In summary, the new algorithms do not achieve any improvements in the worst case. Nevertheless, the uncertified versions of our algorithms admit optimal running times up to logarithmic factors in terms of the combined input and output size. The certified randomized versions satisfy similar complexity bounds in terms of the size of a suitable certificate; such bounds can sometimes be better than the previously known worst case bounds. When performing our expansions at a randomly chosen point in , we also recall that the probability of failure is exponentially small as a function of the bitsize of this point.

Algorithm gcrd lclm Euclidean matrix
Randomized, uncertified
Certified via division
Euclidean certification
Deterministic

Table 1. Complexity bounds for the Euclidean operations on two operators and .

5.4.Generalizations

The algorithms from Section 5.1 extend in a straightforward way to the computation of greatest common right divisors and least common left multiples of more than two operators. For instance, using obvious notations, we obtain the following generalizations of Theorems 21 and 18.

Theorem 26. Let with and , be such that . Assume that and are all non singular at the origin. Then we may compute in time .

Proof. We compute using the following algorithm:

We obtain the result by adding up all complexity bounds.

Remark 27. When taking and using [2], the complexity bound simplifies to . By [6, Theorem 6], we may always take , after which the bound further reduces to . In our randomized setting, this improves upon the bounds from [6, Figure 1].

Remark 28. If we also require a certification of the result, then we may use the trial division technique. This amounts to exact divisions of operators in by . Using the division algorithm from Section 4, and taking and as above, this can be done in time

This is slightly better than the new bound from [6].

Theorem 29. Let and be such that . Assume that and are all non singular at the origin. Then we may compute in time .

Proof. The proof is similar to the one of Theorem 26, except for the way how we compute a basis for . Indeed, we first compute a basis for . This requires a time for the computation of modulo and a time for the remaining linear algebra. We next compute the unique constant matrix such that modulo . Since is non singular, we have at any order, so it suffices to compute up to order in order to obtain up to order .

Remark 30. An interesting question is whether there exists a faster algorithm to compute the orders and of and , without computing and themselves. For this, it suffices to compute the dimensions of and . Assuming that we are at a “non singular point”, the answer is therefore yes: using the techniques from the proofs of Theorems 29 and 26, we may compute in time and in time .

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