Computing Closed-Form Solutions of Integrable Connections Thomas - - PowerPoint PPT Presentation

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

• n 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(x1, . . . , xn) such that ∀ j ∈ {1, . . . , n}, β 2 (j − n) ∂y ∂xj+1 + Tr(Pj Hess(y)) = 0, where β is the Peirce constant (β ∈ {1, 2, 4, 8, −2}), Hess the Hessian matrix and the Pj’s are given symmetric matrices.

Introducing example - G. Letac, W. Bryc (2)

⋄ Case n = 2:    −β

2 ∂y ∂x2 + ∂2y ∂x2

1 − x2

∂2y ∂x2

2

= 2

∂2y ∂x1 ∂x2 + x1 ∂2y ∂x2

2

= ⋄ Case n = 3:        − β ∂y

∂x2 + ∂2y ∂x2

1 − x2

∂2y ∂x2

2 − 2 x3

∂2y ∂x2 ∂x3

= −β

2 ∂y ∂x3 + 2 ∂2y ∂x1 ∂x2 + x1 ∂2y ∂x2

2 − x3

∂2y ∂x2

3

=

∂2y ∂x2

2 + 2

∂2y ∂x1 ∂x3 + 2 x1 ∂2y ∂x2 ∂x3 + x2 ∂2y ∂x2

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

• f 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(x1, . . . , xm) and K = C(x1, . . . , xm), ∂i = ∂/∂xi 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¨

• bner 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 with ∆1 := ∂1 In − A1 . . . ∆m Y = 0 with ∆m := ∂m In − Am where A′

is ∈ Mn(k) and the integrability conditions are satisfied:

∂i(Aj) − Ai Aj = ∂j(Ai) − Aj Ai, ∀ 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 ∂2y + ∂2 1y − x2 ∂2 2y

= 2 ∂1 ∂2y + x1 ∂2

2y

= ⋄ Integrable connection over Q(β) of size 4 in 2 variables: ∂i Y − Ai Y = 0, i = 1, 2, with A1 =        1 −1

2 x1 1 2 β

x2

(−3−β)x1 x12−4 x2

       , A2 =        1 1 −1

2 x1 6+2 β x12−4 x2

       ⋄ Y = (y ∂2y ∂1y ∂2

2y)T

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) Y ′ = A Y , A ∈ Mn(k), denom(A) =

s

• i=1

qi(x)ri+1 ⋄ Algorithm for computing rational solutions (for ex. Barkatou’99): Compute a universal denominator Q = s

i=1 qi(x)mi

Compute polynomial solutions of Z ′ = (A + (Q′/Q) In) Z

Complexity estimate

Y ′ = A Y , A = (ai,j)i,j ∈ Mn(k), denom(A) = s

i=1 qi(x)ri+1

d :=

s

• i=1

(ri + 1) deg(qi) r∞ := max

• max

i,j (1 + deg(num(ai,j)) − deg(den(ai,j))) , 0

• ⋄ Arithmetic (operations in C) complexity estimate (BCEW’12):

Universal denominator: simple form at qi, integer roots of the indicial polynomial: O(n5 maxi(ri) d) Polynomial solutions: degree bound (simple form at ∞), coefficients: O(n5 r2

∞ + n3 N2)

rational solutions of Y ′ = A Y : O(n5 (maxi(ri) d + r2

∞) + n3 N2)

⋄ Main tool: simple form (arithm. compl. in El Bacha’s PhD’11)

Rational solutions of integrable connections (1)

⋄ k = C(x1, . . . , xm), K = C(x1, . . . , xm)      ∆1 Y = 0 with ∆1 := ∂1 In − A1, . . . ∆m Y = 0 with ∆m := ∂m In − Am, Ai ∈ Mn(k) ⋄ Notation: [A1, . . . , Am] 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)

⋄ K1 := C(x2, . . . , xm), K = K1(x1), V := {Y ∈ K n ; ∆1(Y ) = 0} ⋄ V is a K1-vector space stable under the action of each ∆i ⋄ A basis can be computed using an algorithm for OD systems and viewing x2, . . . , xm as transcendental constants Lemma One can compute a non-singular matrix P ∈ Mn(K) such that, ∀ i: Bi := P−1 (Ai P − ∂i(P)) = B11

i

B12

i

B22

i

• ,

B11

i

∈ Ms(K). Moreover, B11

1 = 0 and ∀ i = 2, . . . , m, B11 i

∈ Ms(K1).

