Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Quasi-Optimal Multiplication of Linear Differential Operators
Alexandre Benoit1, Alin Bostan2 and Joris van der Hoeven3
1CNRS, INRIA, UPMC (France)
2INRIA (France)
3CNRS, ´Ecole Polytechnique (France)
ACA 2012
June, 28th 2012
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
I Introduction
3 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product of Linear Differential Operators
P andQ: linear differential operators with polynomial coefficients inK[x]h∂i. The productP Qis given by the relation of composition
∀f ∈K[x], P Q·f =P·(Q·f).
The commutation of this product is given by the Leibniz rule:
∂x=x∂+ 1.
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product of Linear Differential Operators
P andQ: linear differential operators with polynomial coefficients inK[x]h∂i. The productP Qis given by the relation of composition
∀f ∈K[x], P Q·f =P·(Q·f).
The commutation of this product is given by the Leibniz rule:
∂x=x∂+ 1.
4 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
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. . .
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Previous complexity results
Product of operators inK[x]h∂iof orders6rwith polynomial coefficients of degrees6d(i.e bidegrees (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.
6 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexities for Unballanced Product
van der Hoeven 2011 + bound see next talk
Fast algorithms for LCLM or GCRD for operators of bidegrees(r,r)can be reduced to the multiplication of operators with polynomial coefficients of bidegrees(r2,r).
Product of operators of bidegrees(r2,r) Naive algorithm: O(r7)ops Takayama algorithm: O(r˜ 4)ops Van der Hoeven algorithm: O(r2ω)ops
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexities for Unballanced Product
van der Hoeven 2011 + bound see next talk
Fast algorithms for LCLM or GCRD for operators of bidegrees(r,r)can be reduced to the multiplication of operators with polynomial coefficients of bidegrees(r2,r).
Product of operators of bidegrees(r2,r) Naive algorithm: O(r7)ops Takayama algorithm: O(r˜ 4)ops Van der Hoeven algorithm: O(r2ω)ops
7 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Contributions: New Algorithm for Unbalanced Product
New algorithm1 for the product of operators inK[x]h∂iof bidegree(d,r)in O(dr˜ min(d,r)ω−2).
Ifd=r2the complexity becomes
O(r˜ ω+1)(instead ofO(r˜ 4)).
Idea: Modify van der Hoeven algorithm using
Instead of evaluations of operators onxi, evaluations onxiexp(αx) Use multipoint evaluations and interpolations
Use fast algorithm for performing Hermite interpolation
1
[BenoitBostanvanderHoeven, 2012] B. and Bostan and van der Hoeven.
Quasi-Optimal Multiplication of Linear Differential Operators, to appear inFOCS 2012.
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Contributions: New Algorithm for Unbalanced Product
New algorithm1 for the product of operators inK[x]h∂iof bidegree(d,r)in O(dr˜ min(d,r)ω−2).
Ifd=r2the complexity becomes
O(r˜ ω+1)(instead ofO(r˜ 4)).
Idea: Modify van der Hoeven algorithm using
Instead of evaluations of operators onxi, evaluations onxiexp(αx) Use multipoint evaluations and interpolations
Use fast algorithm for performing Hermite interpolation
7 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Contributions: New Algorithm for Unbalanced Product
New algorithm1 for the product of operators inK[x]h∂iof bidegree(d,r)in O(dr˜ min(d,r)ω−2).
Ifd=r2the complexity becomes
O(r˜ ω+1)(instead ofO(r˜ 4)).
Idea: Modify van der Hoeven algorithm using
Instead of evaluations of operators onxi, evaluations onxiexp(αx) Use multipoint evaluations and interpolations
Use fast algorithm for performing Hermite interpolation
1
[BenoitBostanvanderHoeven, 2012] B. and Bostan and van der Hoeven.
Quasi-Optimal Multiplication of Linear Differential Operators, to appear inFOCS 2012.
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
II New Vision of van der Hoeven Algorithm
9 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Skew Product: a Linear Algebra Problem
QP =
r
X
i=0
qi(x)∂iP
We define
S`(P) :=
P
∂P ...
∂`−1P
,
then forQof orderr, we have,
S1(QP) =S1(Q)Sr+1(P). More generally, for any`>0, we have:
S`(QP) =S`(Q)Sr+`(P).
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Skew Product: a Linear Algebra Problem
QP =
r
X
i=0
qi(x)∂iP
We define
S`(P) :=
P
∂P ...
