Algorithmique semi-num´erique rapide des s´eries de Tchebychev Alexandre Benoit Projet Algorithms

INRIA

Soutenance de th`ese Directeur de th`ese : Bruno Salvy 18 juillet 2012

Summary

Goal Using computer algebra, find fast algorithms to compute guaranteed polynomial approximations of functions.

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Summary

Goal Using computer algebra, find fast algorithms to compute guaranteed polynomial approximations of functions.

Framework Approximation theory. Computer algebra:

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Summary

Goal Using computer algebra, find fast algorithms to compute guaranteed polynomial approximations of functions.

Framework Approximation theory. Computer algebra: Solutions of linear differential equations.

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Summary

Goal Using computer algebra, find fast algorithms to compute guaranteed polynomial approximations of functions.

Framework Approximation theory. Computer algebra: Solutions of linear differential equations. Fast algorithms.

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Summary

Goal Using computer algebra, find fast algorithms to compute guaranteed polynomial approximations of functions.

Framework Approximation theory. Computer algebra: Solutions of linear differential equations. Fast algorithms.

How Chebyshev expansion: very good approximation on an interval with good properties for computer algebra.

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Basic Properties of Chebyshev Polynomials

Tn (cos(θ)) = cos(nθ)

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Basic Properties of Chebyshev Polynomials

Tn (cos(θ)) = cos(nθ) Z

1

−1

  0 Tn (x)Tm (x) √ π dx =  π 1 − x2 2

Tn+1 (x) = 2xTn (x) − Tn−1 (x)

if m 6= n if m = 0 otherwise

(1 − x )Tn00 (x) − xTn0 (x) + n2 Tn (x) = 0 2

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Basic Properties of Chebyshev Polynomials

Tn (cos(θ)) = cos(nθ) Z

1

−1

  0 Tn (x)Tm (x) √ π dx =  π 1 − x2 2

Tn+1 (x) = 2xTn (x) − Tn−1 (x)

bn/2c

Tn (x) =

X k=0

(−1)k 2n−2k−1

if m 6= n if m = 0 otherwise

(1 − x )Tn00 (x) − xTn0 (x) + n2 Tn (x) = 0 2

  n n − k n−2k x n−k k

T0 (x) = 1 T1 (x) = x T2 (x) = 2x2 − 1 T3 (x) = 4x3 − 3x T4 (x) = 8x4 − 8x2 + 1 3 / 36

Approximation by Taylor or Chebyshev Series

Two approximations of a function f : by a Taylor series f=

+∞ X

un xn ,

n=0

un =

f (n) (0) , n!

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Approximation by Taylor or Chebyshev Series

Two approximations of a function f : by a Taylor series f=

+∞ X

or by Chebyshev series un xn ,

f=

n=0

f (n) (0) , un = n!

+∞ X

cn Tn (x),

n=0

cn =

1 π

Z

1

f (t) Tn (t) √ dt. 1 − t2 −1

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Chebyshev Truncations are Near-Best Let f be continuous on [−1, 1] with degree n Chebyshev truncation fn and best approximant pn (the polynomial of degree at most n that minimizes kf − pk∞ = sup−1≤x≤1 |f (x) − p(x)|), n ≥ 1. Then 

kf − fn k∞

 4 ≤ 4 + 2 log(n + 1) kf − pn k∞ . π {z } | Λn

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Chebyshev Truncations are Near-Best Let f be continuous on [−1, 1] with degree n Chebyshev truncation fn and best approximant pn (the polynomial of degree at most n that minimizes kf − pk∞ = sup−1≤x≤1 |f (x) − p(x)|), n ≥ 1. Then 

kf − fn k∞

 4 ≤ 4 + 2 log(n + 1) kf − pn k∞ . π {z } | Λn

Λ10 = 4.93... → < 3 bits Λ30 = 5.37... → < 3 bits Λ100 = 5.87... → < 3 bits Λ1000 = 6.80... → < 3 bits

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Chebyshev Truncations are Near-Best Let f be continuous on [−1, 1] with degree n Chebyshev truncation fn and best approximant pn (the polynomial of degree at most n that minimizes kf − pk∞ = sup−1≤x≤1 |f (x) − p(x)|), n ≥ 1. Then 

kf − fn k∞

 4 ≤ 4 + 2 log(n + 1) kf − pn k∞ . π {z } | Λn

Λ10 = 4.93... → < 3 bits Λ30 = 5.37... → < 3 bits

error of the approximation of exp Best approximant of degree 4 5 × 10−3

Λ100 = 5.87... → < 3 bits Λ1000 = 6.80... → < 3 bits

x −1

−1/2

1/2

1

−5 × 10−3 Chebyshev truncation of degree 4 5 / 36

Chebyshev Truncations are Near-Best Let f be continuous on [−1, 1] with degree n Chebyshev truncation fn and best approximant pn (the polynomial of degree at most n that minimizes kf − pk∞ = sup−1≤x≤1 |f (x) − p(x)|), n ≥ 1. Then 

kf − fn k∞

 4 ≤ 4 + 2 log(n + 1) kf − pn k∞ . π {z } | Λn

Λ10 = 4.93... → < 3 bits Λ30 = 5.37... → < 3 bits

error of the approximation of exp Best approximant of degree 4 5 × 10−3

Λ100 = 5.87... → < 3 bits Λ1000 = 6.80... → < 3 bits It’s faster to compute fn instead of pn

x −1

−1/2

1/2

1

−5 × 10−3 Chebyshev truncation of degree 4 5 / 36

Contributions

Fast polynomial conversion, Monomial bases 7→ Chebyshev basis (with Alin Bostan, Chap. 3)

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Contributions

Fast polynomial conversion, Monomial bases 7→ Chebyshev basis (with Alin Bostan, Chap. 3) When a Chebyshev series is solution of a linear differential equation, its coefficients are solutions of a linear recurrence equation Fast algorithm to compute this recurrence using fractions of recurrence operators (with Bruno Salvy, ISSAC’09)

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Contributions

Fast polynomial conversion, Monomial bases 7→ Chebyshev basis (with Alin Bostan, Chap. 3) When a Chebyshev series is solution of a linear differential equation, its coefficients are solutions of a linear recurrence equation Fast algorithm to compute this recurrence using fractions of recurrence operators (with Bruno Salvy, ISSAC’09) Use of this recurrence relation to compute the coefficients numerically (with Mioara Jolde¸s and Marc Mezzarobba, Chap. 6 and 7)

