Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Quasi-Optimal Multiplication of Linear Differential Operators Alexandre Benoit 1 , Alin Bostan 2 and Joris van der Hoeven 3 1 ´ Education nationale (France) 2 INRIA (France) 3 CNRS, ´ Ecole Polytechnique (France) S´ eminaire Caramel November, 23th 2012
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion I Introduction 2 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Product of Linear Differential Operators L and K : linear differential operators with polynomial coefficients in K [ x ] � ∂ � . The product KL is given by the relation of composition ∀ f ∈ K [ x ] , KL · f = K · ( L · f ) . 3 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Product of Linear Differential Operators L and K : linear differential operators with polynomial coefficients in K [ x ] � ∂ � . The product KL is given by the relation of composition ∀ f ∈ K [ x ] , KL · f = K · ( L · f ) . The commutation of this product is given by the Leibniz rule: ∂x = x∂ + 1 . 3 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Complexity of the Product of Linear Differential Operators The product of differential operator is a complexity yardstick. The complexity of more involved, higher-level, operations on linear differential operators can be reduced to that of multiplication: LCLM, GCRD (van der Hoeven 2011) Hadamard product other closure properties for differential operators . . . 4 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Previous complexity results Product of operators in K [ x ] � ∂ � of orders < r with polynomial coefficients of degrees < d (i.e bidegrees less than ( d,r ) ): Naive algorithm: O ( d 2 r 2 min( d,r )) ops Takayama algorithm: ˜ O ( dr min( d,r )) ops Van der Hoeven algorithm (2002): O (( d + r ) ω ) ops using evaluations and interpolations. ω is a feasible exponent for matrix multiplication ( 2 � ω � 3 ) ˜ O indicates that polylogarithmic factors are neglected. 5 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Complexities for Unballanced Product van der Hoeven 2011 + bound given by Bostan et al (ISSAC 2012) Fast algorithms for LCLM or GCRD for operators of bidegrees less than ( r,r ) can be reduced to the multiplication of operators with polynomial coefficients of bidegrees ( r 2 ,r ) . 6 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Complexities for Unballanced Product van der Hoeven 2011 + bound given by Bostan et al (ISSAC 2012) Fast algorithms for LCLM or GCRD for operators of bidegrees less than ( r,r ) can be reduced to the multiplication of operators with polynomial coefficients of bidegrees ( r 2 ,r ) . Product of operators of bidegrees less than ( r 2 ,r ) Naive algorithm: O ( r 7 ) ops Takayama algorithm: ˜ O ( r 4 ) ops Van der Hoeven algorithm: O ( r 2 ω ) ops 6 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Contributions: New Algorithm for Unbalanced Product New algorithm 1 for the product of operators in K [ x ] � ∂ � of bidegree less than ( d,r ) in O ( dr min( d,r ) ω − 2 ) . ˜ 1 [BenoitBostanvanderHoeven, 2012] B. and Bostan and van der Hoeven. Quasi-Optimal Multiplication of Linear Differential Operators, FOCS 2012 . 7 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Contributions: New Algorithm for Unbalanced Product New algorithm 1 for the product of operators in K [ x ] � ∂ � of bidegree less than ( d,r ) in O ( dr min( d,r ) ω − 2 ) . ˜ In the important case d � r , this complexity reads ˜ O ( dr ω − 1 ) . In particular, if d = r 2 the complexity becomes O ( r ω +1 ) ( instead of ˜ ˜ O ( r 4 )) . 