∂`−1P
,
then forQof orderr, we have,
S1(QP) =S1(Q)Sr+1(P).
More generally, for any`>0, we have:
S`(QP) =S`(Q)Sr+`(P).
9 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Skew Product: a Linear Algebra Problem
QP =
r
X
i=0
qi(x)∂iP
We define
S`(P) :=
P
∂P ...
∂`−1P
,
then forQof orderr, we have,
S1(QP) =S1(Q)Sr+1(P).
More generally, for any`>0, we have:
S`(QP) =S`(Q)Sr+`(P).
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Evaluations and Interpolation of Operator
S`(P) :=
P
∂P ...
∂`−1P
=
p0 p1 · · · pr 0 · · · 0
p00 p01+p0 · · · p0r+pr−1 pr 0 ...
... ... . .. 0
p(`−1)0 · · · pr
IfP is an operator of bidegree(d,r), we can computeP fromSd(P)(0)
10 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Evaluations and Interpolation of Operator
S`(P) :=
P
∂P ...
∂`−1P
=
p0 p1 · · · pr 0 · · · 0
p00 p01+p0 · · · p0r+pr−1 pr 0 ...
... ... . .. 0
p(`−1)0 · · · pr
IfP is an operator of bidegree(d,r), we can computeP fromSd(P)(0)
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Fast Computation of S
`(P )(0)
A remark from Bostan, Chyzak and Le Roux (ISSAC 2008)
1 0 0
1 1 0 0
1 2 1 0 0
1 3 3 1 0 0
... . ..
0 · · · 0 p0 p1 · · · pr
... 0 p00 p01 · · · p0r 0 0 . ..
. ..
0 ... p(`−1)0 p(`−1)1 · · · p(`−1)r 0 ... 0
=
p0 p1 · · · pr
p00 p0+p01 · · · p0r+pr−1 pr
. .. ...
p(`−1)0 · · · pr
.
Applications:
Computation ofP fromSd(P)(0)inO((r+d)ω)(inO(rd)˜ using structured matrices)
Computation ofSd(P)(0)fromP inO((r+d)ω)(inO(rd)˜ using structured matrices)
11 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Fast Computation of S
`(P )(0)
A remark from Bostan, Chyzak and Le Roux (ISSAC 2008)
1 0 0
1 1 0 0
1 2 1 0 0
1 3 3 1 0 0
... . ..
0 · · · 0 p0 p1 · · · pr
... 0 p00 p01 · · · p0r 0 0 . ..
. ..
0 ... p(`−1)0 p(`−1)1 · · · p(`−1)r 0 ... 0
=
p0 p1 · · · pr
p00 p0+p01 · · · p0r+pr−1 pr
. .. ...
p(`−1)0 · · · pr
.
Applications:
Computation ofP fromSd(P)(0)inO((r+d)ω)(inO(rd)˜ using structured matrices)
Computation ofSd(P)(0)fromP inO((r+d)ω)(inO(rd)˜ using structured matrices)
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Fast Computation of S
`(P )(0)
A remark from Bostan, Chyzak and Le Roux (ISSAC 2008)
1 0 0
1 1 0 0
1 2 1 0 0
1 3 3 1 0 0
... . ..
0 · · · 0 p0 p1 · · · pr
... 0 p00 p01 · · · p0r 0 0 . ..
. ..
0 ... p(`−1)0 p(`−1)1 · · · p(`−1)r 0 ... 0
=
p0 p1 · · · pr
p00 p0+p01 · · · p0r+pr−1 pr
. .. ...
p(`−1)0 · · · pr
.
Applications:
12 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexity of van der Hoeven Algorithm
Easy bound: IfP andQare of bidegrees(r,d),QP is of bidegree(2r,2d).
Evaluation of S2d+r(P)(0)and S2d(Q)(0)
O(r(r˜ +d))ops
Matrix multiplication S2d(QP)(0) =S2d(Q)(0)· S2d+r(P)(0)
O((d+r)ω)ops
Interpolation. From S2d(QP)(0)to the coefficients ofQP
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)
§
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexity of van der Hoeven Algorithm
Easy bound: IfP andQare of bidegrees(r,d),QP is of bidegree(2r,2d).
Evaluation of S2d+r(P)(0)and S2d(Q)(0)O(r(r˜ +d))ops
Matrix multiplication S2d(QP)(0) =S2d(Q)(0)· S2d+r(P)(0)O((d+r)ω)ops Interpolation. From S2d(QP)(0)to the coefficients ofQP 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)
§
12 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexity of van der Hoeven Algorithm
Easy bound: IfP andQare of bidegrees(r,d),QP is of bidegree(2r,2d).