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Contributions

Fast polynomial conversion, Monomial bases 7→ Chebyshev basis (with Alin Bostan, Chap. 3) When a Chebyshev series is solution of a linear differential equation, its coefficients are solutions of a linear recurrence equation Fast algorithm to compute this recurrence using fractions of recurrence operators (with Bruno Salvy, ISSAC’09) Use of this recurrence relation to compute the coefficients numerically (with Mioara Jolde¸s and Marc Mezzarobba, Chap. 6 and 7) Implementation in the DDMF (ICMS’10)

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Contributions

Fast polynomial conversion, Monomial bases 7→ Chebyshev basis (with Alin Bostan, Chap. 3) When a Chebyshev series is solution of a linear differential equation, its coefficients are solutions of a linear recurrence equation Fast algorithm to compute this recurrence using fractions of recurrence operators (with Bruno Salvy, ISSAC’09) Use of this recurrence relation to compute the coefficients numerically (with Mioara Jolde¸s and Marc Mezzarobba, Chap. 6 and 7) Implementation in the DDMF (ICMS’10)

Generalization of the algorithm for the Chebyshev recurrence to the computation of recurrences for the coefficients of generalized Fourier series (with Bruno Salvy, Chap. 8)

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Contributions

Fast polynomial conversion, Monomial bases 7→ Chebyshev basis (with Alin Bostan, Chap. 3) When a Chebyshev series is solution of a linear differential equation, its coefficients are solutions of a linear recurrence equation Fast algorithm to compute this recurrence using fractions of recurrence operators (with Bruno Salvy, ISSAC’09) Use of this recurrence relation to compute the coefficients numerically (with Mioara Jolde¸s and Marc Mezzarobba, Chap. 6 and 7) Implementation in the DDMF (ICMS’10)

Generalization of the algorithm for the Chebyshev recurrence to the computation of recurrences for the coefficients of generalized Fourier series (with Bruno Salvy, Chap. 8) Fast algorithm for the product of differential operators (with Alin Bostan and Joris van der Hoeven, FOCS’12)

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Outline

1

DDMF

2

Chebyshev Recurrence

3

Computation of the Chebyshev Coefficients by Hadamard Product

4

Numerical Evaluation using the Recurrence Relation

5

Generalized Fourier Series

6

Quasi-Optimal Multiplication of Linear Differential Operators

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I DDMF

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The Special Function erf(x)

08/07/12 17:50

DDMF† Home

Glossary

The Special Function erf (x) [-]

rendering

1. Differential equation

link

The function erf (x) satisfies the differential equation

2

!

" d d2 y (x) x + 2 y (x) = 0 dx dx

with initial values y (0) = 0, (y0) (0) = 2 p1! . [+]

2. Plot

[+]

3. Numerical Evaluation

[+]

4. Symmetry

[+]

5. Taylor Expansion at 0

[+]

6. Local Expansions at Singularities and at Infinity

†[+] ICMS 2010, B., Chyzak, Darasse, Gerhold, Mezzarobba and Salvy 7. Hypergeometric Representation (http://ddmf.msr-inria.inria.fr).

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5. Taylor Expansion at 0 † [+] 6. Local DDMF Expansions at Singularities and at Infinity [+]

7. Hypergeometric Representation

[-]

8. Chebyshev Expansion over [!1; 1] Chebyshev expansion:

erf (x) =

1 X 4!n (!1)n 1F1(1=2 + n; 2 n + 2; ! 1)T2 n+1 (x) 2 : p ! (2 n + 1) n! n=0

First terms and polynomial approximation:

erf (x) = 0:904347 T1 (x) ! 0:0661130 T3 (x) + 0:00472936 T5 (x) + : : : erf (x) " 1:12633280 x ! 0:35903920 x3 + 0:07566976 x5: order = 6 Submit P1 The coefficients cn in the Chebyshev expansion erf (x) = n=0 cnTn(x) satisfy the recurrence

(n2 + 3 n) c (n) + (2 n3 + 12 n2 + 24 n + 16) c (n + 2) + (!n2 ! 5 n ! 4) c (n + 4) = 0: http://ddmf.msr-inria.inria.fr/1.7.2/ddmf?service=SpecialFunction&…ath&mac=Hv49g3JXhLJiYMAimJ3rVrlif0U&sf_id=sf_erf¶meters=q64FAA

† ICMS

2010, B., Chyzak, Darasse, Gerhold, Mezzarobba and Salvy (http://ddmf.msr-inria.inria.fr).

Page 1 of 2

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II Chebyshev Recurrence

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Chebyshev Recurrence

Theorem (Paszkowski 1975) P If cn Tn (x) is solution of a linear differential equation with polynomial coefficients, then the coefficients cn are solution of a linear recurrence with polynomial coefficients.

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5. Taylor Expansion at 0 † [+] 6. Local DDMF Expansions at Singularities and at Infinity [+]

7. Hypergeometric Representation

[-]

8. Chebyshev Expansion over [!1; 1] Chebyshev expansion:

erf (x) =

1 X 4!n (!1)n 1F1(1=2 + n; 2 n + 2; ! 1)T2 n+1 (x) 2 : p ! (2 n + 1) n! n=0

First terms and polynomial approximation:

erf (x) = 0:904347 T1 (x) ! 0:0661130 T3 (x) + 0:00472936 T5 (x) + : : : erf (x) " 1:12633280 x ! 0:35903920 x3 + 0:07566976 x5: order = 6 Submit P1 The coefficients cn in the Chebyshev expansion erf (x) = n=0 cnTn(x) satisfy the recurrence

(n2 + 3 n) c (n) + (2 n3 + 12 n2 + 24 n + 16) c (n + 2) + (!n2 ! 5 n ! 4) c (n + 4) = 0: http://ddmf.msr-inria.inria.fr/1.7.2/ddmf?service=SpecialFunction&…ath&mac=Hv49g3JXhLJiYMAimJ3rVrlif0U&sf_id=sf_erf¶meters=q64FAA

† ICMS

2010, B., Chyzak, Darasse, Gerhold, Mezzarobba and Salvy (http://ddmf.msr-inria.inria.fr).

Page 1 of 2

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Morphisms of Rings of Operators (S · un = un+1 )

Taylor series (f :=

P

un xn )

X X xf = un xn+1 = un−1 xn , X X f0 = nun xn−1 = (n + 1)un+1 xn

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Morphisms of Rings of Operators (S · un = un+1 )

Taylor series (f :=

P

un xn )

X X xf = un xn+1 = un−1 xn , X X f0 = nun xn−1 = (n + 1)un+1 xn

x7→X := S −1 , d dx 7→D

:= (n + 1)S.

