Algorithmique semi-num´erique rapide des s´eries de Tchebychev

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

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Summary

Goal

Using computer algebra, findfast algorithms to compute guaranteed polynomial approximationsof 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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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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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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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.

2 / 36

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)

Z 1

−1

Tn(x)Tm(x)

√1−x² dx=







0 if m6=n π if m= 0

π

2 otherwise

Tn+1(x) = 2xTn(x)−Tn−1(x) (1−x²)T_n⁰⁰(x)−xT_n⁰(x) +n²Tn(x) = 0

Tn(x) =

bn/2c

X

k=0

(−1)^k2^n−2k−1 n n−k

n−k k

x^n−2k

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

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Basic Properties of Chebyshev Polynomials

Tn(cos(θ)) = cos(nθ)

Z 1

−1

Tn(x)Tm(x)

√1−x² dx=







0 if m6=n π if m= 0

π

2 otherwise

Tn+1(x) = 2xTn(x)−Tn−1(x) (1−x²)T_n⁰⁰(x)−xT_n⁰(x) +n²Tn(x) = 0

Tn(x) =

bn/2c

X

k=0

(−1)^k2^n−2k−1 n n−k

n−k k

x^n−2k

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

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Basic Properties of Chebyshev Polynomials

Tn(cos(θ)) = cos(nθ)

Z 1

−1

Tn(x)Tm(x)

√1−x² dx=







0 if m6=n π if m= 0

π

2 otherwise

Tn+1(x) = 2xTn(x)−Tn−1(x) (1−x²)T_n⁰⁰(x)−xT_n⁰(x) +n²Tn(x) = 0

Tn(x) =

bn/2c

X

k=0

(−1)^k2^n−2k−1 n n−k

n−k k

x^n−2k

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

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Approximation by Taylor or Chebyshev Series

Two approximations of a functionf: by a Taylor series

f =

+∞

X

n=0

unxⁿ,

un= f⁽ⁿ⁾(0) n! ,

or by Chebyshev series

f =

+∞

X

n=0

cnTn(x),

cn= 1 π

Z 1

−1

Tn(t) f(t)

√

1−t²dt.

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Approximation by Taylor or Chebyshev Series

Two approximations of a functionf: by a Taylor series

f =

+∞

X

n=0

unxⁿ,

un= f⁽ⁿ⁾(0) n! ,

or by Chebyshev series

f =

+∞

X

n=0

cnTn(x),

cn = 1 π

Z 1

−1

Tn(t) f(t)

√

1−t²dt.

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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 −f_nk∞≤

4 + 4

π²log(n+ 1)

| {z }

Λ_n

kf−p_nk∞.

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

It’s faster to compute f_n instead ofp_n

x

−1 −1/2 1/2 1

−5×10⁻³ 5×10⁻³

Chebyshev truncation of degree4 Best approximant of degree4

error of the approximation ofexp

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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 −f_nk∞≤

4 + 4

π²log(n+ 1)

| {z }

Λ_n

kf−p_nk∞.

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

It’s faster to compute f_n instead ofp_n

x

−1 −1/2 1/2 1

−5×10⁻³ 5×10⁻³

Chebyshev truncation of degree4 Best approximant of degree4

error of the approximation ofexp

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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 −f_nk∞≤

4 + 4

π²log(n+ 1)

| {z }

Λ_n

kf−p_nk∞.

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

It’s faster to compute f_n instead ofp_n

x

−1 −1/2 1/2 1

−5×10⁻³ 5×10⁻³

Chebyshev truncation of degree4 Best approximant of degree4

error of the approximation ofexp

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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 −f_nk∞≤

4 + 4

π²log(n+ 1)

| {z }

Λ_n

kf−p_nk∞.