Evaluation of S2d+r(P)(0)and S2d(Q)(0)O(r(r˜ +d))ops
Matrix multiplication S2d(QP)(0) =S2d(Q)(0)· S2d+r(P)(0)O((d+r)ω)ops Interpolation. From S2d(QP)(0)to the coefficients ofQP 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)
§
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexity of van der Hoeven Algorithm
Easy bound: IfP andQare of bidegrees(r,d),QP is of bidegree(2r,2d).
Evaluation of S2d+r(P)(0)and S2d(Q)(0)O(r(r˜ +d))ops
Matrix multiplication S2d(QP)(0) =S2d(Q)(0)· S2d+r(P)(0)O((d+r)ω)ops Interpolation. From S2d(QP)(0)to the coefficients ofQP 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)
§
12 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Complexity of van der Hoeven Algorithm
Easy bound: IfP andQare of bidegrees(r,d),QP is of bidegree(2r,2d).
Evaluation of S2d+r(P)(0)and S2d(Q)(0)O(r(r˜ +d))ops
Matrix multiplication S2d(QP)(0) =S2d(Q)(0)· S2d+r(P)(0)O((d+r)ω)ops Interpolation. From S2d(QP)(0)to the coefficients ofQP 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)
§
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
III New Algorithm for the Unbalanced Product
14 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product using Multipoint Evaluations and Interpolation
New idea (d > r andd/r∈N):
We replace one multiplication of big matrices by several multiplications of smaller matrices
Evaluations of S2r(P)(0),S2r(P)(1), · · ·,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)
Matrix multiplications: For alli, Sr(QP)(i) =Sr(Q)(i)· S2r(P)(i). O(drω−1)
d/rmultiplications of matrices of size r×r(instead ofr×d)
Interpolations. From Sr(QP)(0),Sr(QP)(1),· · · ,Sr(QP)(2d/r−1)to the coefficients ofQP
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product using Multipoint Evaluations and Interpolation
New idea (d > r andd/r∈N):
We replace one multiplication of big matrices by several multiplications of smaller matrices
Evaluations of S2r(P)(0),S2r(P)(1),· · · ,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)
Matrix multiplications: For alli, Sr(QP)(i) =Sr(Q)(i)· S2r(P)(i). O(drω−1)
d/rmultiplications of matrices of size r×r(instead ofr×d)
Interpolations. From Sr(QP)(0),Sr(QP)(1),· · · ,Sr(QP)(2d/r−1)to the coefficients ofQP
14 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product using Multipoint Evaluations and Interpolation
New idea (d > r andd/r∈N):
We replace one multiplication of big matrices by several multiplications of smaller matrices
Evaluations of S2r(P)(0),S2r(P)(1),· · · ,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)
Matrix multiplications: For alli, Sr(QP)(i) =Sr(Q)(i)· S2r(P)(i).
O(drω−1)
d/rmultiplications of matrices of size r×r(instead ofr×d)
Interpolations. From Sr(QP)(0),Sr(QP)(1),· · · ,Sr(QP)(2d/r−1)to the coefficients ofQP
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product using Multipoint Evaluations and Interpolation
New idea (d > r andd/r∈N):
We replace one multiplication of big matrices by several multiplications of smaller matrices
Evaluations of S2r(P)(0),S2r(P)(1),· · · ,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)
Matrix multiplications: For alli, Sr(QP)(i) =Sr(Q)(i)· S2r(P)(i).
O(drω−1)
d/rmultiplications of matrices of sizer×r(instead ofr×d)
Interpolations. From Sr(QP)(0),Sr(QP)(1),· · · ,Sr(QP)(2d/r−1)to the coefficients ofQP
14 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Product using Multipoint Evaluations and Interpolation
New idea (d > r andd/r∈N):
We replace one multiplication of big matrices by several multiplications of smaller matrices
Evaluations of S2r(P)(0),S2r(P)(1),· · · ,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)
Matrix multiplications: For alli, Sr(QP)(i) =Sr(Q)(i)· S2r(P)(i).
