Generalized Fourier Series for Solutions of Linear Differential Equations

Generalized Fourier Series for Solutions of Linear Differential Equations

Alexandre Benoit,

Joint work with Bruno Salvy

INRIA (France)

June 25, 2010

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

I Introduction

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Generalized Fourier Series

f(x) =X

anψn(x)

Some Examples

sin(x) = 2

∞

X

n=0

(−1)ⁿJ2n(x)

arccos (x) = 1

2πT0(x)−

∞

X

n=0

4

(2n+ 1)²πT2n+1(x) erf(x) = 2

∞

X

n=0

−1 4

ⁿ 1

√π(2n+ 1)n!¹F₁

n+¹₂ 2n+ 2

−x

Applications: Good approximation properties.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Generalized Fourier Series

f(x) =X

anψn(x)

Some Examples

sin(x) = 2

∞

X

n=0

(−1)ⁿJ2n(x)

arccos (x) = 1

2πT0(x)−

∞

X

n=0

4

(2n+ 1)²πT2n+1(x) erf(x) = 2

∞

X

n=0

−1 4

ⁿ 1

√π(2n+ 1)n!¹F₁

n+¹₂ 2n+ 2

−x

More generally (ψn(x))_n∈

Ncan be an orthogonal basis of a Hilbert space.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functionsψn(x) with two special properties Mult byx (Px)

Recx2(xψn(x)) = Recx1(ψn(x))

Example

Monomial polynomials (Mn=xⁿ)

All orthogonal polynomials Bessel functions

Legendre functions

xM_n=M_n+1 2xTn(x) =Tn+1(x) +T_n−1(x)

1

n(xJn+1−xJ_n−1) = 2Jn

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functionsψn(x) with two special properties Mult byx (Px)

Recx2(xψn(x)) = Recx1(ψn(x))

Example

Monomial polynomials (Mn=xⁿ)

All orthogonal polynomials Bessel functions

Legendre functions

Parabolic cylinder functions

xM_n=M_n+1 2xTn(x) =Tn+1(x) +T_n−1(x)

1

n(xJn+1−xJ_n−1) = 2Jn

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functionsψn(x) with two special properties Mult byx (P_x)

Recx2(xψn(x)) = Recx1(ψn(x)) Differentiation (P∂)

Rec∂2(ψ⁰_n(x)) = Rec∂1(ψn(x)) Example

Monomial polynomials Classical orthogonal polynomials

M_n⁰ =nMn−1

1

n+ 1Tn+1⁰ (x)− 1

n−1Tn−1⁰ (x) = 2Tn(x) 2J⁰_n(x) = Jn−1(x)−Jn+1(x)

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functionsψn(x) with two special properties Mult byx (P_x)

Recx2(xψn(x)) = Recx1(ψn(x)) Differentiation (P∂)

Rec∂2(ψ⁰_n(x)) = Rec∂1(ψn(x)) Example

Monomial polynomials Classical orthogonal polynomials Bessel functions

M_n⁰ =nMn−1

1

n+ 1Tn+1⁰ (x)− 1

n−1Tn−1⁰ (x) = 2Tn(x) 2J⁰n(x) = Jn−1(x)−Jn+1(x)

Our framework

Families of functionsψn(x) with two special properties Mult byx (P_x)

Recx2(xψn(x)) = Recx1(ψn(x)) Differentiation (P∂)

Rec_∂2(ψ⁰_n(x)) = Rec_∂1(ψ_n(x)) This is our data-structure forψ_n(x)

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Main Idea

Main Idea

Ifψn(x) satisfies (Px) and (P∂), for any f(x) =P

anψn(x) solution of a linear differentialequation with polynomial coefficients, the coefficientsan

are cancelled by alinear recurrence relationwith polynomial coefficients.

Application:

Efficient numerical computation of the coefficients.

Computation of closed-form for the coefficients (when possible).

Main Idea

Main Idea

Ifψn(x) satisfies (Px) and (P∂), for any f(x) =P

anψn(x) solution of a linear differentialequation with polynomial coefficients, the coefficientsan

are cancelled by alinear recurrence relationwith polynomial coefficients.

Application:

Efficient numerical computation of the coefficients.

Computation of closed-form for the coefficients (when possible).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Previous work

Clenshaw (1957): numerical schemeto compute the coefficients when ψ_n(x) =T_n(x) (Chebyshevseries).

Lewanowicz (1976-2004): algorithms to compute a recurrence relation whenψn is anorthogonal or semi-orthogonal polynomials family.

