generalizedhermitereduction, creativetelescoping...

Generalized Hermite reduction,Creative telescoping,and Definite integration of differentially finite functions

Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno SalvyInria

ISSAC 2018International symposium on symbolic and algebraic computation19 July 2018, New York City

Automatic computation of sums and integrals

n∑i=0

n∑j=0

(i + j

i

)2(4n −2i −2 j

2n −2i

)= (2n +1)

(2n

n

)2

(Blodgelt, 1990)

∫ +∞

0x J1(ax) I1(ax)Y0(x)K0(x)d x =− ln(1−a4)

2πa2 (Glasser, Montaldi, 1994)∫ 1

−1

e−px Tn(x)p1−x2

d x = (−1)nπ In(p)

n∑j=0

n− j∑i=0

q (i+ j )2+ j 2

(q ; q)n−i− j (q ; q)i (q ; q) j=

n∑k=−n

(−1)k q7/2k2+1/2k

(q ; q)n+k (q ; q)n−k(Paule, 1985)

1

Applications in combinatorics Need for faster algorithms

y

x

1

23

4 56

1

2

3 4

56

7

(0,0)

(7,10)

un = #{rook paths from (0, . . . ,0) to (n, . . . ,n) inNd

}• dimension 2

9nun + (−14−10n)un+1 + (2+n)un+2 = 0

• dimension 3 −192n2(1+n)(88+35n)un

+(1+n)(54864+100586n +59889n2 +11305n3)un+1

−(2+n)(43362+63493n +30114n2 +4655n3)un+2

+2(2+n)(3+n)2(53+35n)un+3 = 0• dimension 4

5000n3(1+n)2(2705080+3705334n +1884813n2 +421590n3 +34983n4)un

−(1+n)2(80002536960+282970075928n +·· ·+6386508141n6 +393838614n7)un+1

+2(2+n)(143370725280+500351938492n +·· ·+2636030943n7 +131501097n8)un+2

−(3+n)2(26836974336+80191745800n +100381179794n2 +·· ·+44148546n7)un+3

+2(3+n)2(4+n)3(497952+1060546n +829941n2 +281658n3 +34983n4)un+4 = 0

2

The problem of definite integration

input F (t1, . . . , tn , x)

output G(t1, . . . , tn) = ∫D F (t1, . . . , tn , x)dx

assumption∫

D

∂

∂x

(...

)dx = 0

data structure linear functional equations

linear functionalequations numerical evaluation

asymptotic expansion

proof of identities

closed form

algebraic function

+ × ∑ ∫

3

Previous works

Simple integral, rational function Hermite reduction

input F (t , x) ∈Q(t , x)

output A differential equation forG(t ) = ∮F (t , x)dx

references Ostrogradsky (1845), Hermite (1872), Bostan, Chen, Chyzak, Li (2010a)

F = A0B + ∂

∂x H0

∂∂t F = A1

B + ∂∂x H1

∂2

∂t 2 F = A2B + ∂

∂x H2

......

...

∂r

∂t r F = ArB + ∂

∂x Hr

confinement in==========⇒finite dimension

r∑k=0

ak (t )∂k

∂t kF = 0 + ∂

∂x H

r∑

k=0ak (t ) G (k) = 0

(simple poles)

theorem (Bostan, Chen, Chyzak, Li 2010a)On input of degree d , one can compute the output in O (dω+4)

arithmetic operations. 4

Multiple integral, rational function Griffiths–Dwork reduction

input F (t , x1, . . . , xn) ∈Q(t , x1, . . . , xn)

output A differential equation forG(t ) = ∮F (t , x1, . . . , xn)dx1 · · ·dxn

references Dwork (1962), Griffiths (1969), Bostan, Lairez, Salvy (2013), andLairez (2016)

Compute a0(t ), . . . , ar (t ) ∈Q(t ) such that

r∑k=0

ak (t )∂k

∂t kF =

n∑i=1

∂

∂xi(some rational function)

theorem (Bostan, Lairez, Salvy 2013)One input of degree d , one can compute the output ind 8n+O (1) arithmetic operations.

Generically, the certificate has size > d n2/2 .

5

A differentially finite example

∫ 1

−1

e−px Tn(x)p1−x2︸︷︷︸

=Fn (p,x)

d x = (−1)nπ In(p)

input ∂∂p Fn =−xFn , nFn+1 = ∂

∂x

((x2 −1)Fn

)+ (px2 + (n −1)x −p)Fn ,

(1−x2) ∂2

∂x2 Fn = (2px2+3x−2p) ∂∂x Fn +(p2x2+3px−n2−p2+1)Fn

output p2 ∂2

∂p2 Gn +p ∂∂p Gn − (n2 +p2)Gn = 0

Gn+1 + ∂∂p Gn − n

p Gn = 0

differential finiteness For all i , j ,k Ê 0, there are ai j k and bi j k ∈Q(n, p, x) such that

∂i

∂xi

∂ j

∂p jFn+k (p, x) = ai j k (n, p, x)Fn(p, x)+bi j k (n, p, x)

∂

∂xFn(p, x).

