Chebyshev Expansions for Solutions of Linear Differential Equations

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Chebyshev Expansions for Solutions of Linear Differential Equations

Alexandre Benoit,

Joint work with Bruno Salvy Joint-Centre MSR-INRIA

March 25, 2009

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I Introduction

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Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works

How to evaluate a function f in [−1, 1]?

Two representations off: in Taylor series

f =

+∞

X

n=0

cnxⁿ, cn= f⁽ⁿ⁾(0) n! ,

or in Chebyshev series

f =

+∞

X

n=0

tnTn(x),

tn= 1 π

Z 1

−1

Tn(t) f(t)

√

1−t²dt.

Projects using Chebyshev series to represent functions in Matlab : Chebfun, Miscfun.

How to computetn?

General case: numerical computation of the integral. Slow.

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How to evaluate a function f in [−1, 1]?

f =

+∞

X

n=0

f =

+∞

X

n=0

tnTn(x),

tn= 1 π

Z 1

−1

Tn(t) f(t)

√

1−t²dt.

How to computetn?

General case: numerical computation of the integral. Slow.

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How to evaluate a function f in [−1, 1]?

f =

+∞

X

n=0

f =

+∞

X

n=0

tnTn(x),

tn= 1 π

Z 1

−1

Tn(t) f(t)

√

1−t²dt.

How to computet ?

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Computation of Coefficients with Recurrences

Theorem (60’s)

If f is solution of a linear differential equation with polynomial coefficients, then the Chebyshev coefficients are cancelled by a linear recurrence with polynomial coefficients.

Applications:

Numerical computation of the coefficients.

Computation of closed-form for coefficients. Example (f(x) = arctan(x/2))

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Computation of Coefficients with Recurrences

Theorem (60’s)

Applications:

Computation of closed-form for coefficients. Example (f(x) = arctan(x/2))

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Computation of Coefficients with Recurrences

Theorem (60’s)

Applications:

Computation of closed-form for coefficients.

Example (f(x) = arctan(x/2))

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State of the Art

Clenshaw (1957): numerical scheme to compute the Chebyshev coefficients without computing all these integrals.

Fox and Parker (1968): method for the computation of the Chebyshev recurrence relations for differential equations of small orders.

Paszkowski (1975): algorithm for computing the Chebyshev recurrence relation.

Lewanowicz (1976): algorithm for computing a smaller order Chebyshev recurrence relation in some cases.

Rebillard (1998): new algorithm for computing the Chebyshev recurrence relation.

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New Results (2009)

A simple unified presentation of these algorithms using fractions of recurrence operators.

Complexity analysis of the existing algorithms (orderk, degreek) Paszkowski’s and Lewanowicz’s algorithms: O(k⁴) arithmetic operations inQ.

Rebillard’s algorithm: O(k⁵) arithmetic operations inQ. New fast algorithm: O(k^ω) arithmetic operations. Here,ω is a feasible exponent for matrix multiplication with coefficients inQ (ω≤3).

Implementation of algorithm in Maple.

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New Results (2009)

Complexity analysis of the existing algorithms (orderk, degree k) Paszkowski’s and Lewanowicz’s algorithms: O(k⁴)O(k⁴) arithmetic operations inQ.

Rebillard’s algorithm: O(k⁵)O(k⁵) arithmetic operations inQ.

New fast algorithm: O(k^ω) arithmetic operations. Here,ω is a feasible exponent for matrix multiplication with coefficients inQ (ω≤3).

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New Results (2009)

Complexity analysis of the existing algorithms (orderk, degree k) Paszkowski’s and Lewanowicz’s algorithms: O(k⁴)arithmetic operations inQ.

Rebillard’s algorithm: O(k⁵)arithmetic operations inQ. New fast algorithm: O(k^ω)arithmetic operations. Here,ω is a feasible exponent for matrix multiplication with coefficients inQ (ω≤3).

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New Results (2009)

Complexity analysis of the existing algorithms (orderk, degree k) Paszkowski’s and Lewanowicz’s algorithms: O(k⁴) arithmetic operations inQ.

Rebillard’s algorithm: O(k⁵) arithmetic operations inQ. New fast algorithm: O(k^ω) arithmetic operations. Here,ω is a feasible exponent for matrix multiplication with coefficients inQ (ω≤3).