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Morphisms of Rings of Operators (S · un = un+1 ) Taylor series (f :=

P

un xn )

X X un xn+1 = un−1 xn , xf = X X f0 = nun xn−1 = (n + 1)un+1 xn

x7→X := S −1 , d dx 7→D

(4 + x2 )

:= (n + 1)S.



d dx

2

d + 2x dx

7→(4+S −2 )(n+1)(n+2)S 2 +2S −1 (n + 1)S
= (n + 1) 4(n + 2)S 2 + n 4(n + 2)un+2 + nun = 0

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Morphisms of Rings of Operators (S · un = un+1 ) Monomial Basis xn = Mn (x)

xTn (x) =1/2 (Tn+1 (x) + Tn−1 (x))

xMn (x) = Mn+1 (x),

Tn0 (x) =

Mn0 (x) = nMn−1 (x).



d dx

2 + 2x

n (Tn−1 (x) − Tn+1 (x)) . 2(1 − x2 )

S + S −1 , 2 d (n + 1)S − (n − 1)S −1 2n 7→D := = −1 . 2 dx 2(1 − X ) S −S

x7→X := S −1 , d 7→D := (n + 1)S. dx (4 + x2 )

Chebyshev series

x7→X :=

d dx


(n − 1)(n + 1) (n + 2)S 2 + 18n + (n − 2)S −2 , ((n − 1)S 2 − 2n + (n + 1)S −2 ) 7 (4+S −2 )(n+1)(n+2)S 2 +2S −1 (n+1)S → = (n + 1) 4(n + 2)S 2 + n 4(n + 2)un+2 + nun = 0




(n + 2)cn+2 + 18ncn + (n − 2)cn−2 = 0.

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Morphisms of Rings of Operators (S · un = un+1 ) Monomial Basis xn = Mn (x)

Chebyshev series xTn (x) =1/2 (Tn+1 (x) + Tn−1 (x))

xMn (x) = Mn+1 (x),

n (Tn−1 (x) − Tn+1 (x)) . 2(1 − x2 )

Mn0 (x) = nMn−1 (x).

Tn0 (x) =

x7→X := S −1 , d 7→D := (n + 1)S. dx

x7→X :=

(4 + x2 )



d dx

2 + 2x

S+S −1 , 2

d (n + 1)S − (n − 1)S −1 2n 7→D := = −1 . dx 2(1 − X 2 ) S −S

d dx


(n − 1)(n + 1) (n + 2)S 2 + 18n + (n − 2)S −2 , ((n − 1)S 2 − 2n + (n + 1)S −2 ) 7 (4+S −2 )(n+1)(n+2)S 2 +2S −1 (n+1)S → = (n + 1) 4(n + 2)S 2 + n 4(n + 2)un+2 + nun = 0




(n + 2)cn+2 + 18ncn + (n − 2)cn−2 = 0.

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Morphisms of Rings of Operators (S · un = un+1 ) Monomial Basis xn = Mn (x)

xTn (x) =1/2 (Tn+1 (x) + Tn−1 (x))

xMn (x) = Mn+1 (x), Mn0 (x)

x7→X := S d dx 7→D

(x)−Tn+1 (x)) Tn0 (x) = n(Tn−12(1−x . 2)

= nMn−1 (x). −1

x7→X :=

,

:= (n + 1)S.

(4 + x2 )



d dx

2 + 2x

Chebyshev series

d dx 7→D

:=

S + S −1 , 2 (n+1)S−(n−1)S −1 2(1−X 2 )

=

2n . S −1 − S

d dx


(n − 1)(n + 1) (n + 2)S 2 + 18n + (n − 2)S −2 , ((n − 1)S 2 − 2n + (n + 1)S −2 ) 7 (4+S −2 )(n+1)(n+2)S 2 +2S −1 (n+1)S → = (n + 1) 4(n + 2)S 2 + n 4(n + 2)un+2 + nun = 0




(n + 2)cn+2 + 18ncn + (n − 2)cn−2 = 0.

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Morphisms of Rings of Operators (S · un = un+1 ) Monomial Basis xn = Mn (x) xMn (x) = Mn+1 (x), Mn0 (x) = nMn−1 (x). x7→X := S −1 , d 7→D := (n + 1)S. dx (4 + x2 )



d dx

2

d + 2x dx

7→(4+S −2 )(n+1)(n+2)S 2 +2S −1 (n+1)S
= (n + 1) 4(n + 2)S 2 + n 4(n + 2)un+2 + nun = 0

Chebyshev series xTn (x) =1/2 (Tn+1 (x) + Tn−1 (x)) Tn0 (x) =

n (Tn−1 (x) − Tn+1 (x)) . 2(1 − x2 )

S + S −1 , 2 d (n + 1)S − (n − 1)S −1 2n 7→D := = −1 . 2 dx 2(1 − X ) S −S x7→X :=

(n−1)(n+1)((n+2)S 2 +18n+(n−2)S −2 ) ((n−1)S 2 −2n+(n+1)S −2 )

,

(n + 2)cn+2 + 18ncn + (n − 2)cn−2 = 0.

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Morphisms of Rings of Operators (S · un = un+1 ) Monomial Basis xn = Mn (x)

xTn (x) =1/2 (Tn+1 (x) + Tn−1 (x))

xMn (x) = Mn+1 (x),

Tn0 (x) =

Mn0 (x) = nMn−1 (x).



d dx

2 + 2x

n (Tn−1 (x) − Tn+1 (x)) . 2(1 − x2 )

S + S −1 , 2 d (n + 1)S − (n − 1)S −1 2n 7→D := = −1 . 2 dx 2(1 − X ) S −S

x7→X := S −1 , d 7→D := (n + 1)S. dx (4 + x2 )

Chebyshev series

x7→X :=

d dx


(n − 1)(n + 1) (n + 2)S 2 + 18n + (n − 2)S −2 , ((n − 1)S 2 − 2n + (n + 1)S −2 ) 7 (4+S −2 )(n+1)(n+2)S 2 +2S −1 (n+1)S → = (n + 1) 4(n + 2)S 2 + n 4(n + 2)un+2 + nun = 0




(n + 2)cn+2 + 18ncn + (n − 2)cn−2 = 0.

