Quasi-Optimal Multiplication of Linear Diﬀerential Operators

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Introduction The van der Hoeven Algorithm New Algorithm for the Unbalanced Product (r > d) Reflexion for the Case whend > r Conclusion

Quasi-Optimal Multiplication of Linear Differential Operators

Alexandre Benoit¹, Alin Bostan² and Joris van der Hoeven³

1´Education nationale (France)

2INRIA (France)

3CNRS, ´Ecole Polytechnique (France)

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I Introduction

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IntroductionThe van der Hoeven Algorithm New Algorithm for the Unbalanced Product (r > d) Reflexion for the Case whend > r Conclusion

Product of Linear Differential Operators

LandK: linear differential operators with polynomial coefficients inK[x]h∂i. The productKLis 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.

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Product of Linear Differential Operators

LandK: linear differential operators with polynomial coefficients inK[x]h∂i. The productKLis 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.

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Complexity of the Product of Linear Differential Operators

The product of differential operator is acomplexity 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. . .

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Previous complexity results

Product of operators inK[x]h∂iof orders< rwith polynomial coefficients of degrees< d(i.e bidegrees less than(d,r)):

Naive algorithm: O(d²r²min(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 (26ω63) O˜ indicates that polylogarithmic factors are neglected.

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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(r²,r).

Product of operators of bidegrees less than(r²,r) Naive algorithm: O(r⁷)ops

Takayama algorithm: O(r˜ ⁴)ops Van der Hoeven algorithm: O(r^2ω)ops

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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(r²,r).

Product of operators of bidegrees less than(r²,r) Naive algorithm: O(r⁷)ops

Takayama algorithm: O(r˜ ⁴)ops Van der Hoeven algorithm: O(r^2ω)ops

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Contributions: New Algorithm for Unbalanced Product

New algorithm¹ for the product of operators inK[x]h∂iof bidegree less than(d,r) in

O(dr˜ min(d,r)^ω−2).

In the important cased>r, this complexity readsO(dr˜ ^ω−1). In particular, ifd=r² the complexity becomes

O(r˜ ^ω+1)(instead ofO(r˜ ⁴)).

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Contributions: New Algorithm for Unbalanced Product

New algorithm¹ for the product of operators inK[x]h∂iof bidegree less than(d,r) in

O(dr˜ min(d,r)^ω−2).

In the important cased>r, this complexity readsO(dr˜ ^ω−1).

In particular, ifd=r² the complexity becomes

O(r˜ ^ω+1)(instead ofO(r˜ ⁴)).

1

[BenoitBostanvanderHoeven, 2012] B. and Bostan and van der Hoeven.

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Outline of the proof

Main ideas

Use an evaluation-interpolation strategy on the pointxⁱexp(αx) Use fast algorithm for performing Hermite interpolation

(d, r)←−−−−→^reflection (r, d)allows us to assume that r>d

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II The van der Hoeven Algorithm

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Skew Product: a Linear Algebra Problem

Recall : Lis an operator of bidegree less than (d,r) L(x^`)∈K[x]d+`−1.

L(x^`)i is defined by :

L(x^`) =L(x^`)₀+L(x^`)₁x+· · ·+L(x^`)_d+`−1x^d+`−1

We define :

Φ^k+d,k_L =







L(1)₀ · · · L(x^k−1)₀

... ...

L(1)_k+d−1 · · · L(x^k−1)_k+d−1





∈K^(k+d)×k

we clearly have

Φ^k+2d,k_KL = Φ^k+2d,k+d_K Φ^k+d,k_L , for allk>0.

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Skew Product: a Linear Algebra Problem

Recall : Lis an operator of bidegree less than (d,r) L(x^`)∈K[x]d+`−1.

L(x^`)i is defined by :

L(x^`) =L(x^`)₀+L(x^`)₁x+· · ·+L(x^`)_d+`−1x^d+`−1 We define :

Φ^k+d,k_L =







L(1)₀ · · · L(x^k−1)₀

... ...