1 [BenoitBostanvanderHoeven, 2012] B. and Bostan and van der Hoeven. Quasi-Optimal Multiplication of Linear Differential Operators, FOCS 2012 . 7 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Outline of the proof Main ideas Use an evaluation-interpolation strategy on the point x i exp( αx ) Use fast algorithm for performing Hermite interpolation reflection ( d, r ) ← − − − − → ( r, d ) allows us to assume that r � d 8 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion II The van der Hoeven Algorithm 9 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Skew Product: a Linear Algebra Problem Recall : L is an operator of bidegree less than ( d,r ) L ( x ℓ ) ∈ K [ x ] d + ℓ − 1 . L ( x ℓ ) i is defined by : L ( x ℓ ) = L ( x ℓ ) 0 + L ( x ℓ ) 1 x + · · · + L ( x ℓ ) d + ℓ − 1 x d + ℓ − 1 10 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Skew Product: a Linear Algebra Problem Recall : L is an operator of bidegree less than ( d,r ) L ( x ℓ ) ∈ K [ x ] d + ℓ − 1 . L ( x ℓ ) i is defined by : L ( x ℓ ) = L ( x ℓ ) 0 + L ( x ℓ ) 1 x + · · · + L ( x ℓ ) d + ℓ − 1 x d + ℓ − 1 We define : L ( x k − 1 ) 0 L (1) 0 · · · . . Φ k + d,k ∈ K ( k + d ) × k . . = . . L L ( x k − 1 ) k + d − 1 L (1) k + d − 1 · · · we clearly have Φ k +2 d,k = Φ k +2 d,k + d Φ k + d,k , for all k � 0 . KL K L 10 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Study of Φ L We denote L = l 0 ( ∂ ) + xl 1 ( ∂ ) + · · · + x d − 1 l d − 1 ( ∂ ) ( l i ∈ K [ ∂ ] r ) l ( k − 1) l 0 (0) l ′ 0 (0) · · · (0) 0 l 1 (0) ( l ′ 1 + l 0 )(0) . . . . . . . . . . . . Φ k + d,k := l d − 1 (0) ( l ′ d − 1 + l d − 2 )(0) L 0 l d − 1 (0) . . ... . . . 0 . 0 · · · 0 l d − 1 (0) 11 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Study of Φ L We denote L = l 0 ( ∂ ) + xl 1 ( ∂ ) + · · · + x d − 1 l d − 1 ( ∂ ) ( l i ∈ K [ ∂ ] r ) l ( k − 1) l 0 (0) l ′ 0 (0) · · · (0) 0 l 1 (0) ( l ′ 1 + l 0 )(0) . . . . . . . . . . . . Φ k + d,k := l d − 1 (0) ( l ′ d − 1 + l d − 2 )(0) L 0 l d − 1 (0) . . ... . . . 0 . 0 · · · 0 l d − 1 (0) If L is an operator of bidegree ( r,d ) , we can compute L from Φ r + d,r L 11 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Algorithm Using Evaluations-Interpolation KL is an operator of bidegree less than (2 d, 2 r ) . Then the operator KL can be recovered from the matrix Φ 2 r +2 d, 2 r KL 12 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Algorithm Using Evaluations-Interpolation KL is an operator of bidegree less than (2 d, 2 r ) . Then the operator KL can be recovered from the matrix Φ 2 r +2 d, 2 r KL We deduce an algorithm to compute KL . (Evaluation) Computation of Φ 2 r +2 d, 2 r + d and of Φ 2 r + d, 2 r from K and L . 1 K L (Inner multiplication) Computation of the matrix product. 2 (Interpolation) Recovery of KL from Φ 2 r +2 d, 2 r . 3 KL 12 / 23
Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product ( r > d ) Reflexion for the Case when d > r Conclusion Algorithm Using Evaluations-Interpolation KL is an operator of bidegree less than (2 d, 2 r ) . Then the operator KL can be recovered from the matrix Φ 2 r +2 d, 2 r KL We deduce an algorithm to compute KL . (Evaluation) Computation of Φ 2 r +2 d, 2 r + d and of Φ 2 r + d, 2 r from K and L . 1 K L (Inner multiplication) Computation of the matrix product. O (( d + r ) ω ) ops 2 (Interpolation) Recovery of KL from Φ 2 r +2 d, 2 r . 3 KL 12 / 23
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