Generalized Hermite reduction, Creative telescoping, and Definite - PowerPoint PPT Presentation

Generalized Hermite reduction, Creative telescoping, and Definite integration of differentially finite functions Alin Bostan, Frédéric Chyzak, Pierre Lairez , and Bruno Salvy Inria ISSAC 2018 International symposium on symbolic and algebraic computation 19 July 2018, New York City

Automatic computation of sums and integrals (Blodgelt, 1990) (Paule, 1985) (Glasser, Montaldi, 1994) 1 � � 2 � � � � 2 n n i + j 4 n − 2 i − 2 j 2 n � � = (2 n + 1) i 2 n − 2 i n i = 0 j = 0 � +∞ x J 1 ( ax ) I 1 ( ax ) Y 0 ( x ) K 0 ( x ) dx = − ln(1 − a 4 ) 2 π a 2 0 � 1 e − px T n ( x ) dx = ( − 1) n π I n ( p ) � 1 − x 2 − 1 n − j q ( i + j ) 2 + j 2 ( − 1) k q 7/2 k 2 + 1/2 k n n � � � = ( q ; q ) n − i − j ( q ; q ) i ( q ; q ) j ( q ; q ) n + k ( q ; q ) n − k j = 0 i = 0 k =− n

Applications in combinatorics 3 • dimension 4 • dimension 3 • dimension 2 Need for faster algorithms 7 6 5 4 2 2 1 6 5 4 3 1 2 � rook paths from (0,...,0) to ( n ,..., n ) in N d � u n = # 9 nu n + ( − 14 − 10 n ) u n + 1 + (2 + n ) u n + 2 = 0 (7,10) y − 192 n 2 (1 + n )(88 + 35 n ) u n + (1 + n )(54864 + 100586 n + 59889 n 2 + 11305 n 3 ) u n + 1 − (2 + n )(43362 + 63493 n + 30114 n 2 + 4655 n 3 ) u n + 2 + 2(2 + n )(3 + n ) 2 (53 + 35 n ) u n + 3 = 0 x (0,0) 5000 n 3(1 + n )2(2705080 + 3705334 n + 1884813 n 2 + 421590 n 3 + 34983 n 4) un − (1 + n )2(80002536960 + 282970075928 n +···+ 6386508141 n 6 + 393838614 n 7) un + 1 + 2(2 + n )(143370725280 + 500351938492 n +···+ 2636030943 n 7 + 131501097 n 8) un + 2 − (3 + n )2(26836974336 + 80191745800 n + 100381179794 n 2 +···+ 44148546 n 7) un + 3 + 2(3 + n )2(4 + n )3(497952 + 1060546 n + 829941 n 2 + 281658 n 3 + 34983 n 4) un + 4 = 0

The problem of definite integration data structure linear functional equations algebraic function closed form proof of identities asymptotic expansion numerical evaluation equations linear functional 3 assumption input F ( t 1 ,..., t n , x ) � output G ( t 1 ,..., t n ) = D F ( t 1 ,..., t n , x )d x � ∂ � � ... d x = 0 ∂ x D � � + ×

Previous works

Simple integral, rational function . . . . . . . Hermite reduction . confinement in finite dimension  (simple poles) theorem (Bostan, Chen, Chyzak, Li 2010a) arithmetic operations. . 4 references Ostrogradsky (1845), Hermite (1872), Bostan, Chen, Chyzak, Li (2010a) input F ( t , x ) ∈ Q ( t , x ) � output A difgerential equation for G ( t ) = F ( t , x )d x A 0 ∂ F = + ∂ x H 0 B A 1 ∂ ∂ ∂ t F ∂ x H 1 = + a k ( t ) ∂ k r B � ∂ ∂ t k F = 0 + ∂ x H ∂ 2 A 2 ∂ = = = = = = = = = = ⇒ ∂ t 2 F = + ∂ x H 2 k = 0 B r � a k ( t ) G ( k ) = 0 � ∂ r A r ∂ ∂ t r F ∂ x H r = + k = 0 B On input of degree d , one can compute the output in O ( d ω + 4 )

