Computing Closed-Form Solutions of Integrable Connections Thomas Cluzeau thomas.cluzeau@xlim.fr http://www.ensil.unilim.fr/~cluzeau/ Joint work with M. Barkatou, C. El Bacha and J.-A. Weil Algorithms Project’s Seminar INRIA Rocquencourt, March 26, 2012
Introducing example - G. Letac, W. Bryc (1) ⋄ Problem in probability theory: find all probability distributions µ on real symmetric matrices of order n such that if X and Y are independent with the same distribution µ , then X + Y = S and S − 1 X 2 S − 1 = Z are independent. ⋄ Under some restrictions, the problem can be reduced to ( Bryc-Letac’12 ): Find y ( x 1 , . . . , x n ) such that β 2 ( j − n ) ∂ y ∀ j ∈ { 1 , . . . , n } , + Tr ( P j Hess ( y )) = 0 , ∂ x j +1 where β is the Peirce constant ( β ∈ { 1 , 2 , 4 , 8 , − 2 } ), Hess the Hessian matrix and the P j ’s are given symmetric matrices.
Introducing example - G. Letac, W. Bryc (2) ⋄ Case n = 2: ∂ x 2 + ∂ 2 y ∂ 2 y − β ∂ y 1 − x 2 = 0 2 ∂ x 2 ∂ x 2 2 ∂ 2 y ∂ 2 y 2 ∂ x 1 ∂ x 2 + x 1 = 0 ∂ x 2 2 ⋄ Case n = 3: ∂ x 2 + ∂ 2 y ∂ 2 y ∂ 2 y − β ∂ y 1 − x 2 2 − 2 x 3 = 0 ∂ x 2 ∂ x 2 ∂ x 2 ∂ x 3 ∂ 2 y ∂ 2 y ∂ 2 y − β ∂ y ∂ x 3 + 2 ∂ x 1 ∂ x 2 + x 1 2 − x 3 = 0 2 ∂ x 2 ∂ x 2 3 ∂ 2 y ∂ 2 y ∂ 2 y ∂ 2 y 2 + 2 ∂ x 1 ∂ x 3 + 2 x 1 ∂ x 2 ∂ x 3 + x 2 = 0 ∂ x 2 ∂ x 2 3 ⋄ Problem: compute “solutions” of such linear systems of PDEs
Contributions ⋄ Remark : the latter systems are D-finite ( Chyzak-Salvy’98 ) ⋄ In this talk, we provide algorithms for computing: rational solutions hyperexponential solutions of such D -finite linear systems of PDEs. ⋄ Maple implementation available at http://www.ensil.unilim.fr/~cluzeau/PDS.html ⋄ Complexity analysis
Outline of the talk 1 D -finite linear systems of PDEs 2 Rational solutions 3 Hyperexponential solutions 4 Implementation 5 Conclusions
I D -finite linear systems of PDEs
Notations and a definition ⋄ C computable field of char. zero, C its algebraic closure ⋄ k = C ( x 1 , . . . , x m ) and K = C ( x 1 , . . . , x m ), ∂ i = ∂/∂ x i Definition U universal differential extension of k containing all solutions of linear systems of PDEs over k (existence, e.g., Kolchin’73 ). A linear system of PDEs is said to be D -finite if its solution space in U is of finite dimension over C . ⋄ Algorithms to test if a given system is D -finite exist ( Chyzak-Salvy ’98 - Gr¨ obner or Janet basis computations) Implementation: OreModules ( Chyzak-Quadrat-Robertz )
Integrable connections Definition Integrable connection over k of size n in m variables: ∆ 1 Y = 0 ∆ 1 := ∂ 1 I n − A 1 with . . . ∆ m Y = 0 ∆ m := ∂ m I n − A m with where A ′ i s ∈ M n ( k ) and the integrability conditions are satisfied: ∂ i ( A j ) − A i A j = ∂ j ( A i ) − A j A i , ∀ i , j ∈ { 1 , . . . , m } ⋄ Every D -finite linear system of PDEs can be written as an integrable connection ( Chyzak-Salvy ’98), implementation in OreModules ( Chyzak-Quadrat-Robertz )