6

Challenges

minimality Wewant to find all relations satified by the integrals

bounds Wewant to understand and control:

• the size of the output,• the computational complexity of the algorithm.

certificateless Wewant to avoiding computing the certificate (otherwise, thecomplexity gets out of control):

• the certificate is much bigger than the output• not possible to compute it with good complexity• it is often useless

We give up:

• simple certification of the output• case where

∫D

∂∂x (...)dx 6= 0

7

Creative telescoping

principle Find all relations∑( j ,k)∈B

c j ,k (n, p)∂

∂p jFn+k (p, x) = ∂

∂x

(u(n, p, x)Fn(p, x)+ v(n, p, x)

∂

∂xFn(p, x)

)

∑

( j ,k)∈Bc j ,k (n, p)

∂

∂p jGn+k (p) = 0.

equivalently Find all B ⊂N2 and(

c j k

)∈Q(n, p)B such that

∂

∂xu = −a200 v + ∑

(i , j )∈Bc j k a0 j k

∂

∂xv =− u −b200 v + ∑

(i , j )∈Bc j k b0 j k ,

has a rational solution u, v ∈Q(n, p, x).

8

Four generations of algorithms 1st generation

problem Find all B ⊂N2 and(

c j k

)∈Q(n, p)B s.t. ∃u, v ∈Q(n, p, x)

∂∂x u =−a200 v + ∑

(i , j )∈Bc j k a0 j k and ∂

∂x v =− u −b200 v + ∑(i , j )∈B

c j k b0 j k .

elimination Only look for solutions with u, v ∈Q(n, p).Akin to polynomial elimination.

Fasenmyer (1949); see also Takayama (1990), Galligo (1985), etc.

minimality bounds certificateless

9

Four generations of algorithms 2nd generation


c j k

)∈Q(n, p)B s.t. ∃u, v ∈Q(n, p, x)

∂∂x u =−a200 v + ∑


∂x v =− u −b200 v + ∑(i , j )∈B

c j k b0 j k .

rational solutions Iteratively solve the differential system (Abramov 1989; Barkatou1999) with increasing support B (FGLM-like).

Chyzak (2000) ; see also Picard (1906), Zeilberger (1990)


10

Four generations of algorithms 3rd generation


c j k

)∈Q(n, p)B s.t. ∃u, v ∈Q(n, p, x)

∂∂x u =−a200 v + ∑


∂x v =− u −b200 v + ∑(i , j )∈B

c j k b0 j k .

linear algebra Predict the denominator of solutions u, v ∈Q(n, p, x),reduce to linear algebra overQ(n, p).

Lipshitz (1988), Apagodu, Zeilberger (2006), Koutschan (2010)


11

Four generations of algorithms 4th generation


c j k

)∈Q(n, p)B s.t. ∃u, v ∈Q(n, p, x)

∂∂x u =−a200 v + ∑


∂x v =− u −b200 v + ∑(i , j )∈B

c j k b0 j k .

reduction of pole order Generalization of Hermite’s reduction

Bostan, Chen, Chyzak, Li (2010b), Chen, Kauers, Singer (2012) andChen, Kauers, Koutschan (2016), Bostan, Chen, Chyzak, Li, Xin (2013),Chen, Huang, Kauers, Li (2015) and Huang (2016), Bostan, Dumont,Salvy (2016), Chen, Hoeij, Kauers, Koutschan (2018), Hoeven (2017)


12

New algorithm

Obstructions to integrability


c j k

)∈Q(n, p)B s.t. ∃u, v ∈Q(n, p, x)

(∗) ∂∂x u =−a200 v + ∑


∂x v =− u −b200 v + ∑(i , j )∈B

c j k b0 j k .

2G/4G hybrid algorithm For all ( j ,k) ∈N2, produce an obstruction λ j k such that

λ j k = 0 ⇔

∂∂x u = −a200 v +a0 j k

∂∂x v =− u −b200 v +b0 j k

has a solution.

By linearity, (∗) has a solution if and only if∑( j ,k)∈B

c j k λ j k = 0.

13

Lagrange identity

differential operator L : f (x) 7→∑i

ai (x)di

dxif (x)

adjoint operator L∗ : f (x) 7→∑i

(−1)i di

dxi

(ai (x) f (x)

)Lagrange’s identity uL( f ) = L∗(u) f + d

dx

(. . .