Rational solutions of integrable connections (3)

⋄ v1, . . . , vs K1-basis of V, V = (v1 . . . vs) ∈ Mn×s(K) Theorem (BCEW’12) Y = V Γ ∈ K n rat. sol. of [A1, . . . , Am] iff Γ ∈ K s

1 rat. sol. of

     ˜ ∆2 Γ = 0 with ˜ ∆2 := ∂2 Is − B11

2 ,

. . . ˜ ∆m Γ = 0 with ˜ ∆m := ∂m Is − B11

m ,

No more x1! Recursive algorithm (with efficient method for computing B11

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 ∂i0(q) = 0 ⇒ q | denom(Ai0) (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) Y ′ = A Y , A ∈ Mn(k), denom(A) =

s

• i=1

qi(x)ri+1 Definition Exponential solution: exp(

• f dx) z, where f ∈ K and z ∈ K n.

⋄ 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 , A =

1 xr+1 (A0+A1 x+A2 x2+· · · ), r ∈ N, Ai ∈ Mn(C)

Definition Non-ramified local exponential part at x = 0: polynomial ˜ f in 1/x ˜ f = αp+1 xp+1 + αp xp + · · · + α1 x , where 0 ≤ p ≤ r and α′

is ∈ C such that there exists a formal local

solution of the system of the form exp( ˜ f dx) ˜ z, where ˜ z is a vector of formal power series in x. ⋄ Arithmetic cost (BCEW’12): O(n5 r3 min(n, r)) op. in an alg.

• ext. of C of degree ≤ n (super-reduction, Barkatou-Pfluegel’09)

Complexity estimate

Y ′ = A Y , A = (ai,j)i,j ∈ Mn(k), denom(A) = s

i=1 qi(x)ri+1

d :=

s

• i=1

(ri + 1) deg(qi) r∞ := max

• max

i,j (1 + deg(num(ai,j)) − deg(den(ai,j))) , 0

• Exponential solutions of Y ′ = A Y (BCEW’12):

O(n5 (maxi(ri)2 d

i min(n, ri) + r3 ∞ min(n, r∞))) op. in an

• alg. ext. of C of degree ≤ n

O(nδ+3 N2) op. in an alg. ext. of C of degree ≤ nδ δ! (δ: number of singularities, N: degree bound for all the computed polynomial solutions)

Hyperexponential solutions of integrable connections (1)

     ∆1 Y = 0 with ∆1 := ∂1 In − A1, . . . ∆m Y = 0 with ∆m := ∂m In − Am, Ai ∈ Mn(C(x1, . . . , xm)) ⋄ K = C(x1, . . . , xm) Definition L differential extension of K having the same field of constants. ⊙ u = 0 ∈ L hyperexponential over K: ∀ i, fi := ∂i(u)/u ∈ K. ⊙ hyperexponential solution: solution u z with u hyperexponential

• ver K and z ∈ K n .

⋄ u hyperexponential over K ⇒ ∂j(fi) = ∂i(fj), ∀ i, j ⋄ u z hyperexp. sol. of [A1, . . . , Am] ⇒ z rat. sol. of [A1 − f1 In, . . . , Am − fm In]

Hyperexponential solutions of integrable connections (2)

⋄ Recursive algorithm as for rational solutions

• Exp. sol. of Y ′ = A1 Y computed with algorithm for OD
• systems. Let u z be such a solution

fi := ∂i(u)/u ∈ K and ∆i,u := ∂i − (Ai − fi In) w1, . . . , ws basis of Wu = {w ∈ K n ; ∆1,u(w) = 0}, complete it into a basis of K n matrix P = (Wu ˜ W ) Theorem (BCEW’12) Y = u Wu Γu hyperexp. sol. of [A1, . . . , Am] iff Γu hyperexp. sol.

• f [B11

2 , . . . , B11 m ] where Bi = P−1 ((Ai − fi In) P − ∂i(P)) and

B11

i

∈ Ms(K1) denotes the first s × s submatrix of Bi. ⋄ Complexity: worst case estimate to be improved ⋄ Discard local exp. parts involving non-rat. functions of xj’s, j = 1

IV Implementation

Maple package IntegrableConnections

⋄ Algorithms are implemented in a Maple package called IntegrableConnections Available with some examples at http://www.ensil.unilim.fr/~cluzeau/PDS.html Main procedures: RationalSolutions (& Eigenring), HyperexponentialSolutions Some adaptations of Isolde code (Barkatou-Pfluegel) Demo.

V Conclusions

Contributions and Perspectives

⋄ Summary of the contributions: Complexity estimates for computing rat. and exp. solutions of OD systems (in the literature of OD systems, Grigoriev’90) Algorithms for computing rational and hyperexponential solutions of integrable connections Implementation available (IntegrableConnections) ⋄ Perspectives: Precise complexity analysis of algorithms for integrable connections Algorithms for other types of solutions and factorization