Λ10= 4.93...→<3 bits Λ30= 5.37...→<3 bits Λ₁₀₀= 5.87...→<3 bits Λ1000= 6.80...→<3 bits It’s faster to compute fn

instead ofp_n

x

−1 −1/2 1/2 1

−5×10⁻³ 5×10⁻³

Chebyshev truncation of degree4 Best approximant of degree4

error of the approximation ofexp

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Contributions

Fastpolynomial conversion, Monomial bases7→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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Contributions

Fast polynomial conversion, Monomial bases7→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 tocompute this recurrenceusing 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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Contributions

Fast polynomial conversion, Monomial bases7→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 tocompute the coefficients numerically(with Mioara Jolde¸s and Marc Mezzarobba, Chap. 6 and 7)

Implementation in theDDMF(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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Contributions

Fast polynomial conversion, Monomial bases7→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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Contributions

Fast polynomial conversion, Monomial bases7→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 ofgeneralized Fourier series (with Bruno Salvy, Chap. 8)

Fast algorithm for theproduct of differential operators(with Alin Bostan and Joris van der Hoeven,FOCS’12)

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Contributions

Fast polynomial conversion, Monomial bases7→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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DDMF

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The Special Function

1. Differential equation

The function satisfies the differential equation

with initial values , .

2. Plot

3. Numerical Evaluation 4. Symmetry

5. Taylor Expansion at 0

6. Local Expansions at Singularities and at Infinity 7. Hypergeometric Representation

8. Chebyshev Expansion over

Chebyshev expansion:

First terms and polynomial approximation:

order =

The coefficients in the satisfy the

recurrence

erf ( ) x

erf( )x

2 y y

!d dx ( )x

"

x+ d²

dx² ( ) = 0x y( ) = 00 (y⁰) 0( ) = 2^p1_!

[ ! 1; 1 ]

erf( ) =x X¹ :

n=0

2 p!(2n+ 1)n!

4^!n(!1)ⁿ1 1F(1=2+n;2n+ 2;!1)T2 +1n ( )x

erf( ) = 0x :904347T₁( )x !0:0661130T₃( ) + 0x :00472936T₅( ) +x : : : erf( )x "1:12633280x!0:35903920x³+ 0:07566976x⁵:

cn erf( ) =x P₁ T(x)

n=0cn n

n :

( ²+ 3n)c n( ) + 2( n³+ 12n²+ 24n+ 1 )6 c n( + 2) +(!n²!5n!4)c n( + 4) = 0

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Chebyshev expansion

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

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DDMF

08/07/12 17:50 The Special Function erf(x)

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The Special Function

1. Differential equation

The function satisfies the differential equation

with initial values , .

2. Plot

3. Numerical Evaluation 4. Symmetry

5. Taylor Expansion at 0

6. Local Expansions at Singularities and at Infinity 7. Hypergeometric Representation

8. Chebyshev Expansion over

Chebyshev expansion:

First terms and polynomial approximation:

order =

The coefficients in the satisfy the

recurrence

erf ( ) x

erf( )x

2 y y

!d dx ( )x

"

x+ d²

dx² ( ) = 0x y( ) = 00 (y⁰) 0( ) = 2^p1_!

[ ! 1; 1 ]

erf( ) =x X¹ :

n=0

2 p!(2n+ 1)n!

4^!n(!1)ⁿ1 1F(1=2+n;2n+ 2;!1)T2 +1n ( )x

erf( ) = 0x :904347T₁( )x !0:0661130T₃( ) + 0x :00472936T₅( ) +: : :x erf( )x "1:12633280x!0:35903920x³+ 0:07566976x⁵:

cn erf( ) =x P₁ T(x)

n=0cn n

n :

( ²+ 3n)c n( ) + 2( n³+ 12n²+ 24n+ 1 )6 c n( + 2) +(!n²!5n!4)c n( + 4) = 0

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Chebyshev expansion

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II Chebyshev Recurrence

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Chebyshev Recurrence

Theorem (Paszkowski 1975)

IfP

cnTn(x)is solution of a linear differential equation with polynomial coefficients, then the coefficientscn are solution of a linear recurrence with polynomial coefficients.

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DDMF

08/07/12 17:50 The Special Function erf(x)

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The Special Function

1. Differential equation

The function satisfies the differential equation

with initial values , .