L(1)_k+d−1 · · · L(x^k−1)_k+d−1





∈K^(k+d)×k

we clearly have

Φ^k+2d,k_KL = Φ^k+2d,k+d_K Φ^k+d,k_L , for allk>0.

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Study of Φ

_L

We denoteL=l0(∂) +xl1(∂) +· · ·+x^d−1l_d−1(∂)(li∈K[∂]r)

Φ^k+d,k_L :=







l0(0) l⁰₀(0) · · · l^(k−1)₀ (0) l1(0) (l⁰₁+l0)(0)

... ... ... ...

l_d−1(0) (l⁰_d−1+l_d−2)(0) 0 l_d−1(0)

... 0 . .. ...

0 · · · 0 ld−1(0)







IfLis an operator of bidegree(r,d), we can computeLfromΦ^r+d,r_L

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Study of Φ

_L

We denoteL=l0(∂) +xl1(∂) +· · ·+x^d−1l_d−1(∂)(li∈K[∂]r)

Φ^k+d,k_L :=







l0(0) l⁰₀(0) · · · l^(k−1)₀ (0) l1(0) (l⁰₁+l0)(0)

... ... ... ...

l_d−1(0) (l⁰_d−1+l_d−2)(0) 0 l_d−1(0)

... 0 . .. ...

0 · · · 0 ld−1(0)







IfLis an operator of bidegree(r,d), we can computeLfromΦ^r+d,r_L

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Algorithm Using Evaluations-Interpolation

KLis an operator of bidegree less than(2d,2r).

Then the operatorKLcan be recovered from the matrix Φ^2r+2d,2r_KL

We deduce an algorithm to computeKL.

1 (Evaluation) Computation ofΦ^2r+2d,2r+d_K and ofΦ^2r+d,2r_L fromK andL.

2 (Inner multiplication) Computation of the matrix product.

O((d+r)^ω)ops

3 (Interpolation) Recovery ofKLfromΦ^2r+2d,2r_KL .

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Algorithm Using Evaluations-Interpolation

2 (Inner multiplication) Computation of the matrix product.

O((d+r) )ops

3 (Interpolation) Recovery ofKLfromΦ^2r+2d,2r_KL .

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Algorithm Using Evaluations-Interpolation

2 (Inner multiplication) Computation of the matrix product. O((d+r)^ω)ops

3 (Interpolation) Recovery ofKLfromΦ^2r+2d,2r.

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Fast Evaluation and Interpolation

A remark from Bostan, Chyzak and Le Roux (ISSAC 2008)







l0 l1 · · · ld

l⁰₀ l0+l⁰₁ · · · l⁰_d+ld−1 ld

. .. ...

l^(`−1)₀ · · · l_d







=







1 0 0

1 1 0 0

1 2 1 0 0

1 3 3 1 0 0

... . ..













0 · · · 0 l₀ l₁ · · · l_d

... 0 l⁰₀ l₁⁰ · · · l⁰_d 0 0 . ..

. ..

0 ... l₀^(`−1) l₁^(`−1) · · · l_d^(`−1) 0 ... 0







Computation ofΦ^r+d,r_L fromLin O((r+d)^ω)(inO(rd)˜ using structured matrices)

Computation ofLfromφ_r+d,r(L)inO((r+d)^ω)(inO(rd)˜ using structured matrices)

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Fast Evaluation and Interpolation







l0 l1 · · · ld

l⁰₀ l0+l⁰₁ · · · l⁰_d+ld−1 ld

. .. ...

l^(`−1)₀ · · · l_d







=







1 0 0

1 1 0 0

1 2 1 0 0

1 3 3 1 0 0

... . ..













0 · · · 0 l₀ l₁ · · · l_d

... 0 l⁰₀ l₁⁰ · · · l⁰_d 0 0 . ..

. ..