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Morphisms of Rings of Operators 2 Definition The Chebyshev morphism ϕ is defined by:
1 ϕ(x) = S + S −1 and ϕ 2



d dx

 =

2n . −S + S −1

Theorem (BenoitSalvy2009)
P d i f ∈ Ck, L = pi dx a linear differential operator of order k such that L · f = 0. Suppose that either of the following holds: Z 1 (k) f (x) √ (i). dx is convergent; 1 − x2 −1 Z 1 (1 − x2 )k f (k) (x) √ (ii). dx is convergent and (1 − x2 )i |pi , i = 0, . . . ,k. 1 − x2 −1 Then, the Chebyshev coefficients of f are cancelled by any numerator of ϕ(L). 13 / 36

New and Fast Algorithm†

Theorem If the order is at most k and the degrees of pi are at most k, Paszkowski (1975) and Lewanowicz (1976): O(k 4 ) arithmetic operations. New: O(k ω ) arithmetic operations. ω is a feasible exponent for matrix multiplication (2 ≤ ω ≤ 3)

†

[BenoitSalvy, 2012] A.B., Bruno Salvy, Chebyshev Expansions for Solutions of Linear Differential Equations, ISSAC 2009. 14 / 36

New and Fast Algorithm†

Theorem If the order is at most k and the degrees of pi are at most k, Paszkowski (1975) and Lewanowicz (1976): O(k 4 ) arithmetic operations. New: O(k ω ) arithmetic operations. ω is a feasible exponent for matrix multiplication (2 ≤ ω ≤ 3) Idea for the new algorithm: Compute the numerator of a fraction of recurrence operators using a divide-and-conquer method.

†

[BenoitSalvy, 2012] A.B., Bruno Salvy, Chebyshev Expansions for Solutions of Linear Differential Equations, ISSAC 2009. 14 / 36

III Computation of the Chebyshev Coefficients by Hadamard Product

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Hadamard Product for Chebyshev Expansions† Hyp: f is analytic in the closed unit disk. Then, there exists un and cn such that f (x) =

X n∈N

† extension

of Thacher 1964

un xn =

X

cn Tn (x).

n∈N

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5. Taylor Expansion at 0 † [+] 6. Local DDMF Expansions at Singularities and at Infinity [+]

7. Hypergeometric Representation

[-]

8. Chebyshev Expansion over [!1; 1] Chebyshev expansion:

erf (x) =

1 X 4!n (!1)n 1F1(1=2 + n; 2 n + 2; ! 1)T2 n+1 (x) 2 : p ! (2 n + 1) n! n=0

First terms and polynomial approximation:

erf (x) = 0:904347 T1 (x) ! 0:0661130 T3 (x) + 0:00472936 T5 (x) + : : : erf (x) " 1:12633280 x ! 0:35903920 x3 + 0:07566976 x5: order = 6 Submit P1 The coefficients cn in the Chebyshev expansion erf (x) = n=0 cnTn(x) satisfy the recurrence

(n2 + 3 n) c (n) + (2 n3 + 12 n2 + 24 n + 16) c (n + 2) + (!n2 ! 5 n ! 4) c (n + 4) = 0: http://ddmf.msr-inria.inria.fr/1.7.2/ddmf?service=SpecialFunction&…ath&mac=Hv49g3JXhLJiYMAimJ3rVrlif0U&sf_id=sf_erf¶meters=q64FAA

† ICMS

2010, B., Chyzak, Darasse, Gerhold, Mezzarobba and Salvy (http://ddmf.msr-inria.inria.fr).

Page 1 of 2

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Hadamard Product for Chebyshev Expansions† Hyp: f is analytic in the closed unit disk. Then, there exists un and cn such that f (xt) =

X n∈N

† extension

of Thacher 1964

un xn tn =

X

cn (t)Tn (x).

n∈N

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Hadamard Product for Chebyshev Expansions† Hyp: f is analytic in the closed unit disk. Then, there exists un and cn such that f (xt) =

X n∈N

un xn tn =

X

cn (t)Tn (x).

n∈N

Inner product with Tn (x) Z P Z 1 k k k X 2 1 2X x T (x) k∈N uk x t Tn (x) k √ √ n cn (t) = dx = uk t dx = uk gn,k tk . 2 π −1 π 1 − x2 1 − x −1 k∈N | {z } k∈N independant of f

† extension

of Thacher 1964

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Hadamard Product for Chebyshev Expansions† Hyp: f is analytic in the closed unit disk. Then, there exists un and cn such that f (xt) =

X n∈N

un xn tn =

X

cn (t)Tn (x).

n∈N

Inner product with Tn (x) Z P Z 1 k k k X 2 1 2X x T (x) k∈N uk x t Tn (x) k √ √ n cn (t) = dx = uk t dx = uk gn,k tk . 2 π −1 π 1 − x2 1 − x −1 k∈N | {z } k∈N independant of f

  X X 2tn k 1−2k−n 2k + n gn (t) = gn,k t = 2 t2k+n = . √ n√ k+n (1 + 1 − t2 ) 1 − t2 k∈N k∈N

† extension

of Thacher 1964

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Hadamard Product for Chebyshev Expansions† Hyp: f is analytic in the closed unit disk. Then, there exists un and cn such that f (xt) =

X n∈N

un xn tn =

X

cn (t)Tn (x).

n∈N

Inner product with Tn (x) Z P Z 1 k k k X 2 1 2X x T (x) k∈N uk x t Tn (x) k √ √ n cn (t) = dx = uk t dx = uk gn,k tk . 2 π −1 π 1 − x2 1 − x −1 k∈N | {z } k∈N independant of f

  X X 2tn k 1−2k−n 2k + n gn (t) = gn,k t = 2 t2k+n = . √ n√ k+n (1 + 1 − t2 ) 1 − t2 k∈N k∈N cn (t) = f (t) gn (t)

† extension

of Thacher 1964

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Application to the Closed Form of Coefficients

P un xn is m-hypergeometric, i.e. un+m /un is rational, then f gk Idea: If f = is 2m-hypergeometric.