2. Plot

3. Numerical Evaluation 4. Symmetry

5. Taylor Expansion at 0

6. Local Expansions at Singularities and at Infinity 7. Hypergeometric Representation

8. Chebyshev Expansion over

Chebyshev expansion:

First terms and polynomial approximation:

order =

The coefficients in the satisfy the

recurrence

erf ( ) x

erf( )x

2 y y

!d dx ( )x

"

x+ d²

dx² ( ) = 0x y( ) = 00 (y⁰) 0( ) = 2^p1_!

[ ! 1; 1 ]

erf( ) =x X¹ :

n=0

2 p!(2n+ 1)n!

4^!n(!1)ⁿ1 1F(1=2+n;2n+ 2;!1)T2 +1n ( )x

erf( ) = 0x :904347T₁( )x !0:0661130T₃( ) + 0x :00472936T₅( ) +: : :x erf( )x "1:12633280x!0:35903920x³+ 0:07566976x⁵:

cn erf( ) =x P₁ T(x)

n=0cn n

n :

( ²+ 3n)c n( ) + 2( n³+ 12n²+ 24n+ 1 )6 c n( + 2) +(!n²!5n!4)c n( + 4) = 0

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Chebyshev expansion

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

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Morphisms of Rings of Operators (S · u

= u

)

Taylor series (f := P unxⁿ) xf=X

unxⁿ⁺¹=X

un−1xⁿ, f⁰=X

nunxⁿ⁻¹=X

(n+ 1)un+1xⁿ

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Morphisms of Rings of Operators (S · u

= u

)

Taylor series (f := P unxⁿ) xf=X

unxⁿ⁺¹=X

un−1xⁿ, f⁰=X

nunxⁿ⁻¹=X

(n+ 1)un+1xⁿ

x7→X :=S⁻¹,

d

dx7→D:= (n+ 1)S.

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Morphisms of Rings of Operators (S · u

= u

)

Taylor series (f := P unxⁿ) xf=X

unxⁿ⁺¹=X

un−1xⁿ, f⁰=X

nunxⁿ⁻¹=X

(n+ 1)un+1xⁿ

x7→X :=S⁻¹,

d

dx7→D:= (n+ 1)S.

(4 +x²)

d

dx 2

+ 2x_dx^d

7→(4+S⁻²)(n+1)(n+2)S²+2S⁻¹(n+ 1)S

= (n+ 1) 4(n+ 2)S²+n 4(n+ 2)un+2+nun= 0

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Morphisms of Rings of Operators (S · u

= u

)

Monomial Basisxⁿ=M_n(x) xMn(x) =Mn+1(x), M_n⁰(x) =nMn−1(x).

Chebyshev series

xTn(x) =1/2 (Tn+1(x) +Tn−1(x)) T_n⁰(x) =n(T_n−1(x)−T_n+1(x))

2(1−x²) . x7→X :=S⁻¹,

d

dx7→D:= (n+ 1)S.

x7→X := S+S⁻¹

2 ,

d

dx7→D:= (n+ 1)S−(n−1)S⁻¹

2(1−X²) = 2n S⁻¹−S.

(4 +x²)

d

dx 2

+ 2xd dx 7→(4+S⁻²)(n+1)(n+2)S²+2S⁻¹(n+1)S

= (n+ 1) 4(n+ 2)S²+n 4(n+ 2)un+2+nun= 0

(n−1)(n+ 1) (n+ 2)S²+ 18n+ (n−2)S⁻² ((n−1)S²−2n+ (n+ 1)S⁻²) ,

(n+ 2)cn+2+ 18ncn+ (n−2)cn−2= 0.

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Morphisms of Rings of Operators (S · u

= u

)

Monomial Basisxⁿ=Mn(x) xM_n(x) =M_n+1(x), M_n⁰(x) =nM_n−1(x).

Chebyshev series

xT_n(x) =1/2 (T_n+1(x) +T_n−1(x)) T_n⁰(x) =n(T_n−1(x)−Tn+1(x))

2(1−x²) . x7→X :=S⁻¹,

d

dx7→D:= (n+ 1)S.

x7→X := ^S+S₂⁻¹, d

dx7→D:= (n+ 1)S−(n−1)S⁻¹

2(1−X²) = 2n S⁻¹−S.