Multiple integral, rational function Lairez (2016) theorem (Bostan, Lairez, Salvy 2013) (some rational function) Grifgiths–Dwork reduction 5 references Dwork (1962), Grifgiths (1969), Bostan, Lairez, Salvy (2013), and input F ( t , x 1 ,..., x n ) ∈ Q ( t , x 1 ,..., x n ) � output A difgerential equation for G ( t ) = F ( t , x 1 ,..., x n )d x 1 ··· d x n Compute a 0 ( t ),..., a r ( t ) ∈ Q ( t ) such that r a k ( t ) ∂ k n ∂ � � ∂ t k G = ∂ x i k = 0 i = 1 One input of degree d , one can compute the output in d 8 n + O (1) arithmetic operations. Generically, the certificate has size > d n 2 /2 .

A differentially finite example input 6 � 1 e − px T n ( x ) dx = ( − 1) n π I n ( p ) = 0 � 1 − x 2 − 1 � �� � = F n ( p , x ) � � ( x 2 − 1) F n + ( px 2 + ( n − 1) x − p ) F n , ∂ nF n + 1 = ∂ ∂ p F n = − xF n , ∂ x (1 − x 2 ) ∂ 2 ∂ x 2 F n = (2 px 2 + 3 x − 2 p ) ∂ ∂ x F n + ( p 2 x 2 + 3 px − n 2 − p 2 + 1) F n output p 2 ∂ 2 ∂ p G n − ( n 2 + p 2 ) G n = 0 ∂ p 2 G n + p ∂ G n + 1 + ∂ ∂ p G n − n p G n = 0 difgerential finiteness For all i , j , k � 0 , there are a i jk and b i jk ∈ Q ( n , p , x ) such that ∂ i ∂ j ∂ p j F n + k ( p , x ) = a i jk ( n , p , x ) F n ( p , x ) + b i jk ( n , p , x ) ∂ ∂ x F n ( p , x ). ∂ x i

Challenges • not possible to compute it with good complexity • case where • simple certification of the output We give up: minimality We want to find all relations satified by the integrals • it is ofuen useless • the certificate is much bigger than the output complexity gets out of control): certificateless We want to avoiding computing the certificate (otherwise, the • the computational complexity of the algorithm. • the size of the output, bounds We want to understand and control: 7 � ∂ ∂ x (...)d x �= 0 D

Creative telescoping principle Find all relations 8 � � c j , k ( n , p ) ∂ = ∂ u ( n , p , x ) F n ( p , x ) + v ( n , p , x ) ∂ � ∂ p j F n + k ( p , x ) ∂ x F n ( p , x ) ∂ x ( j , k ) ∈ B c j , k ( n , p ) ∂ � ∂ p j G n + k ( p ) = 0. � ( j , k ) ∈ B � � equivalently Find all B ⊂ N 2 and ∈ Q ( n , p ) B such that c jk  ∂ �  ∂ x u = − a 200 v + c jk a 0 jk     ( i , j ) ∈ B ∂ �   ∂ x v = − u − b 200 v + c jk b 0 jk ,    ( i , j ) ∈ B has a rational solution u , v ∈ Q ( n , p , x ) .

Four generations of algorithms and    certificateless bounds minimality  Fasenmyer (1949); see also Takayama (1990), Galligo (1985), etc. Akin to polynomial elimination. 1st generation 9 � � problem Find all B ⊂ N 2 and ∈ Q ( n , p ) B s.t. ∃ u , v ∈ Q ( n , p , x ) c jk � � ∂ ∂ ∂ x u = − a 200 v + c jk a 0 jk ∂ x v = − u − b 200 v + c jk b 0 jk . ( i , j ) ∈ B ( i , j ) ∈ B elimination Only look for solutions with u , v ∈ Q ( n , p ) .

Four generations of algorithms and    certificateless bounds minimality  Chyzak (2000) ; see also Picard (1906), Zeilberger (1990) rational solutions Iteratively solve the difgerential system (Abramov 1989; Barkatou 2nd generation 10 � � problem Find all B ⊂ N 2 and ∈ Q ( n , p ) B s.t. ∃ u , v ∈ Q ( n , p , x ) c jk � � ∂ ∂ ∂ x u = − a 200 v + c jk a 0 jk ∂ x v = − u − b 200 v + c jk b 0 jk . ( i , j ) ∈ B ( i , j ) ∈ B 1999) with increasing support B (FGLM-like).