Example: Bryc-Letac system for n = 2 − β � 2 ∂ 2 y + ∂ 2 1 y − x 2 ∂ 2 2 y = 0 2 ∂ 1 ∂ 2 y + x 1 ∂ 2 2 y = 0 ⋄ Integrable connection over Q ( β ) of size 4 in 2 variables: ∂ i Y − A i Y = 0 , i = 1 , 2 , with 0 0 1 0 0 1 0 0 − 1 0 0 0 2 x 1 0 0 0 1 A 1 = , A 2 = 1 − 1 0 2 β 0 x 2 0 0 0 2 x 1 ( − 3 − β ) x 1 6+2 β 0 0 0 0 0 0 x 12 − 4 x 2 x 12 − 4 x 2 ∂ 2 2 y ) T ⋄ Y = ( y ∂ 2 y ∂ 1 y
Existing works ⋄ Algorithmic studies of D -finite linear systems of PDEs: Chyzak’00 , Oaku-Takayama-Tsai’01 : rational solutions of holonomic systems Li-Schwarz-Tsarev’03 : factorization, hyperexp. solutions Barkatou-Cluzeau-Weil’05 : factorization in char. p Wu’05 , Li-Singer-Wu-Zheng’06 : Picard-Vessiot extensions, factorization, hyperexp. solutions over Laurent-Ore algebras ⋄ Strategy of our work: Consider integrable connections Proceed recursively: benefit from algorithms for ordinary differential (OD) systems
II Rational solutions
Rational solutions of OD systems (1) ⋄ C computable field of char. zero, C its algebraic closure, k = C ( x ) and K = C ( x ) s Y ′ = A Y , � q i ( x ) r i +1 A ∈ M n ( k ) , denom ( A ) = i =1 ⋄ Algorithm for computing rational solutions (for ex. Barkatou’99 ): Compute a universal denominator Q = � s i =1 q i ( x ) m i Compute polynomial solutions of Z ′ = ( A + ( Q ′ / Q ) I n ) Z
Complexity estimate Y ′ = A Y , denom ( A ) = � s i =1 q i ( x ) r i +1 A = ( a i , j ) i , j ∈ M n ( k ) , s � d := ( r i + 1) deg( q i ) i =1 � � r ∞ := max max i , j (1 + deg( num ( a i , j )) − deg( den ( a i , j ))) , 0 ⋄ Arithmetic (operations in C ) complexity estimate ( BCEW’12 ): Universal denominator: simple form at q i , integer roots of the indicial polynomial : O ( n 5 max i ( r i ) d ) Polynomial solutions: degree bound (simple form at ∞ ), coefficients: O ( n 5 r 2 ∞ + n 3 N 2 ) � rational solutions of Y ′ = A Y : O ( n 5 (max i ( r i ) d + r 2 ∞ ) + n 3 N 2 ) ⋄ Main tool: simple form (arithm. compl. in El Bacha’s PhD’11 )
Rational solutions of integrable connections (1) ⋄ k = C ( x 1 , . . . , x m ), K = C ( x 1 , . . . , x m ) ∆ 1 Y = 0 with ∆ 1 := ∂ 1 I n − A 1 , . . A i ∈ M n ( k ) . ∆ m Y = 0 ∆ m := ∂ m I n − A m , with ⋄ Notation : [ A 1 , . . . , A m ] Definition Rational solution: vector Y ∈ K n such that ∆ i ( Y ) = 0 , ∀ i . ⋄ Recursive process: Compute V := { Y ∈ K n ; ∆ 1 ( Y ) = 0 } Reduce the size ( m and n ) of the problem