(4 +x²)

d

dx 2

+ 2xd dx 7→(4+S⁻²)(n+1)(n+2)S²+2S⁻¹(n+1)S

= (n+ 1) 4(n+ 2)S²+n 4(n+ 2)un+2+nun= 0

(n−1)(n+ 1) (n+ 2)S²+ 18n+ (n−2)S⁻² ((n−1)S²−2n+ (n+ 1)S⁻²) ,

(n+ 2)cn+2+ 18ncn+ (n−2)cn−2= 0.

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Morphisms of Rings of Operators (S · u

= u

)

Monomial Basisxⁿ=M_n(x) xMn(x) =Mn+1(x), M_n⁰(x) =nMn−1(x).

Chebyshev series

xTn(x) =1/2 (Tn+1(x) +Tn−1(x)) T_n⁰(x) =^n(Tn−1_2(1−x^(x)−T2ⁿ⁺¹) ^(x)).

x7→X:=S⁻¹,

d

dx7→D:= (n+ 1)S.

x7→X:= S+S⁻¹

2 ,

d

dx7→D:=(n+1)S−(n−1)S⁻¹

2(1−X²) = 2n S⁻¹−S.

(4 +x²)

d

dx 2

+ 2xd dx 7→(4+S⁻²)(n+1)(n+2)S²+2S⁻¹(n+1)S

= (n+ 1) 4(n+ 2)S²+n 4(n+ 2)un+2+nun= 0

(n−1)(n+ 1) (n+ 2)S²+ 18n+ (n−2)S⁻² ((n−1)S²−2n+ (n+ 1)S⁻²) ,

(n+ 2)cn+2+ 18ncn+ (n−2)cn−2= 0.

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Morphisms of Rings of Operators (S · u

= u

)

Monomial Basisxⁿ=Mn(x) xMn(x) =Mn+1(x), M_n⁰(x) =nM_n−1(x).

Chebyshev series

xTn(x) =1/2 (Tn+1(x) +T_n−1(x)) T_n⁰(x) =n(Tn−1(x)−Tn+1(x))

2(1−x²) . x7→X :=S⁻¹,

d

dx7→D:= (n+ 1)S.

x7→X := S+S⁻¹

2 ,

d

dx7→D:= (n+ 1)S−(n−1)S⁻¹

2(1−X²) = 2n S⁻¹−S.

(4 +x²) d

dx

2

+ 2x_dx^d

7→(4+S⁻²)(n+1)(n+2)S²+2S⁻¹(n+1)S

= (n+ 1) 4(n+ 2)S²+n 4(n+ 2)un+2+nun= 0

(n−1)(n+1)(^(n+2)S2+18n+(n−2)S⁻²)

((n−1)S²−2n+(n+1)S⁻²) , (n+ 2)cn+2+ 18ncn+ (n−2)cn−2= 0.

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Morphisms of Rings of Operators (S · u

= u

)

Monomial Basisxⁿ=M_n(x) xMn(x) =Mn+1(x), M_n⁰(x) =nMn−1(x).

Chebyshev series

xTn(x) =1/2 (Tn+1(x) +Tn−1(x)) T_n⁰(x) =n(T_n−1(x)−T_n+1(x))

2(1−x²) . x7→X :=S⁻¹,

d

dx7→D:= (n+ 1)S.

x7→X := S+S⁻¹

2 ,

d

dx7→D:= (n+ 1)S−(n−1)S⁻¹

2(1−X²) = 2n S⁻¹−S.

(4 +x²)

d

dx 2

+ 2xd dx 7→(4+S⁻²)(n+1)(n+2)S²+2S⁻¹(n+1)S

= (n+ 1) 4(n+ 2)S²+n 4(n+ 2)un+2+nun= 0

(n−1)(n+ 1) (n+ 2)S²+ 18n+ (n−2)S⁻² ((n−1)S²−2n+ (n+ 1)S⁻²) ,

(n+ 2)cn+2+ 18ncn+ (n−2)cn−2= 0.

13 / 36

Morphisms of Rings of Operators 2

Definition

The Chebyshev morphism ϕis defined by:

ϕ(x) = 1

2 S+S⁻¹ andϕ

d dx

= 2n

−S+S⁻¹.