Rebillard and Zakrajˇsek (2006): General algorithmcomputing a recurrence relation whenψn is a family of hypergeometric polynomials

Benoit and Salvy (2009) : Simple unified presentationand complexity analysis of the previous algorithms usingFractions of recurrence relations whenψn=Tn. New and fast algorithm to compute the Chebyshev recurrence.

New Results (2010)

Simple unified presentation of the previous algorithms using Pairs of recurrence relations.

New general algorithmcomputing the recurrence relation of the coefficients for a Generalized Fourier Series whenψn(x) satisfies (Px) and (P∂).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

II Pairs of Recurrence Relations

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Examples: Chebyshev case (f (x ) = P

u

T

(x ))

Basic rules:

xf =X

anTn (Px)

−−→ an=un−1+un+1

2 f⁰=X

b_nT_n (P_∂)

−−→ b_n−1−b_n+1= 2nu_n.

Combine:

f⁰+ 2xf =X

c_nT_n (P_∂+ 2P_x)

−−−−−−−→ c_n−1−c_n+1= Rec₁(u_n). Application: Chebyshev series forexp(−x²).

(f⁰+ 2xf)⁰ =X

dnTn (P∂)

−−→ d_n−1−dn+1= 2ncn,

→ Rec2(dn) = Rec3(un), (f⁰+ 2xf)⁰−2f =X

enTn → Rec4(en) = Rec5(un). Application: Chebyshev series forerf(x).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Examples: Chebyshev case (f (x ) = P

u

T

(x ))

Basic rules:

xf =X

anTn (Px)

−−→ an=un−1+un+1

2 f⁰=X

b_nT_n (P_∂)

−−→ b_n−1−b_n+1= 2nu_n. Combine:

f⁰+ 2xf =X

c_nT_n (P_∂+ 2P_x)

−−−−−−−→ c_n−1−c_n+1= Rec₁(u_n).

Application: Chebyshev series forexp(−x²).

(f⁰+ 2xf)⁰ =X

dnTn (P∂)

−−→ d_n−1−dn+1= 2ncn,

→ Rec2(dn) = Rec3(un), (f⁰+ 2xf)⁰−2f =X

enTn → Rec4(en) = Rec5(un). Application: Chebyshev series forerf(x).

Examples: Chebyshev case (f (x ) = P

u

T

(x ))

Basic rules:

xf =X

anTn (Px)

−−→ an=un−1+un+1

2 f⁰=X

b_nT_n (P_∂)

−−→ b_n−1−b_n+1= 2nu_n. Combine:

f⁰+ 2xf =X

c_nT_n (P_∂+ 2P_x)

−−−−−−−→ c_n−1−c_n+1= Rec₁(u_n).

Application: Chebyshev series forexp(−x²).

(f⁰+ 2xf)⁰=X

dnTn (P∂)

−−→ d_n−1−dn+1= 2ncn,

→ Rec2(dn) = Rec3(un),

0 0− X

→

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33))

GivenRec1 andRec2, there exists a recurrence relation Recand a pair

Recg₁,Recg₂

such that for all sequences(u_n)_n∈

N:

Rec (un) =Recg1◦Rec1(un) =Recg2◦Rec2(un)

The LCLM is the recurrence relation Rec withminimal order. Computation : Euclidean algorithm.

Operation 1: Addition

Rec₁(a_n) = Rec₃(u_n), Rec₂(b_n) = Rec₄(u_n)

→Rec(an+bn) =

Recg1◦Rec3+Recg2◦Rec4

(un). Operation 2: Composition

Rec1(an) = Rec3(un), Rec4(bn) = Rec2(an)

→Recg1◦Rec2(un) =Recg2◦Rec3(bn).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33))

GivenRec1 andRec2, there exists a recurrence relation Recand a pair

Recg₁,Recg₂

such that for all sequences(u_n)_n∈

N:

Rec (un) =Recg1◦Rec1(un) =Recg2◦Rec2(un) The LCLM is the recurrence relation Rec withminimal order.

Computation : Euclidean algorithm. Operation 1: Addition

Rec₁(a_n) = Rec₃(u_n), Rec₂(b_n) = Rec₄(u_n)

→Rec(an+bn) =

Recg1◦Rec3+Recg2◦Rec4

(un). Operation 2: Composition

Rec1(an) = Rec3(un), Rec4(bn) = Rec2(an)

→Recg1◦Rec2(un) =Recg2◦Rec3(bn).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33))

GivenRec1 andRec2, there exists a recurrence relation Recand a pair

Recg₁,Recg₂

such that for all sequences(u_n)_n∈

N:

Rec (un) =Recg1◦Rec1(un) =Recg2◦Rec2(un) The LCLM is the recurrence relation Rec withminimal order.

Computation : Euclidean algorithm.