).

corollary 1 M( f ) = M∗(1) f + ddx

(. . .

), for any diff. op. M .

corollary 2 If L( f ) = 0 then L∗(u) f = ddx

(. . .

)for any u(x)

corollary 3 If L is the minimal annihilating operator of f ,then for any differential operator M ,M( f ) “is a derivative” ⇔ ∃y ∈ K (x), M∗(1) = L∗(y).

14

Generalized Hermite reduction

Hermite reduction Generalized Hermite red.

input u ∈ K (x) u ∈ K (x) and M ∈ K [x]⟨ ddx ⟩

output v ∈ K (x) v ∈ K (x)

prop. 1 u − v ∈ ddx K (x) u − v ∈ M (K (x))

prop. 2 u = ddx (. . . ) ⇒ v = 0 u = M(. . . ) ⇒ v = 0

15

Testing integrability with GHR

input γ(x) a “function”L, the minimal annihilating operator of γf ∈ K (x)⟨ d

dx ⟩ ·γ, the function space generated by γ

output ∃g ∈ K (x)⟨ ddx ⟩ ·γ, f = d

dxγ

algorithm write f = u(x)γ+ ddx (. . . ) . corollary 1

v(x) ←GHR(u,L∗)

return v = 0 . corollary 3

16

GHR powered variant of Chyzak's algorithm

input I a D-finite ideal and f ∈A/I

output generators of the telescoping idealT f w.r.t. ∂∂x

algorithm γ← a cyclic vector ofA/I with respect to ∂∂x

L ← the minimal operator annihilating γL ← [1];G ← {};Q ← {}

while µ← pop(L ) doif µ is a not multiple of the leading term of an element ofG then

write µ · f = uµ(x)γ+ ∂∂x (. . . )

λµ ←GHR(uµ,L∗)

if ∃ a K -linear rel. between λµ and { λν | ν ∈Q} then( aν )ν∈Q ← coeff. of the relation λµ u =∑

ν∈Q aν λν

Add µ−∑ν∈Q aνν toG

elseadd µ toQ ; enqueue δ1µ, . . . ,δeµ inL .

returnG 17

Timings Sample of a benchmark with> 100 instances

∫2Jm+n(2t x)Tm−n(x)p

1−x2dx [diff. t , shift n and m] (1)∫ 1

0C (λ)

n (x)C (λ)m (x)C (λ)

`(x)(1−x2)λ−

12 dx [shift n, m, `] (2)∫ ∞

0x J1(ax)I1(ax)Y0(x)K0(x)dx [diff. a] (3)∫

n2+x+1n2+1

((x+1)2

(x−4)(x−3)2(x2−5)3

)n√x2 −5e

x3+1x(x−3)(x−4)2 dx [shift n] (4)∫

C (µ)m (x)C (ν)

n (x)(1−x2)ν−1/2 dx [shift n, m, µ, ν] (5)∫x`C (µ)

m (x)C (ν)n (x)(1−x2)ν−1/2 dx [shift `, m, n, µ, ν] (6)∫

(x +a)γ+λ−1(a −x)β−1C (γ)m (x/a)C (λ)

n (x/a)dx, [diff. a, shift n,m,β,γ,λ] (7)

18

Timings Results

Integral (1) (2) (3) (4) (5) (6) (7)

New algorithm (Maple) 13 s > 1h > 1h 1.5 s 1.5 s 165 s 53 s

ChyzakK 19 s 253 s 45 s 232 s 516 s >1h >1h

KoutschanK 1.9 s* 2.3 s 5.3 s >1h 2.3 s* 5.4 s 2.2 s*

* Nonminimal output.K Uses Koutschan’s HolonomicFunctions (Mathematica package).

conclusion It really works! New algorithm for D-finite integrationNew proof of the D-finiteness of the telescoping ideal of a D-finite function2G/4G unification

future work Better understanding of the practical performanceGeneralization to discrete sums

19

References i

Abramov, S. A. (1989). “Rational solutions of linear differential and difference equations withpolynomial coefficients”. In: Zh. Vychisl. Mat. i Mat. Fiz. 29.11, pp. 1611–1620, 1757.

Apagodu, M., D. Zeilberger (2006). “Multi-Variable Zeilberger and Almkvist-ZeilbergerAlgorithms and the Sharpening of Wilf- Zeilberger Theory”. In: Adv. in Appl. Math. 37.2,pp. 139–152.

Barkatou, M. A. (1999). “On Rational Solutions of Systems of Linear Differential Equations”. In:Journal of Symbolic Computation 28.4-5, pp. 547–567.