Theorem (BenoitSalvy2009)

f∈ C^k,L=P pi d

dx

i

a linear differential operator of orderk such thatL·f= 0.

Suppose that either of the following holds:

(i).

Z 1

−1

f^(k)(x)

√1−x²dxis convergent;

(ii).

Z 1

−1

(1−x²)^kf^(k)(x)

√1−x² dxis convergent and(1−x²)ⁱ|pi,i= 0, . . . ,k.

Then, the Chebyshev coefficients off are cancelled by anynumerator ofϕ(L).

14 / 36

New and Fast Algorithm

Theorem

If the order is at mostk and the degrees ofpi are at mostk,

Paszkowski (1975) and Lewanowicz (1976): O(k⁴)arithmetic operations.

New: O(k^ω)arithmetic operations.

ω is a feasible exponent for matrix multiplication (2≤ω≤3)

Ideafor 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, ISSAC2009.

14 / 36

New and Fast Algorithm

Theorem

If the order is at mostk and the degrees ofpi are at mostk,

Paszkowski (1975) and Lewanowicz (1976): O(k⁴)arithmetic operations.

New: O(k^ω)arithmetic operations.

ω is a feasible exponent for matrix multiplication (2≤ω≤3)

Ideafor 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, ISSAC2009.

15 / 36

III Computation of the Chebyshev Coefficients

by Hadamard Product

16 / 36

Hadamard Product for Chebyshev Expansions

Hyp: f is analytic in the closed unit disk. Then, there existsun andcn such that f(x) =X

n∈N

unxⁿ =X

n∈N

cnTn(x).

Inner product withT_n(x) c_n(t) = 2

π Z 1

−1

P

k∈Nu_kx^kt^kT_n(x)

√1−x² dx= 2 π

X

k∈N

u_kt^k Z 1

−1

x^kT_n(x)

√1−x²dx

| {z }

independant off

=X

k∈N

u_kg_n,kt^k.

gn(t) =X

k∈N

gn,kt^k=X

k∈N

2^1−2k−n

2k+n k+n

t^2k+n= 2tⁿ (1 +√

1−t²)ⁿ√ 1−t². cn(t) =f(t)gn(t)

†extension of Thacher 1964

9 / 36

DDMF

08/07/12 17:50 The Special Function erf(x)

Page 1 of 2 http://ddmf.msr-inria.inria.fr/1.7.2/ddmf?service=SpecialFunction&…ath&mac=Hv49g3JXhLJiYMAimJ3rVrlif0U&sf_id=sf_erf&parameters=q64FAA

The Special Function

1. Differential equation

The function satisfies the differential equation

with initial values , .

2. Plot

3. Numerical Evaluation 4. Symmetry

5. Taylor Expansion at 0

6. Local Expansions at Singularities and at Infinity 7. Hypergeometric Representation

8. Chebyshev Expansion over

Chebyshev expansion:

First terms and polynomial approximation:

order =

The coefficients in the satisfy the

recurrence

erf ( ) x

erf( )x

2 y y

!d dx ( )x

"

x+ d²

dx² ( ) = 0x y( ) = 00 (y⁰) 0( ) = 2^p1_!

[ ! 1; 1 ]

erf( ) =x X¹ :

n=0

2 p!(2n+ 1)n!

4^!n(!1)ⁿ1 1F(1=2+n;2n+ 2;!1)T2 +1n ( )x

erf( ) = 0x :904347T₁( )x !0:0661130T₃( ) + 0x :00472936T₅( ) +: : :x erf( )x "1:12633280x!0:35903920x³+ 0:07566976x⁵:

cn erf( ) =x P₁ T(x)

n=0cn n

n :

( ²+ 3n)c n( ) + 2( n³+ 12n²+ 24n+ 1 )6 c n( + 2) +(!n²!5n!4)c n( + 4) = 0

Home Glossary

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6 Submit

link rendering

Chebyshev expansion

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

16 / 36

Hadamard Product for Chebyshev Expansions

Hyp: f is analytic in the closed unit disk. Then, there existsun andcn such that f(xt) =X

n∈N

unxⁿtⁿ =X

n∈N

cn(t)Tn(x).