Operation 1: Addition

Rec₁(a_n) = Rec₃(u_n), Rec₂(b_n) = Rec₄(u_n)

→Rec(an+bn) =

Recg1◦Rec3+Recg2◦Rec4

(un). Operation 2: Composition

Rec1(an) = Rec3(un), Rec4(bn) = Rec2(an)

→Recg1◦Rec2(un) =Recg2◦Rec3(bn).

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33))

GivenRec1 andRec2, there exists a recurrence relation Recand a pair

Recg₁,Recg₂

such that for all sequences(u_n)_n∈

N:

Rec (un) =Recg1◦Rec1(un) =Recg2◦Rec2(un) The LCLM is the recurrence relation Rec withminimal order.

Computation : Euclidean algorithm.

Operation 1: Addition

Rec₁(a_n) = Rec₃(u_n), Rec₂(b_n) = Rec₄(u_n)

→Rec(an+bn) =

Recg1◦Rec3+Recg2◦Rec4

(un).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Main Result

Main Result : Morphism

There exists a morphism such that iff =Punψn(x) andg =Pvnψn(x) are related byL(f) =g (La linear differential operator), then:

ϕ(L) = (Rec1,Rec2) with Rec1(un) = Rec2(vn) In particular ifL(f) = 0, thenRec1(un) = 0.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Definition of the Morphism ϕ

f =X

unψn(x) g =X

vnψn(x)

Rec_x2(xψ_n(x)) = Rec_x1(ψ_n(x))

ifxf =g, then

Recx2(un) = Recx1(vn) ϕ(x)

Rec∂2(ψ_n⁰(x)) = Rec∂1(ψn(x))

iff⁰=g, then

Rec∂1(un) = Rec∂2(vn) ϕ(∂)

Example for Chebyshev series: 2xTn(x) =Tn+1(x) +Tn−1(x)

T_n+1⁰ (x)

n+ 1 −T_n−1⁰ (x)

n−1 = 2T_n(x)

u_n+1+u_n−1= 2v_n 2un= 1

n(vn−1−vn+1) ϕ

Example for Bessel series 1

n(xJn+1−xJn−1) = 2Jn

2J⁰_n(x) = J_n−1(x)−J_n+1(x)

2un= v_n+1

n+ 1 + v_n−1 n−1 un+1−un−1= 2vn

ϕ

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Definition of the Morphism ϕ

f =X

unψn(x) g =X

vnψn(x)

Rec_x2(xψ_n(x)) = Rec_x1(ψ_n(x))

ifxf =g, then

Recx2(un) = Recx1(vn) ϕ(x)

Rec∂2(ψ_n⁰(x)) = Rec∂1(ψn(x))

iff⁰=g, then

Rec∂1(un) = Rec∂2(vn) ϕ(∂)

Example for Chebyshev series:

2xTn(x) =Tn+1(x) +T_n−1(x) T_n+1⁰ (x)

n+ 1 −T_n−1⁰ (x)

n−1 = 2T_n(x)

u_n+1+u_n−1= 2v_n 2un=1

n(vn−1−vn+1) ϕ

Example for Bessel series

1 2u = v_n+1

+ v_n−1

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

General Algorithm

Recall

Definition ofϕ(x) andϕ(∂)

Algorithm to compute addition and composition between two pairs

General Algorithm

We deduce from this morphism a general Horner-likealgorithm to compute the recurrence relationsatisfy by the coefficients of a generalized Fourier series solution of a linear differential equation.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

General Algorithm

Recall

Definition ofϕ(x) andϕ(∂)

Algorithm to compute addition and composition between two pairs General Algorithm

We deduce from this morphism a general Horner-likealgorithm to compute the recurrence relationsatisfy by the coefficients of a generalized Fourier series solution of a linear differential equation.

III Recurrences of Smaller Order

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1,Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair“

Recg1,Recg2

”

with the following relations for all sequences (un)_n∈

Nand (vn)_n∈

N: Rec◦Recg1(un) = Rec1(un) Rec◦Recg2(vn) = Rec2(vn)

The orders of the recurrence relationsRecgi are at most those of Reci. Aim:

Find cases where for all coefficients of Generalized Fourier series (un,vn), we have :

Rec1(un) = Rec2(vn) =⇒Recg1(un) =Recg2(vn)

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1,Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair“

Recg1,Recg2

”

with the following relations for all sequences (un)_n∈

Nand (vn)_n∈

N: Rec◦Recg1(un) = Rec1(un) Rec◦Recg2(vn) = Rec2(vn)

Theordersof the recurrence relationsRecgi areat mostthose of Reci.