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Application to the Closed Form of Coefficients

P un xn is m-hypergeometric, i.e. un+m /un is rational, then f gk Idea: If f = is 2m-hypergeometric. Corollary (Luke, 1969):  pFq

 X a1 ; . . . ; ap xt = ck (t)Tk (x), b1 ; . . . ; bq k∈N

ck (t) = 2 (a1 )k . . . (ap )k tk 2pF2q+1 2k (b1 )k . . . (bq )k k!

a +k a +k+1 a1 +k a1 +k+1 ; . . . ; p2 ; p 2 2 ; 2 b +k b +k+1 b1 +k b1 +k+1 ; . . . ; q2 ; q 2 ; k + 2 ; 2

! t2 . 1 4q−p+1

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Applications to Special Functions Function

Chebyshev expansion

exp(xt)

P∞

0

n∈N (−1)

n

n=0

sin(xt) erf(xt) Si(xt)

2 P

P

J2n+1 (t)T2n+1 (x)

(−1)n t2n+1 √2 n∈N π 4n (2n+1)n! 1F1

4−n (−1)n n∈N (2n+1)!(2n+1) 1F1

P

2 In (t)Tn (x)







n+ 12 2n+2

n+ 12 2n+2,n+ 32

 − t2 T2n+1 (x)

 1 2 − 4 t T2n+1 (x)

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Applications to Special Functions Function

Chebyshev expansion

exp(xt)

P∞

0

n∈N (−1)

n

n=0

sin(xt)

2 P

erf(xt)

4−n (−1)n n∈N (2n+1)!(2n+1) 1F1

P

0

1

n∈N 81n n!

+

J2n+1 (t)T2n+1 (x)

(−1)n t2n+1 √2 n∈N π 4n (2n+1)n! 1F1

P

Si(xt)

Ai(xt)

P

2 In (t)Tn (x)

−









n+ 12 2n+2

n+ 12 2n+2,n+ 32

32/3 1 144 Γ(n+4/3) 2 F5

√ 3 3 2 3 Γ(n+2/3) 2 F5



 − t2 T2n+1 (x)

 1 2 − 4 t T2n+1 (x)

1/2 n+4/3;1/2 n+5/6 t6 4/3;5/3;n+1;n+4/3;n+5/3 1296



! 6  1/2n+2/3;1/2 n+1/6 t T3n (x) 1/3;2/3;n+1;n+2/3;n+1/3 1296 +··· 18 / 36

Application to Numerical Computation f (x) =

X n∈N

cn (1)Tn (x), with cn (t) = f (t)

(1 +

√

2tn √ 1 − t2 )n 1 − t2

19 / 36

Application to Numerical Computation f (x) =

X n∈N

cn (1)Tn (x), with cn (t) = f (t)

(1 +

√

2tn √ 1 − t2 )n 1 − t2

P∞ cn (t) = k=0 cn,k tk is solution of a linear differential equation, then cn,k are solutions of a linear recurrence

19 / 36

Application to Numerical Computation f (x) =

X n∈N

cn (1)Tn (x), with cn (t) = f (t)

(1 +

√

2tn √ 1 − t2 )n 1 − t2

P∞ cn (t) = k=0 cn,k tk is solution of a linear differential equation, then cn,k are solutions of a linear recurrence −1 ˜ Application: Algorithm for the computation of cn (1) in O(log( ) + n) bit operations INPUT: n,  OUTPUT: e cn (1) such that |e cn (1) − cn (1)| < 

19 / 36

Application to Numerical Computation f (x) =

X n∈N

cn (1)Tn (x), with cn (t) = f (t)

(1 +

√

2tn √ 1 − t2 )n 1 − t2

P∞ cn (t) = k=0 cn,k tk is solution of a linear differential equation, then cn,k are solutions of a linear recurrence −1 ˜ Application: Algorithm for the computation of cn (1) in O(log( ) + n) bit operations INPUT: n,  OUTPUT: e cn (1) such that |e cn (1) − cn (1)| < 

˜ Computation of the recurrence satisfied by cn,k , plus initial conditions (O(n) ops) (Using binary splitting)

Use ofP the Mezzarobba-Salvy algorithm to compute N such ∞ that | n=N +1 cn,k | <  (O(log(−1 )) ops) PN −1 ˜ ) = O(log( ˜ Computation of k=0 cn,k using the recurrence (O(N )) ops) (Using binary splitting) 19 / 36

Application to Numerical Computation f (x) =

X n∈N

cn (1)Tn (x), with cn (t) = f (t)

(1 +

√

2tn √ 1 − t2 )n 1 − t2

P∞ cn (t) = k=0 cn,k tk is solution of a linear differential equation, then cn,k are solutions of a linear recurrence −1 ˜ Application: Algorithm for the computation of cn (1) in O(log( ) + n) bit operations INPUT: n,  OUTPUT: e cn (1) such that |e cn (1) − cn (1)| < 

˜ Computation of the recurrence satisfied by cn,k , plus initial conditions (O(n) ops) (Using binary splitting)

Use ofP the Mezzarobba-Salvy algorithm to compute N such ∞ that | n=N +1 cn,k | <  (O(log(−1 )) ops) PN −1 ˜ ) = O(log( ˜ Computation of k=0 cn,k using the recurrence (O(N )) ops) (Using binary splitting) ˜ log(−1 )). To compute ` coefficients, the complexity becomes O(` 19 / 36

IV Numerical Evaluation using the Recurrence Relation

20 / 36

Direct Application of the Recurrence Taylor: exp(x) = un Rec: un+1 = n+1

P

1 n x n!

u0 = 1

1/0! = 1

u1 = 1

1/1! = 1

u2 = 0.5

1/2! = 0.5

.. . u50 ≈ 3.28 × 10−65

.. . 1/50! ≈ 3.28 × 10−65

21 / 36

Direct Application of the Recurrence Taylor: exp(x) = un Rec: un+1 = n+1

P

1 n x n!

P Chebyshev: exp(x) = In (1)Tn (x) Rec: un+1 = −2nun + un−1

u0 = 1

1/0! = 1

u0 = 1.266

I0 (1) ≈ 1.266

u1 = 1

1/1! = 1

u1 = 0.565

I1 (1) ≈ 0.565

1/2! = 0.5

u2 ≈ 0.136

I2 (1) ≈ 0.136

u2 = 0.5 .. . u50 ≈ 3.28 × 10−65

.. .

.. .

.. .

1/50! ≈ 3.28 × 10−65

21 / 36

Direct Application of the Recurrence Taylor: exp(x) = un Rec: un+1 = n+1

P

1 n x n!

P Chebyshev: exp(x) = In (1)Tn (x) Rec: un+1 = −2nun + un−1

u0 = 1

1/0! = 1

u0 = 1.266

I0 (1) ≈ 1.266

u1 = 1

1/1! = 1

u1 = 0.565

I1 (1) ≈ 0.565

1/2! = 0.5

u2 ≈ 0.136

I2 (1) ≈ 0.136

u2 = 0.5 .. . u50 ≈ 3.28 × 10−65

.. .