Inner product withT_n(x) c_n(t) = 2

π Z 1

−1

P

k∈Nu_kx^kt^kT_n(x)

√1−x² dx= 2 π

X

k∈N

u_kt^k Z 1

−1

x^kT_n(x)

√1−x²dx

| {z }

independant off

=X

k∈N

u_kg_n,kt^k.

gn(t) =X

k∈N

gn,kt^k=X

k∈N

2^1−2k−n

2k+n k+n

t^2k+n= 2tⁿ (1 +√

1−t²)ⁿ√ 1−t². cn(t) =f(t)gn(t)

†extension of Thacher 1964

16 / 36

Hadamard Product for Chebyshev Expansions

Hyp: f is analytic in the closed unit disk. Then, there existsun andcn such that f(xt) =X

n∈N

unxⁿtⁿ =X

n∈N

cn(t)Tn(x).

Inner product withT_n(x) c_n(t) = 2

π Z 1

−1

P

k∈Nu_kx^kt^kT_n(x)

√1−x² dx= 2 π

X

k∈N

u_kt^k Z 1

−1

x^kT_n(x)

√1−x²dx

| {z }

independant off

=X

k∈N

u_kg_n,kt^k.

gn(t) =X

k∈N

gn,kt^k=X

k∈N

2^1−2k−n

2k+n k+n

t^2k+n= 2tⁿ (1 +√

1−t²)ⁿ√ 1−t². cn(t) =f(t)gn(t)

†extension of Thacher 1964

16 / 36

Hadamard Product for Chebyshev Expansions

Hyp: f is analytic in the closed unit disk. Then, there existsun andcn such that f(xt) =X

n∈N

unxⁿtⁿ =X

n∈N

cn(t)Tn(x).

Inner product withT_n(x) c_n(t) = 2

π Z 1

−1

P

k∈Nu_kx^kt^kT_n(x)

√1−x² dx= 2 π

X

k∈N

u_kt^k Z 1

−1

x^kT_n(x)

√1−x²dx

| {z }

independant off

=X

k∈N

u_kg_n,kt^k.

gn(t) =X

k∈N

gn,kt^k=X

k∈N

2^1−2k−n

2k+n k+n

t^2k+n = 2tⁿ (1 +√

1−t²)ⁿ√ 1−t².

cn(t) =f(t)gn(t)

†extension of Thacher 1964

16 / 36

Hadamard Product for Chebyshev Expansions

Hyp: f is analytic in the closed unit disk. Then, there existsun andcn such that f(xt) =X

n∈N

unxⁿtⁿ =X

n∈N

cn(t)Tn(x).

Inner product withT_n(x) c_n(t) = 2

π Z 1

−1

P

k∈Nu_kx^kt^kT_n(x)

√1−x² dx= 2 π

X

k∈N

u_kt^k Z 1

−1

x^kT_n(x)

√1−x²dx

| {z }

independant off

=X

k∈N

u_kg_n,kt^k.

gn(t) =X

k∈N

gn,kt^k=X

k∈N

2^1−2k−n

2k+n k+n

t^2k+n = 2tⁿ (1 +√

1−t²)ⁿ√ 1−t². cn(t) =f(t)gn(t)

†extension of Thacher 1964

17 / 36

Application to the Closed Form of Coefficients

Idea: Iff =Punxⁿ is m-hypergeometric, i.e. un+m/un is rational, then fgk

is 2m-hypergeometric.

Corollary(Luke, 1969):

pFq

a1;. . .;ap

b1;. . .;bq

xt

=X

k∈N

ck(t)Tk(x),

ck(t) = 2 2^k

(a1)k. . .(ap)k

(b₁)_k. . .(b_q)_k t^k k!^2pF2q+1

a₁+k

2 ;^a1+k+1₂ ;. . .;^ap₂^+k;^ap+k+1₂

b₁+k

2 ;^b1+k+1₂ ;. . .;^bq+k₂ ;^bq+k+1₂ ;k+ 1

t² 4^q−p+1

! .