Aim:

Find cases where for all coefficients of Generalized Fourier series (un,vn), we have :

Rec1(un) = Rec2(vn) =⇒Recg1(un) =Recg2(vn)

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1,Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair“

Recg1,Recg2

”

with the following relations for all sequences (un)_n∈

Nand (vn)_n∈

N: Rec◦Recg1(un) = Rec1(un) Rec◦Recg2(vn) = Rec2(vn)

The orders of the recurrence relationsRecgi are at most those of Reci. Aim:

Find cases where for all coefficients of Generalized Fourier series (un,vn), we have :

Rec (u ) = Rec (v ) =⇒Rec (u ) =Rec (v )

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

GCD for reduction of order

Theorem

Given L a linear differential operator, f =P

unψn(x), g=P

vnψn(x) such that L(f) =g and a pair(Rec1,Rec2) =ϕ(L). We have

Recg1(un) =Recg2(vn)

ifψn is any of:

Classical orthogonal polynomials Bessel functions

Hypergeometric polynomials

Application: Adaptation of the previous algorithm At the end of the previous algorithm, add a final step:

Remove theGCLD of the two recurrence relations of the pair.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

GCD for reduction of order

Theorem

Given L a linear differential operator, f =P

unψn(x), g=P

vnψn(x) such that L(f) =g and a pair(Rec1,Rec2) =ϕ(L). We have

Recg1(un) =Recg2(vn) ifψn is any of:

Classical orthogonal polynomials Bessel functions

Hypergeometric polynomials

Application: Adaptation of the previous algorithm At the end of the previous algorithm, add a final step:

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Example of reduction for Chebyshev series

p1−x²=X

n∈N

4

π(2n+ 1)T_2n(x) =X

n∈N

c_nT_n(x)

√

1−x²is the solution of the differential equation:

xy(x) + (1−x²)y⁰(x) = 0

With the general algorithm we obtain the pair of recurrence relations :

Rec₁(u_n) = (n+3)u_n+2−2nu_n+(n−3)u_n−2and Rec₂(v_n) = 2 (−v_n+1+v_n−1). We deduce : (n+ 3)c_n+2−2nc_n+ (n−3)c_n−2= 0.

Recg1(un) = (n+ 2)un+1−(n−2)un−1andRecg2(vn) =−2vn+1+ 2vn−1. We deduce : (n+ 2)cn+1−(n−2)c_n−1= 0.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Example of reduction for Chebyshev series

p1−x²=X

n∈N

4

π(2n+ 1)T_2n(x) =X

n∈N

c_nT_n(x)

√

1−x²is the solution of the differential equation:

xy(x) + (1−x²)y⁰(x) = 0

With the general algorithm we obtain the pair of recurrence relations :

Rec₁(u_n) = (n+3)u_n+2−2nu_n+(n−3)u_n−2and Rec₂(v_n) = 2 (−v_n+1+v_n−1). We deduce : (n+ 3)c_n+2−2nc_n+ (n−3)c_n−2= 0.

Recg1(un) = (n+ 2)un+1−(n−2)un−1andRecg2(vn) =−2vn+1+ 2vn−1. We deduce : (n+ 2)cn+1−(n−2)c_n−1= 0.

Example of reduction for Chebyshev series

p1−x²=X

n∈N

4

π(2n+ 1)T_2n(x) =X

n∈N

c_nT_n(x)

√

1−x²is the solution of the differential equation:

xy(x) + (1−x²)y⁰(x) = 0

With the general algorithm we obtain the pair of recurrence relations :

Rec₁(u_n) = (n+3)u_n+2−2nu_n+(n−3)u_n−2and Rec₂(v_n) = 2 (−v_n+1+v_n−1). We deduce : (n+ 3)c_n+2−2nc_n+ (n−3)c_n−2= 0.

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

IV Conclusion

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Conclusion

Contributions:

Use of Pairs of recurrence relations.

New general algorithm.

Use of the GCLD to reduce order of the recurrence.

Perspectives:

Computation of the recurrence ofminimal order. Numerical computation of the coefficients. Closed form for the coefficients.

Example erf(x) =

∞

X

n=0

24⁻ⁿ(−1)ⁿ1F₁(n+ 1/2; 2n+ 2;−1)

√π(2n+ 1)n! T2n+1(x).

Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Conclusion

Contributions:

Use of Pairs of recurrence relations.

New general algorithm.

Use of the GCLD to reduce order of the recurrence.

Perspectives:

Computation of the recurrence ofminimal order.

Numerical computation of the coefficients.

Closed form for the coefficients.

Example erf(x) =

∞

X

n=0

24⁻ⁿ(−1)ⁿ1F1(n+ 1/2; 2n+ 2;−1)

√π(2n+ 1)n! T2n+1(x).