0 ... l₀^(`−1) l₁^(`−1) · · · l_d^(`−1) 0 ... 0







Computation ofLfromφ_r+d,r(L)inO((r+d)^ω)(inO(rd)˜ using structured matrices)

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Fast Evaluation and Interpolation







l0 l1 · · · ld

l⁰₀ l0+l⁰₁ · · · l⁰_d+ld−1 ld

. .. ...

l^(`−1)₀ · · · l_d







=







1 0 0

1 1 0 0

1 2 1 0 0

1 3 3 1 0 0

... . ..













0 · · · 0 l₀ l₁ · · · l_d

... 0 l⁰₀ l₁⁰ · · · l⁰_d 0 0 . ..

. ..

0 ... l₀^(`−1) l₁^(`−1) · · · l_d^(`−1) 0 ... 0







Applications:

Computation ofΦ^r+d,r_L fromLin O((r+d)^ω)(inO(rd)˜ using structured matrices)

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Complexity of van der Hoeven Algorithm

Easy bound: IfLandKare of bidegrees less than (d,r),KLis of bidegree less than(2d,2r).

Evaluation ofΦ^2r+d,2r_L andΦ^2r+2d,2r_K

O(rd)˜ ops

Matrix multiplicationΦ^2r+2d,2r_KL = Φ^2d_K · Φ^2d+r_L

O((d+r)^ω)ops

Interpolation. FromΦ^2d+2r,2r_KL to the coefficients ofKL

O(dr)˜ ops Complexity of van der Hoeven algorithm ifd=r: O(r˜ ^ω)

©

Complexity of van der Hoeven algorihtm ifd=r²: O(r˜ ^2ω)

§

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Complexity of van der Hoeven Algorithm

Evaluation ofΦ^2r+d,2r_L andΦ^2r+2d,2r_K O(rd)˜ ops

Matrix multiplicationΦ^2r+2d,2r_KL = Φ^2d_K · Φ^2d+r_L O((d+r)^ω)ops Interpolation. FromΦ^2d+2r,2r_KL to the coefficients ofKLO(dr)˜ ops

Complexity of van der Hoeven algorithm ifd=r: O(r˜ ^ω)

©

§

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Complexity of van der Hoeven Algorithm

Matrix multiplicationΦ^2r+2d,2r_KL = Φ^2d_K · Φ^2d+r_L O((d+r)^ω)ops Interpolation. FromΦ^2d+2r,2r_KL to the coefficients ofKLO(dr)˜ ops Complexity of van der Hoeven algorithm ifd=r: O(r˜ ^ω)

©

§

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Complexity of van der Hoeven Algorithm

Matrix multiplicationΦ^2r+2d,2r_KL = Φ^2d_K · Φ^2d+r_L O((d+r)^ω)ops Interpolation. FromΦ^2d+2r,2r_KL to the coefficients ofKLO(dr)˜ ops Complexity of van der Hoeven algorithm ifd=r: O(r˜ ^ω)

©

§

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III New Algorithm for the Unbalanced Product

(r > d)

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Operate on Exponential Polynomials

Lalso operates onK[x]e^αxfor everyα∈K

L = X

i

L_i(x)∂ⁱ

we have:

L(P e^αx) = L_nα(P) L_nα = X

i

Li(x)(∂+α)ⁱ

Φ^k+d,k_L

nα :=







l0(α) l⁰₀(α) · · · l^(k−1)₀ (α) l1(α) (l⁰₁+l0)(α)

... ... ... ...

l_d−1(α) (l⁰_d−1+l_d−2)(α) 0 l_d−1(α)

... 0 . .. ...

0 · · · 0 ld−1(α)







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Operate on Exponential Polynomials

Lalso operates onK[x]e^αxfor everyα∈K More specifically, writing

L = X

i

L_i(x)∂ⁱ

we have:

L(P e^αx) = L_nα(P) L_n_α = X

i

L_i(x)(∂+α)ⁱ

Φ^k+d,k_L

nα :=







l0(α) l⁰₀(α) · · · l^(k−1)₀ (α) l1(α) (l⁰₁+l0)(α)

... ... ... ...

l_d−1(α) (l⁰_d−1+l_d−2)(α) 0 l_d−1(α)

... 0 . .. ...