Bostan, A., S. Chen, F. Chyzak, Z. Li (2010a). “Complexity of Creative Telescoping for BivariateRational Functions”. In: Proceedings of the 35th International Symposium on Symbolic andAlgebraic Computation. ISSAC 2010 (Munich). New York, NY, USA: ACM, pp. 203–210.

– (2010b). “Complexity of creative telescoping for bivariate rational functions”. In: ISSAC’10.ACM, pp. 203–210.

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References ii

Bostan, A., S. Chen, F. Chyzak, Z. Li, G. Xin (2013). “Hermite reduction and creative telescopingfor hyperexponential functions”. In: ISSAC’13. ACM, pp. 77–84.

Bostan, A., L. Dumont, B. Salvy (2016). “Efficient algorithms for mixed creative telescoping”. In:ISSAC’16. ACM, pp. 127–134.

Bostan, A., P. Lairez, B. Salvy (2013). “Creative Telescoping for Rational Functions Using theGriffiths–Dwork Method”. In: Proceedings of the 38th International Symposium on Symbolicand Algebraic Computation. ISSAC 2013 (Boston). New York, NY, USA: ACM, pp. 93–100.

Chen, S., M. van Hoeij, M. Kauers, C. Koutschan (2018). “Reduction-based creative telescopingfor fuchsian D-finite functions”. In: J. Symbolic Comput. 85, pp. 108–127.

Chen, S., H. Huang, M. Kauers, Z. Li (2015). “A modified Abramov-Petkovšek reduction andcreative telescoping for hypergeometric terms”. In: ISSAC’15. ACM, pp. 117–124.

Chen, S., M. Kauers, C. Koutschan (2016). “Reduction-based creative telescoping for algebraicfunctions”. In: ISSAC’16. ACM, pp. 175–182.

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References iii

Chen, S., M. Kauers, M. F. Singer (2012). “Telescopers for rational and algebraic functions viaresidues”. In: ISSAC’12. ACM, pp. 130–137.

Chyzak, F. (2000). “An Extension of Zeilberger’s Fast Algorithm to General HolonomicFunctions”. In: Discrete Math. 217.1-3. Formal power series and algebraic combinatorics(Vienna, 1997), pp. 115–134.

Dwork, B. (1962). “On the Zeta Function of a Hypersurface”. In: Inst. Hautes Études Sci. Publ.Math. 12, pp. 5–68.

Fasenmyer, M. C. (1949). “A Note on Pure Recurrence Relations”. In: Amer. Math. Monthly 56,pp. 14–17.

Galligo, A. (1985). “Some Algorithmic Questions on Ideals of Differential Operators”. In:EUROCAL ’85, Vol.\ 2 (Linz, 1985). Vol. 204. Lecture Notes in Comput. Sci. Berlin: Springer,pp. 413–421.

Griffiths, P. A. (1969). “On the Periods of Certain Rational Integrals”. In: Ann. of Math. 2nd ser. 90,pp. 460–541.

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References iv

Hermite, C. (1872). “Sur l’intégration Des Fractions Rationnelles”. In: Ann. Sci. École Norm. Sup.2nd ser. 1, pp. 215–218.

Hoeven, J. van der (2017). Constructing reductions for creative telescoping. Technical Report,HAL 01435877, http://hal.archives-ouvertes.fr/hal-01435877/.

Huang, H. (2016). “New bounds for hypergeometric creative telescoping”. In: ISSAC’16. ACM,pp. 279–286.

Koutschan, C. (2010). HolonomicFunctions, User’s Guide. 10-01. RISC Report Series, Universityof Linz, Austria.

Lairez, P. (2016). “Computing Periods of Rational Integrals”. In:Mathematics of Computation85.300, pp. 1719–1752.

Lipshitz, L. (1988). “The Diagonal of a D-Finite Power Series Is D-Finite”. In: J. Algebra 113.2,pp. 373–378.

Ostrogradsky, M. (1845). “De l’intégration des fractions rationnelles”. In: Bull. classephys.-math. Acad. Impériale des Sciences Saint-Pétersbourg 4, pp. 145–167, 286–300.

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http://hal.archives-ouvertes.fr/hal-01435877/

References v

Takayama, N. (1990). “An Algorithm of Constructing the Integral of a Module — an InfiniteDimensional Analog of Gröbner Basis”. In: Proceedings of the 15th International Symposiumon Symbolic and Algebraic Computation. Tokyo, Japan: ACM, pp. 206–211.

Zeilberger, D. (1990). “A fast algorithm for proving terminating hypergeometric identities”. In:Discrete Math. 80.2, pp. 207–211.

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