.. .

67 1/50! ≈ 3.28 × 10−65 u50 ≈ 4.450 × 10

.. . I50 (1) ≈ 2.934 × 10−80

21 / 36

Direct Application of the Recurrence Taylor: exp(x) = un Rec: un+1 = n+1

P

1 n x n!

P Chebyshev: exp(x) = In (1)Tn (x) Rec: un+1 = −2nun + un−1

u0 = 1

1/0! = 1

u0 = 1.266

I0 (1) ≈ 1.266

u1 = 1

1/1! = 1

u1 = 0.565

I1 (1) ≈ 0.565

1/2! = 0.5

u2 ≈ 0.136

I2 (1) ≈ 0.136

u2 = 0.5 .. . u50 ≈ 3.28 × 10−65

.. .

.. .

.. .

67 1/50! ≈ 3.28 × 10−65 u50 ≈ 4.450 × 10

Study of the Chebyshev recurrence If un is solution, then there exists another solution vn ∼ u1n

I50 (1) ≈ 2.934 × 10−80

Newton polygon of a Chebyshev recurrence

n

κ−2 = · · · = κ2 κ−3 κ3 S 21 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X n∈N

In (t)Tn (x) =

XX

cn,k tk Tn (x).

n∈N k∈N

22 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X n∈N

In (t)Tn (x) =

XX

cn,k tk Tn (x).

n∈N k∈N

We can compute t In (t) − 2(n + 1) In+1 (t) − t In+2 (t) = 0.

22 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X

In (t)Tn (x) =

n∈N

XX

cn,k tk Tn (x).

n∈N k∈N

We can compute X X X t cn,k tk − 2(n + 1) cn+1,k tk − t cn+2,k tk = 0. k∈N

k∈N

k∈N

22 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X

In (t)Tn (x) =

n∈N

XX

cn,k tk Tn (x).

n∈N k∈N

We can compute X X X t cn,k tk − 2(n + 1) cn+1,k tk − t cn+2,k tk = 0 k∈N

k∈N

mod tN +2 .

k∈N

22 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X

In (t)Tn (x) =

n∈N

XX

cn,k tk Tn (x).

n∈N k∈N

We can compute X X X t cn,k tk − 2(n + 1) cn+1,k tk − t cn+2,k tk = 0 k∈N

k∈N

mod tN +2 .

k∈N

We deduce a non-homogeneous linear recurrence in n t

N X

cn,k tk − 2(n + 1)

k=0

N X

cn+1,k tk − t

k=0

N X

cn+2,k tk = 2(n + 1)cn+1,N +1 tN +1 .

k=0

satisfied by N X

cn,k tk .

k=0

22 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X

XX

In (t)Tn (x) =

n∈N

cn,k tk Tn (x).

n∈N k∈N

We can compute X X X t cn,k tk − 2(n + 1) cn+1,k tk − t cn+2,k tk = 0 k∈N

k∈N

mod tN +2 .

k∈N

We deduce a non-homogeneous linear recurrence in n N X k=0

cn,k − 2(n + 1)

N X

cn+1,k −

k=0

N X

cn+2,k = 2(n + 1)cn+1,N +1 .

k=0

satisfied by N X

cn,k .

k=0

22 / 36

Hadamard Product and Linear Recurrence exp(xt) =

X

XX

In (t)Tn (x) =

n∈N

cn,k tk Tn (x).

n∈N k∈N

We can compute X X X t cn,k tk − 2(n + 1) cn+1,k tk − t cn+2,k tk = 0 k∈N

k∈N

mod tN +2 .

k∈N

We deduce a non-homogeneous linear recurrence in n N X k=0

cn,k − 2(n + 1)

N X

cn+1,k −

k=0

N X

cn+2,k = 2(n + 1)cn+1,N +1 .

k=0

satisfied by N X

cn,k .

k=0

Application: Algorithm to compute the first ` coefficients e cn (1) = −1 ˜ that |e cn (1) − f | <  in O(` + log( )) arithmetic operations.

PN

k=0 cn,k

such 22 / 36

Miller’s Method Example y(x) = ex =

P∞

n=0 cn

Tn (x)

cn+1 + 2n cn − cn−1 = 0

23 / 36

Miller’s Method Example y(x) = ex =

P∞

n=0 cn

cn+1 + 2n cn − cn−1 = 0

Tn (x)

2 solutions: In (1) and Kn (1)

u0 u1 u2

.. .

u50 u51 u52

c0 c1 c2

.. .

c50 c51 c52

23 / 36

Miller’s Method Example y(x) = ex =

P∞

n=0 cn

cn+1 + 2n cn − cn−1 = 0

Tn (x)

2 solutions: In (1) and Kn (1)

u0 u1 u2

c0 c1 c2

.. .

u50 ≈ u51 = u52 =

1.02 · 102 1 0

.. .

c50 c51 c52

23 / 36

Miller’s Method Example y(x) = ex =

P∞

n=0 cn

cn+1 + 2n cn − cn−1 = 0

Tn (x)

2 solutions: In (1) and Kn (1)

u0 ≈ −4.40 · 1081 u1 ≈ 1.96 · 1081 u2 ≈ −4.72 · 1080 .. . u50 ≈ u51 = u52 =

1.02 · 102 1 0

c0 c1 c2

.. .

c50 c51 c52

23 / 36

Miller’s Method Example y(x) = ex =

P∞

n=0 cn

cn+1 + 2n cn − cn−1 = 0

Tn (x)

2 solutions: In (1) and Kn (1)

S=

50 X

un Tn (0) ≈ −3.48 · 1081

n=0

u0 ≈ −4.40 · 1081 u1 ≈ 1.96 · 1081 u2 ≈ −4.72 · 1080 .. . u50 ≈ u51 = u52 =

1.02 · 102 1 0

c0 c1 c2

.. .

c50 c51 c52

23 / 36

Miller’s Method Example y(x) = ex =

P∞

n=0 cn

cn+1 + 2n cn − cn−1 = 0

Tn (x)

2 solutions: In (1) and Kn (1)

S=

50 X

un Tn (0) ≈ −3.48 · 1081

n=0

u0 ≈ −4.40 · 1081 u1 ≈ 1.96 · 1081 u2 ≈ −4.72 · 1080 .. . u50 ≈ u51 = u52 =

1.02 · 102 1 0

gn := un /S

c0 ≈ 1.27 c1 ≈ −5.65 · 10−1 c2 ≈ 1.36 · 10−1 .. . c50 ≈ c51 ≈ c52 ≈

2.93 · 10−80 2.88 · 10−82 0

23 / 36

Algorithm from the Second method† Algorithm Input: a differential equation of order r with boundary conditions Output: a polynomial approximation of degree N of the solution compute the Chebyshev recurrence (order 2s ≥ 2r); for i from 1 to s using the recurrence relation backwards, compute the first N coefficients of the sequence u[i] starting with the initial conditions   u[i] (N + 2s), · · · ,u[i] (N + i), · · · ,u[i] (N + 1) = (0, · · · ,1, · · · ,0) ;

combine the s sequences u[i] according to the r boundary conditions and the s − r symmetry relations.