O(drω−1)
d/rmultiplications of matrices of sizer×r(instead ofr×d)
Interpolations. From Sr(QP)(0),Sr(QP)(1),· · · ,Sr(QP)(2d/r−1) to the coefficients ofQP
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Hermite Evaluations and Interpolations
Fast algorithm for evaluations and interpolations inO(n)˜ ops (Chin (1976)).
Application:
Evaluations and interpolation of Sr(P)(i)fori∈[0..2d/r−1]inO(rd)˜ ops
15 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Hermite Evaluations and Interpolations
Fast algorithm for evaluations and interpolations inO(n)˜ ops (Chin (1976)).
Application:
Evaluations and interpolation of Sr(P)(i)fori∈[0..2d/r−1]inO(rd)˜ ops
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Hermite Evaluations and Interpolations
Fast algorithm for evaluations and interpolations inO(n)˜ ops (Chin (1976)).
Application:
16 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
New Algorithm when d > r
Evaluations of Sr(P)(0),Sr(P)(1), · · ·,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)
O(dr)˜
Matrix multiplications: For alliSr(QP)(i) =Sr(Q)(i)· S2r(P)(i)O(drω−1) Interpolation. From Sr(QP)(0),Sr(QP)(1),· · ·,Sr(QP)(2d/r−1)to the coefficients ofQP
O(dr)˜
Conplexity of algorithm whend > r: O(dr˜ ω−1)arithmetic operations
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
New Algorithm when d > r
Evaluations of Sr(P)(0),Sr(P)(1), · · ·,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)O(dr)˜
Matrix multiplications: For alliSr(QP)(i) =Sr(Q)(i)· S2r(P)(i)O(drω−1) Interpolation. From Sr(QP)(0),Sr(QP)(1),· · ·,Sr(QP)(2d/r−1)to the coefficients ofQP O(dr)˜
Conplexity of algorithm whend > r: O(dr˜ ω−1)arithmetic operations
16 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
New Algorithm when d > r
Evaluations of Sr(P)(0),Sr(P)(1), · · ·,S2r(P)(2d/r−1)and Sr(Q)(0),Sr(Q)(1), · · ·,Sr(Q)(2d/r−1)O(dr)˜
Matrix multiplications: For alliSr(QP)(i) =Sr(Q)(i)· S2r(P)(i)O(drω−1) Interpolation. From Sr(QP)(0),Sr(QP)(1),· · ·,Sr(QP)(2d/r−1)to the coefficients ofQP O(dr)˜
Conplexity of algorithm whend > r: O(dr˜ ω−1)arithmetic operations
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Application when r > d
ϕis the morphism fromK[x]h∂ito itself such that:
ϕ(∂) =x, ϕ(x) =−∂.
Application, new algorithm for the product whenr > d
compute the canonical forms (xat left and ∂at right) ofϕ(P)andϕ(Q), new algorithm inO(dr)˜
compute the productM =ϕ(P)ϕ(Q)of operatorsϕ(P)andϕ(Q) inO(dω−1r)using the previous algorithm return the (canonical form of the) operatorP Q=ϕ−1(M)
new algorithm inO(dr)˜
Complexity of the product inO(d˜ ω−1r)arithmetic operations whenr > d
17 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Application when r > d
ϕis the morphism fromK[x]h∂ito itself such that:
ϕ(∂) =x, ϕ(x) =−∂.
Application, new algorithm for the product whenr > d
compute the canonical forms (xat left and ∂at right) ofϕ(P)andϕ(Q), new algorithm inO(dr)˜
compute the productM =ϕ(P)ϕ(Q)of operatorsϕ(P)andϕ(Q) inO(dω−1r)using the previous algorithm return the (canonical form of the) operatorP Q=ϕ−1(M)
new algorithm inO(dr)˜
Complexity of the product inO(d˜ ω−1r)arithmetic operations whenr > d
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
Application when r > d
ϕis the morphism fromK[x]h∂ito itself such that:
ϕ(∂) =x, ϕ(x) =−∂.
Application, new algorithm for the product whenr > d
compute the canonical forms (xat left and ∂at right) ofϕ(P)andϕ(Q), new algorithm inO(dr)˜
compute the productM =ϕ(P)ϕ(Q)of operatorsϕ(P)andϕ(Q) inO(dω−1r)using the previous algorithm return the (canonical form of the) operatorP Q=ϕ−1(M)
new algorithm inO(dr)˜
18 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
IV Conclusion
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
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 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
19 / 19 Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
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 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
Introduction New Vision of van der Hoeven Algorithm New Algorithm for the Unbalanced Product Conclusion
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 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