Four generations of algorithms and    certificateless bounds minimality  Lipshitz (1988), Apagodu, Zeilberger (2006), Koutschan (2010) 3rd generation 11 � � problem Find all B ⊂ N 2 and ∈ Q ( n , p ) B s.t. ∃ u , v ∈ Q ( n , p , x ) c jk � � ∂ ∂ ∂ x u = − a 200 v + c jk a 0 jk ∂ x v = − u − b 200 v + c jk b 0 jk . ( i , j ) ∈ B ( i , j ) ∈ B linear algebra Predict the denominator of solutions u , v ∈ Q ( n , p , x ) , reduce to linear algebra over Q ( n , p ) .

Four generations of algorithms 4th generation    certificateless bounds minimality Salvy (2016), Chen, Hoeij, Kauers, Koutschan (2018), Hoeven (2017) Chen, Huang, Kauers, Li (2015) and Huang (2016), Bostan, Dumont, Chen, Kauers, Koutschan (2016), Bostan, Chen, Chyzak, Li, Xin (2013),  Bostan, Chen, Chyzak, Li (2010b), Chen, Kauers, Singer (2012) and reduction of pole order Generalization of Hermite’s reduction 12 and � � problem Find all B ⊂ N 2 and ∈ Q ( n , p ) B s.t. ∃ u , v ∈ Q ( n , p , x ) c jk � � ∂ ∂ ∂ x u = − a 200 v + c jk a 0 jk ∂ x v = − u − b 200 v + c jk b 0 jk . ( i , j ) ∈ B ( i , j ) ∈ B

New algorithm

Obstructions to integrability and has a solution. 13 � � problem Find all B ⊂ N 2 and ∈ Q ( n , p ) B s.t. ∃ u , v ∈ Q ( n , p , x ) c jk � � ∂ ∂ ( ∗ ) ∂ x u = − a 200 v + c jk a 0 jk ∂ x v = − u − b 200 v + c jk b 0 jk . ( i , j ) ∈ B ( i , j ) ∈ B 2G/4G hybrid algorithm For all ( i , j ) ∈ N 2 , produce an obstruction λ jk such that  ∂ ∂ x u = − a 200 v + a 0 jk  λ jk = 0 ⇔ ∂  ∂ x v = − u − b 200 v + b 0 jk By linearity, ( ∗ ) has a solution if and only if � c jk λ jk = 0. ( j , k ) ∈ B

Lagrange identity . 14 a i ( x ) d i � difgerential operator L : f ( x ) �→ d x i f ( x ) i ( − 1) i d i adjoint operator L ∗ : f ( x ) �→ � � � a i ( x ) f ( x ) d x i i Lagrange’s identity uL ( f ) = L ∗ ( u ) f + d � � ... d x � � corollary 1 M ( f ) = M ∗ (1) f + d ... , for any difg. op. M . d x � � corollary 2 If L ( f ) = 0 then L ∗ ( u ) f = d ... for any u ( x ) d x corollary 3 If L is the minimal annihilating operator of f , then for any difgerential operator M , ∃ y ∈ K ( x ), M ∗ (1) = L ∗ ( y ) . M ( f ) “is a derivative” ⇔

Generalized Hermite reduction output prop. 2 prop. 1 Hermite reduction 15 Generalized Hermite red. input u ∈ K ( x ) and M ∈ K [ x ] 〈 d u ∈ K ( x ) d x 〉 v ∈ K ( x ) v ∈ K ( x ) u − v ∈ d d x K ( x ) u − v ∈ M ( K ( x )) u = d d x (...) ⇒ v = 0 u = M (...) ⇒ v = 0

algorithm Testing integrability with GHR 16 input γ ( x ) a “function” L , the minimal annihilating operator of γ f ∈ K ( x ) 〈 d d x 〉· γ , the function space generated by γ output ∃ g ∈ K ( x ) 〈 d d x 〉· γ , f = d d x γ write f = u ( x ) γ + d d x (...) ⊲ corollary 1 v ( x ) ← GHR ( v , L ∗ ) return v = 0 ⊲ corollary 3