17 / 36

Application to the Closed Form of Coefficients

Idea: Iff =Punxⁿ is m-hypergeometric, i.e. un+m/un is rational, then fgk

is 2m-hypergeometric.

Corollary(Luke, 1969):

pFq

a1;. . .;ap

b1;. . .;bq

xt

=X

k∈N

ck(t)Tk(x),

ck(t) = 2 2^k

(a1)k. . .(ap)k

(b₁)_k. . .(b_q)_k t^k k!^2pF2q+1

a₁+k

2 ;^a1+k+1₂ ;. . .;^ap₂^+k;^ap+k+1₂

b₁+k

2 ;^b1+k+1₂ ;. . .;^bq+k₂ ;^bq+k+1₂ ;k+ 1

t² 4^q−p+1

! .

18 / 36

Applications to Special Functions

Function Chebyshev expansion

exp(xt) P∞

n=0

02 In(t)Tn(x)

sin(xt) 2P

n∈N(−1)ⁿJ2n+1(t)T2n+1(x)

erf(xt) P

n∈N

√2 π

(−1)ⁿ 4ⁿ

t²ⁿ⁺¹ (2n+1)n! 1F₁

n+¹₂ 2n+2

−t²

T_2n+1(x)

Si(xt) P

n∈N

4⁻ⁿ(−1)ⁿ (2n+1)!(2n+1) 1F1

_n+1 2

2n+2,n+³₂

−¹₄t²

T2n+1(x)

Ai(xt) P

n∈N 0 1

81ⁿn! −³₁₄₄^2/3_{Γ(n+4/3) 2}¹ F_{5 1/2n+4/3;1/2n+5/6}

4/3;5/3;n+1;n+4/3;n+5/3

t⁶ 1296

+³

√3 3

2 Γ(n+2/3) 2F5

1/2n+2/3;1/2n+1/6 1/3;2/3;n+1;n+2/3;n+1/3

t⁶ 1296

! T3n(x) +· · ·

18 / 36

Applications to Special Functions

Function Chebyshev expansion

exp(xt) P∞

n=0

02 In(t)Tn(x)

sin(xt) 2P

n∈N(−1)ⁿJ2n+1(t)T2n+1(x)

erf(xt) P

n∈N

√2 π

(−1)ⁿ 4ⁿ

t²ⁿ⁺¹ (2n+1)n! 1F₁

n+¹₂ 2n+2

−t²

T_2n+1(x)

Si(xt) P

n∈N

4⁻ⁿ(−1)ⁿ (2n+1)!(2n+1) 1F1

_n+1 2

2n+2,n+³₂

−¹₄t²

T2n+1(x)

Ai(xt) P

n∈N 0 1

81ⁿn! −³₁₄₄^2/3_{Γ(n+4/3) 2}¹ F_{5 1/2n+4/3;1/2n+5/6}

4/3;5/3;n+1;n+4/3;n+5/3

t⁶ 1296

+³

√3 3

2 Γ(n+2/3) 2F5

1/2n+2/3;1/2n+1/6 1/3;2/3;n+1;n+2/3;n+1/3

t⁶ 1296

! T3n(x) +· · ·

19 / 36

Application to Numerical Computation

f(x) =X

n∈N

cn(1)Tn(x), with cn(t) =f(t) 2tⁿ (1 +√

1−t²)ⁿ√ 1−t²

cn(t) =P∞

k=0cn,kt^k is solution of a linear differential equation, thencn,k are solutions of a linear recurrence

Application: Algorithm for the computation ofcn(1)inO(log(˜ ⁻¹) +n)bit operations

INPUT:n,

OUTPUT:ecn(1)such that |ecn(1)−cn(1)|<

Computation of the recurrence satisfied byc_n,k, plus initial conditions

(O(n)˜ ops)

(Using binary splitting)

Use of the Mezzarobba-Salvy algorithm to computeN such that|P∞

n=N+1cn,k|<

(O(log(⁻¹))ops)

Computation ofPN

k=0cn,k using the recurrence

(O(N˜ ) = ˜O(log(⁻¹))ops)

(Using binary splitting)

To compute` coefficients, the complexity becomesO(`˜ log(⁻¹)).