†

[BenoitMezzarobbaJode¸s, 2012] A.B., Mioara Jolde¸s and Marc Mezzarobba, Rigorous uniform approximation of D-finite functions using Chebyshev expansions, In preparation.

24 / 36

Algorithm from the Second method† Algorithm Input: a differential equation of order r with boundary conditions Output: a polynomial approximation of degree N of the solution compute the Chebyshev recurrence (order 2s ≥ 2r); for i from 1 to s using the recurrence relation backwards, compute the first N coefficients of the sequence u[i] starting with the initial conditions   u[i] (N + 2s), · · · ,u[i] (N + i), · · · ,u[i] (N + 1) = (0, · · · ,1, · · · ,0) ;

combine the s sequences u[i] according to the r boundary conditions and the s − r symmetry relations. Theorem This algorithm runs in linear time. †

[BenoitMezzarobbaJode¸s, 2012] A.B., Mioara Jolde¸s and Marc Mezzarobba, Rigorous uniform approximation of D-finite functions using Chebyshev expansions, In preparation.

24 / 36

Quality of the Approximation 1.5e–142 4e–52

2e–97 1e–142

2e–52

x/2 √e x+16

–1 –0.8 –0.6 –0.4 –0.2

1e–97

0.2 0.4 0.6 0.8 x

1

–1 –0.8 –0.6 –0.4 –0.2

5e–143 0.2 0.4 0.6 0.8 x

1

0.2 0.4 0.6 0.8 x

1

0.2 0.4 0.6 0.8 x

1

0.2 0.4 x 0.6 0.8

1

–5e–143

–1e–97

–2e–52

–1 –0.8 –0.6 –0.4 –0.2

–1e–142 –2e–97

–4e–52

–1.5e–142

8e–44

4e–168 1e–102

6e–44

3 cos x − sin x 2

4e–44

2e–168

5e–103

2e–44 –1 –0.8 –0.6 –0.4 –0.2 –2e–44

0.2 0.4 0.6 0.8 x

1

–1 –0.8 –0.6 –0.4 –0.2 0

0.2 0.4 0.6 0.8 x

1

–5e–103

–4e–44 –6e–44

–2e–168

–1e–102 –4e–168

–8e–44

1e–23

8e–16

3e–08

6e–16

2e–08

5e–24

4e–16 1e–08 –1 –0.8 –0.6 –0.4 –0.2

e1/(1+2x

2

–1 –0.8 –0.6 –0.4 –0.2 0

0.2 0.4 x 0.6 0.8

) –1e–08

1

2e–16 –1 –0.8 –0.6 –0.4 –0.2 0 –2e–16

0.2 0.4 x 0.6 0.8

–4e–16 –2e–08

–1 –0.8 –0.6 –0.4 –0.2 0 –5e–24

–6e–16

–3e–08

–8e–16

–4e–08

–1e–15

degree = 30

1

degree = 60

–1e–23

degree = 90 25 / 36

Validation of the Polynomial

Algorithm Input: Differential operator, initial conditions and a polynomial of degree d Output: R such that ||f − p||∞ < R and R is not too large

26 / 36

Validation of the Polynomial

Algorithm Input: Differential operator, initial conditions and a polynomial of degree d Output: R such that ||f − p||∞ < R and R is not too large log10 x/2 √e x+16

3 cos x−sin x 2

e1/(1+2x

2

)

(bounds computed) kf −pk∞

4,8

0,58

0, 57

3,1

3,7

4,1

0,57 0,56 0,56 degree = 30 degree = 60 degree = 90

26 / 36

V Generalized Fourier Series

27 / 36

Generalized Fourier Series

f (x) =

X

an ψn (x)

Some Examples

sin(x) = 2

∞ X

(−1)n J2n (x)

n=0 ∞ X 4 1 T0 (x) − arccos (x) = 2 T2n+1 (x) 2π n=0 (2 n + 1) π n   ∞  X 1 1 n + 12 √ F erf (x) = 2 − − x 1 1 4 2n + 2 π (2 n + 1) n! n=0

28 / 36

Generalized Fourier Series

f (x) =

X

an ψn (x)

Some Examples

sin(x) = 2

∞ X

(−1)n J2n (x)

n=0 ∞ X 4 1 T0 (x) − arccos (x) = 2 T2n+1 (x) 2π n=0 (2 n + 1) π n   ∞  X 1 1 n + 12 √ F erf (x) = 2 − − x 1 1 4 2n + 2 π (2 n + 1) n! n=0

Aim: general algorithm for these series

28 / 36

Framework Families of functions ψn (x) with two special properties Mult by x (Px ) Recx2 (xψn (x)) = Recx1 (ψn (x))

Examples Monomial polynomials (Mn = xn ) All orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions

29 / 36

Framework Families of functions ψn (x) with two special properties Mult by x (Px ) Recx2 (xψn (x)) = Recx1 (ψn (x))

Examples Monomial polynomials (Mn = xn ) All orthogonal polynomials Bessel functions Legendre functions

xMn = Mn+1 2xTn (x) = Tn+1 (x) + Tn−1 (x) 1 (xJn+1 − xJn−1 ) = 2Jn n

Parabolic cylinder functions

29 / 36

Framework Families of functions ψn (x) with two special properties Mult by x (Px ) Recx2 (xψn (x)) = Recx1 (ψn (x)) Differentiation (P∂ ) Rec∂2 (ψn0 (x)) = Rec∂1 (ψn (x))

Examples Monomial polynomials Classical orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions 29 / 36

Framework Families of functions ψn (x) with two special properties Mult by x (Px ) Recx2 (xψn (x)) = Recx1 (ψn (x)) Differentiation (P∂ ) Rec∂2 (ψn0 (x)) = Rec∂1 (ψn (x))