0 · · · 0 ld−1(α)







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Operate on Exponential Polynomials

Lalso operates onK[x]e^αxfor everyα∈K More specifically, writing

L = X

i

L_i(x)∂ⁱ

we have:

L(P e^αx) = L_nα(P) L_n_α = X

i

L_i(x)(∂+α)ⁱ

Φ^k+d,k_L

nα :=





l0(α) l⁰₀(α) · · · l^(k−1)₀ (α) l1(α) (l⁰₁+l0)(α)

... ... ... ...

l_d−1(α) (l⁰_d−1+l_d−2)(α) 0 l_d−1(α)

... 0 . .. ...





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Hermite Evaluations and Interpolations

Fast algorithm for evaluations and interpolations inO(n)˜ ops (Chin (1976)).

Application:

Evaluations and interpolation ofΦ^2d,d_L

nαi fori∈[0..r/d−1]in O(rd)˜ ops

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Hermite Evaluations and Interpolations

Evaluations and interpolation ofΦ^2d,d_L

nαi fori∈[0..r/d−1]in O(rd)˜ ops

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Hermite Evaluations and Interpolations

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Product using Multipoint Evaluations and Interpolation

We suppose r > d Idea

Forp=dr/de, choose distinctα0, . . . , α_p−1, and let Loperates on Vk = K[x]ke^α⁰^x⊕ · · · ⊕K[x]ke^α^p−1^x

We replace one multiplication of big matrices by several multiplications of smaller matrices

Evaluations ofΦ_L

nαi andΦ_K

nαi, forifrom0top−1(O(dr)) Matrix multiplications: For alli,Φ^4d,2d_KL

nαi = Φ^4d,3d_K

nαi · Φ^3d,2d_L

nαi. O(rd^ω−1) r/dmultiplications of matrices of size d×d(instead ofr×d)

Interpolations. FromΦ^4d,2d_KL

nαi toKLO(dr)˜

Conplexity of algorithm whenr > d: O(rd˜ ^ω−1)arithmetic operations

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Product using Multipoint Evaluations and Interpolation

Evaluations ofΦ^3d,2d_L

nαi andΦ^4d,3d_K

nαi, forifrom0top−1(O(dr))˜

Matrix multiplications: For alli,Φ^4d,2d_KL

nαi = Φ^4d,3d_K

nαi · Φ^3d,2d_L

nαi. O(rd^ω−1) r/dmultiplications of matrices of size d×d(instead ofr×d)

nαi toKLO(dr)˜

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Product using Multipoint Evaluations and Interpolation

nαi

andΦ^4d,3d_K

nαi

, forifrom0top−1(O(dr))˜ Matrix multiplications: For alli,Φ^4d,2d_KL

nαi = Φ^4d,3d_K

nαi· Φ^3d,2d_L

nαi.

O(rd )

r/dmultiplications of matrices of size d×d(instead ofr×d) Interpolations. FromΦ^4d,2d_KL

nαi toKLO(dr)˜

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Product using Multipoint Evaluations and Interpolation

nαi andΦ^4d,3d_K

nαi, forifrom0top−1(O(dr))˜ Matrix multiplications: For alli,Φ^4d,2d_KL

nαi = Φ^4d,3d_K

nαi· Φ^3d,2d_L

nαi. O(rd^ω−1) r/dmultiplications of matrices of sized×d(instead ofr×d)

nαi toKLO(dr)˜

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Product using Multipoint Evaluations and Interpolation

nαi andΦ^4d,3d_K

nαi = Φ^4d,3d_K

nαi· Φ^3d,2d_L

nαi toKLO(dr)˜

Conplexity of algorithm whenr > d: O(rd )arithmetic operations

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Product using Multipoint Evaluations and Interpolation

nαi andΦ^4d,3d_K

nαi = Φ^4d,3d_K

nαi· Φ^3d,2d_L

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IV Reflexion for the Case when d > r

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Computing the Reflexion

The reflexionϕis the morphism fromK[x]h∂ito itself such that:

ϕ(∂) =x, ϕ(x) =−∂.