Rational solutions of integrable connections (2) ⋄ K 1 := C ( x 2 , . . . , x m ), K = K 1 ( x 1 ), V := { Y ∈ K n ; ∆ 1 ( Y ) = 0 } ⋄ V is a K 1 -vector space stable under the action of each ∆ i ⋄ A basis can be computed using an algorithm for OD systems and viewing x 2 , . . . , x m as transcendental constants Lemma One can compute a non-singular matrix P ∈ M n ( K ) such that, ∀ i: � B 11 B 12 � B i := P − 1 ( A i P − ∂ i ( P )) = B 11 i i , ∈ M s ( K ) . B 22 i 0 i Moreover, B 11 1 = 0 and ∀ i = 2 , . . . , m, B 11 ∈ M s ( K 1 ) . i
Rational solutions of integrable connections (3) ⋄ v 1 , . . . , v s K 1 -basis of V , V = ( v 1 . . . v s ) ∈ M n × s ( K ) Theorem ( BCEW’12) Y = V Γ ∈ K n rat. sol. of [ A 1 , . . . , A m ] iff Γ ∈ K s 1 rat. sol. of ˜ ˜ ∆ 2 := ∂ 2 I s − B 11 ∆ 2 Γ = 0 2 , with . . No more x 1 ! . ˜ ˜ ∆ m := ∂ m I s − B 11 ∆ m Γ = 0 m , with � Recursive algorithm (with efficient method for computing B 11 i ’s) ⋄ Complexity: worst case estimate (op. in k ) � to be improved! ⋄ Denominators: q irred. factor of the denom. of a rat. sol. such that ∂ i 0 ( q ) � = 0 ⇒ q | denom ( A i 0 ) ( BCEW’12 )
III Hyperexponential solutions
Exponential solutions of ordinary differential systems (1) ⋄ C computable field of char. zero, C its algebraic closure, k = C ( x ) and K = C ( x ) s Y ′ = A Y , � q i ( x ) r i +1 A ∈ M n ( k ) , denom ( A ) = i =1 Definition f d x ) z , where f ∈ K and z ∈ K n . � Exponential solution: exp( ⋄ Algorithm for computing exponential solutions ( Pfluegel’01 ): Compute the non-ramified local exponential parts at each sing. For each combination, compute polynomial solutions ⋄ Bottlenecks: large number of comb. & computations in algebraic extensions of C of large degree
Exponential parts and complexity estimate Y ′ = A Y , x r +1 ( A 0 + A 1 x + A 2 x 2 + · · · ) , r ∈ N , A i ∈ M n ( C ) 1 A = Definition Non-ramified local exponential part at x = 0: polynomial ˜ f in 1 / x f = α p +1 x p +1 + α p x p + · · · + α 1 ˜ x , where 0 ≤ p ≤ r and α ′ i s ∈ C such that there exists a formal local � ˜ solution of the system of the form exp( f d x ) ˜ z , where ˜ z is a vector of formal power series in x . ⋄ Arithmetic cost ( BCEW’12 ): O ( n 5 r 3 min( n , r )) op. in an alg. ext. of C of degree ≤ n (super-reduction, Barkatou-Pfluegel’09 )
Complexity estimate Y ′ = A Y , denom ( A ) = � s i =1 q i ( x ) r i +1 A = ( a i , j ) i , j ∈ M n ( k ) , s � d := ( r i + 1) deg( q i ) i =1 � � r ∞ := max max i , j (1 + deg( num ( a i , j )) − deg( den ( a i , j ))) , 0 � Exponential solutions of Y ′ = A Y ( BCEW’12 ): O ( n 5 (max i ( r i ) 2 d � i min( n , r i ) + r 3 ∞ min( n , r ∞ ))) op. in an alg. ext. of C of degree ≤ n O ( n δ +3 N 2 ) op. in an alg. ext. of C of degree ≤ n δ δ ! ( δ : number of singularities, N : degree bound for all the computed polynomial solutions)
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