Examples Monomial polynomials Classical orthogonal polynomials Bessel functions

Mn0 = nMn−1 1 1 0 0 Tn+1 (x) − Tn−1 (x) = 2Tn (x) n+1 n−1 0 2Jn (x) = Jn−1 (x) − Jn+1 (x)

Legendre functions Parabolic cylinder functions 29 / 36

Framework

Families of functions ψn (x) with two special properties Mult by x (Px ) Recx2 (xψn (x)) = Recx1 (ψn (x)) Differentiation (P∂ ) Rec∂2 (ψn0 (x)) = Rec∂1 (ψn (x)) This is our data-structure for ψn (x)

29 / 36

New algorithm to compute the recurrence† Main Idea P If ψn (x) satisfies (Px ) and (P∂ ), for any f (x) = an ψn (x) solution of a linear differential equation with polynomial coefficients, the coefficients an are solutions of a linear recurrence relation with polynomial coefficients.

†

[BenoitSalvy, 2012] A.B. and Bruno Salvy. Generalized Fourier Series for Solutions of Linear Differential Equations, In preparation. 30 / 36

New algorithm to compute the recurrence† Main Idea P If ψn (x) satisfies (Px ) and (P∂ ), for any f (x) = an ψn (x) solution of a linear differential equation with polynomial coefficients, the coefficients an are solutions of a linear recurrence relation with polynomial coefficients. New Contribution New general algorithm computing the recurrence relation of the coefficients of a Generalized Fourier Series when ψ(x) satisfies (Px ) and (P∂ ).

†

[BenoitSalvy, 2012] A.B. and Bruno Salvy. Generalized Fourier Series for Solutions of Linear Differential Equations, In preparation. 30 / 36

New algorithm to compute the recurrence† Main Idea P If ψn (x) satisfies (Px ) and (P∂ ), for any f (x) = an ψn (x) solution of a linear differential equation with polynomial coefficients, the coefficients an are solutions of a linear recurrence relation with polynomial coefficients. New Contribution New general algorithm computing the recurrence relation of the coefficients of a Generalized Fourier Series when ψ(x) satisfies (Px ) and (P∂ ). L(x,∂) 7→ numer(L(X,D))

†

[BenoitSalvy, 2012] A.B. and Bruno Salvy. Generalized Fourier Series for Solutions of Linear Differential Equations, In preparation. 30 / 36

VI Quasi-Optimal Multiplication of Linear Differential Operators

31 / 36

Product of Linear Differential Operators

The product of differential operators is a complexity yardstick: That of more involved, higher-level, operations on linear differential operators can be reduced to it: LCLM, GCRD (van der Hoeven 2011) closure properties for differential operators.

32 / 36

van der Hoeven’s Algorithm (2002)

 Φk+d,k L

 = 

L(1)0 .. .

···

L(1)k+d−1

···

L(xk−1 )0 .. . L(xk−1 )k+d−1

  (k+d)×k ∈K

we clearly have Φk+2d,k KL

=

k+2d,k+d k+d,k ΦK ΦL ,

for all k > 0.

Algorithm: when K and L are of degrees d and order r. Evaluation Computation of Φ2r+2d,2r+d and Φ2r+d,2r from K and L K L ˜ (O((r + d)2 ) ops). Inner multiplication Computation of the matrix product (O((r + d)ω ) ops). ˜ Interpolation Recovery of KL from Φ2r+2d,2r (O((r + d)2 ) ops). KL

33 / 36

New and Fast Algorithm for the Product†

i ,k+d,k Φα L

L(eαi x ) · · ·

=

L(xk−1 eαi x )




∈ K(k+d)×k

we clearly have i ,k+2d,k Φα KL

=

i ,k+2d,k+d i ,k+d,k Φα Φα , K L

for all k > 0.

Algorithm: when K and L are of degrees d and order r (r > d). αi ,3d,2d i ,4d,3d Evaluation For i = 0, . . . , r/d, computation of Φα and ΦL from K K and L.

Inner multiplication Computation of O(r/d) products of matrices with size d × d. i ,4d,2d Interpolation Recovery of KL from Φα . KL

†

[BenoitBostanvanderHoeven, 2012] A.B., Alin Bostan and Joris van der Hoeven. Quasi-Optimal Multiplication of Linear Differential Operators, FOCS 2012. 34 / 36

New and Fast Algorithm for the Product† i ,k+d,k Φα L

L(eαi x ) · · ·

=

L(xk−1 eαi x )




∈ K(k+d)×k

we clearly have i ,k+2d,k Φα KL

=

i ,k+2d,k+d i ,k+d,k Φα Φα , K L

for all k > 0.

Algorithm: when K and L are of degrees d and order r (r > d). αi ,3d,2d i ,4d,3d Evaluation For i = 0, . . . , r/d, computation of Φα and ΦL from K K ˜ and L (O(rd) ops).

Inner multiplication Computation of O(r/d) products of matrices with size d × d (O(rdω−1 ) ops). ˜ Interpolation Recovery of KL from Φαi ,4d,2d (O(rd) ops). KL

†

[BenoitBostanvanderHoeven, 2012] A.B., Alin Bostan and Joris van der Hoeven. Quasi-Optimal Multiplication of Linear Differential Operators, FOCS 2012. 34 / 36

VII Conclusion

35 / 36

Summary and Perspectives

Summary Fast algorithm for the computation of coefficients of Chebyshev expansions. Use of the Chebyshev recurrence relation for this computation. New tools to compute this recurrence relation: Fraction of recurrence operators. New and fast algorithm for the product of operators.

36 / 36

Summary and Perspectives

Summary Fast algorithm for the computation of coefficients of Chebyshev expansions. Use of the Chebyshev recurrence relation for this computation. New tools to compute this recurrence relation: Fraction of recurrence operators. New and fast algorithm for the product of operators. Perspectives Use the same idea to compute the coefficients of other generalized Fourier series. Use the fast algorithm for the product of operators to design new and fast algorithms for linear differential or recurrence operators.

36 / 36

Chebyshev Series vs Taylor Series II Convergence Domains: Taylor series: disk centered at x0 = 0, avoiding the singularities of f

Chebyshev series: elliptic disk with foci at ±1, avoids the singularities of f

Taylor series cannot converge over entire [−1,1] unless all singularities lie outside the unit circle. X Chebyshev series converges over entire [−1,1] as soon as there are no real singularities in [−1,1]. 37 / 36