Given

L = X

i,j

pi,j∂^jxⁱ, compute qi,j withL = X

i,j

qi,jxⁱ∂^j.

We have :

ϕ(L) = X

i,j

(−1)^jpi,jx^j∂ⁱ.

Theorem

GivenL∈K[x, ∂]d,r, we may computeϕ(L)in timeO˜((d+r) min (d, r)). Proof :

Show that

i!q_i,j=X

k>0

j+k k

(i+k)!p_i+k,j+k

Reduce to the computation ofO(d˜ +r)Taylor shifts of lengthmin(d, r).

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Computing the Reflexion

ϕ(∂) =x, ϕ(x) =−∂.

Given

L = X

i,j

p_i,j∂^jxⁱ, compute q_i,j withL = X

i,j

q_i,jxⁱ∂^j.

We have :

ϕ(L) = X

i,j

(−1)^jp_i,jx^j∂ⁱ.

Theorem

GivenL∈K[x, ∂]d,r, we may computeϕ(L)in timeO˜((d+r) min (d, r)). Proof :

Show that

i!q_i,j=X

k>0

j+k k

(i+k)!p_i+k,j+k

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Computing the Reflexion

ϕ(∂) =x, ϕ(x) =−∂.

Given

L = X

i,j

q_i,jxⁱ∂^j.

We have :

ϕ(L) = X

i,j

Theorem

GivenL∈K[x, ∂]d,r, we may computeϕ(L)in timeO˜((d+r) min (d, r)).

Proof : Show that

i!q_i,j=X

k>0

j+k k

(i+k)!p_i+k,j+k

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Computing the Reflexion

ϕ(∂) =x, ϕ(x) =−∂.

Given

L = X

i,j

q_i,jxⁱ∂^j.

We have :

ϕ(L) = X

i,j

Theorem

GivenL∈K[x, ∂]d,r, we may computeϕ(L)in timeO˜((d+r) min (d, r)).

Proof : Show that

i!q_i,j=Xj+k k

(i+k)!p_i+k,j+k

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New Algorithm for the Case when d > r

Idea : ifLis an operator of bidegree(d,r), thenϕ(L)is an operator of bidegree(r,d).

Application: New algorithm

compute the canonical forms (xat left and ∂at right) ofϕ(L)andϕ(K), new algorithm inO(dr)˜

compute the productM =ϕ(L)ϕ(K)of operatorsϕ(L)andϕ(K) inO(r^ω−1d)using the previous algorithm

return the (canonical form of the) operatorKL=ϕ⁻¹(M) new algorithm inO(dr)˜

Complexity of the product inO(r˜ ^ω−1d)arithmetic operations whend > r

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New Algorithm for the Case when d > r

Complexity of the product inO(r d)arithmetic operations whend > r

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New Algorithm for the Case when d > r

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V Conclusion

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Contribution: better algorithm for the product of differential operator:

Previous: O((d+r)^ω)arithmetic operations

New algorithm: O(rdmin(r,d)^ω−2)arithmetic operations

The same algorithm works also for product of ,θ operators, recurrence operators or q-difference operators.

Perspective: Use of this fast product to improve algorithms to compute: differential operator canceling Hadamard product of series

differential operator canceling product of series

differential operator obtained by substitution with an algebraic function

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differential operator canceling Hadamard product of series differential operator canceling product of series

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Perspective: Use of this fast product to improve algorithms to compute:

differential operator canceling Hadamard product of series differential operator canceling product of series