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II Fractions of Recurrence Operators

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

Taylor series (f := P cnxⁿ) xf =X

cnxⁿ⁺¹=X cn−1xⁿ, f⁰=X

ncnxⁿ⁻¹=X

(n+ 1)cn+1xⁿ

x7→X :=S⁻¹, d

dx7→D:= (n+ 1)S.

(4 +x²)

„d dx

«2

+ 2x d 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)cn+2+ncn= 0

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

ncnxⁿ⁻¹=X

(n+ 1)cn+1xⁿ

x7→X :=S⁻¹,

d

dx7→D := (n+ 1)S.

(4 +x²)

„d dx

«2

+ 2x d 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)cn+2+ncn= 0

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

ncnxⁿ⁻¹=X

(n+ 1)cn+1xⁿ

x7→X :=S⁻¹, d

dx7→D:= (n+ 1)S.

(4 +x²)

„d dx

«2

+ 2x d 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)cn+2+ncn= 0

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

ncnxⁿ⁻¹=X

(n+ 1)cn+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)cn+2+ncn= 0

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

Monomial Basisxⁿ=Mn(x) xM_n(x) =M_n+1(x), (M_n(x))⁰ =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+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 d 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)c +nc = 0

(n−1)(n+ 1)`

(n+ 2)S²+ 18n+ (n−2)S⁻²´ ((n−1)S²−2n+ (n+ 1)S⁻²) , (n+ 2)tn+2+ 18ntn+ (n−2)tn−2= 0.

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

Monomial Basisxⁿ=Mn(x) xM_n(x) =M_n+1(x), (Mn(x))⁰ =nM_n−1(x).

Chebyshev series

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

+ 2x d 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)cn+2+ncn= 0

(n−1)(n+ 1)`

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

Chebyshev series

xT_n(x) =1/2 (T_n+1(x) +T_n−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

+ 2x d 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)cn+2+ncn= 0

(n−1)(n+ 1)`

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

Monomial Basisxⁿ=Mn(x) xMn(x) =Mn+1(x), (Mn(x))⁰ =nM_n−1(x).

Chebyshev series

xTn(x) =1/2 (Tn+1(x) +T_n−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

+ 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)cn+2+ncn= 0

(n−1)(n+1)(^(n+2)S²+18n+(n−2)S⁻²)

((n−1)S²−2n+(n+1)S⁻²) , (n+ 2)tn+2+ 18ntn+ (n−2)tn−2= 0.

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Morphisms of Rings of Operators (S · u

_n

= u

_n+1

)

Chebyshev series

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 d 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)c +nc = 0

(n−1)(n+ 1)`

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Introduction Fractions of Recurrence OperatorsAlgorithms Conclusion and Future Works

Ore Polynomials: Framework for Recurrence Operators

Pai(n)un+i is represented byP

ai(n)Sⁱ. These polynomials are non-commutative.

Multiplication defined by: Sn= (n+ 1)S.

Ring denotedQ(n)hSi.

Main property: the degree in S of a product is the sum of the degrees of its factors.

Algorithm for (left or right) euclidian division.

Algorithm for (left or right) gcd,lcmand cofactors. (Ore 33)

Example (gcld algorithm)

INPUT recurrence operatorsAandB OUTPUT The “greatest”G such that A=GA˜ andB=GB˜

Example (lclm algorithm)

INPUT recurrence operators AandB OUTPUT U andV such that UA=VB

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Ore Polynomials: Framework for Recurrence Operators

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Ore Polynomials: Framework for Recurrence Operators

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Fractions of Recurrence Operators (Ore 1933)

The ringQ(n)hSipossesses a field of fractions.

Field of fractions ofQ(n)hSidefined by:

A B = C

D ⇔ ∃(U,V) such thatUA=VC andUB =VD.

Addition:

A B +C

D = UA UB +VC

VD = UA+VC UB ,

Multiplication:

D C ·A

B = VD VC ·UA

UB = UA VC.

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Fractions of Recurrence Operators (Ore 1933)

A B = C

Addition:

A B +C

D = UA UB +VC

VD = UA+VC UB ,

Multiplication:

D C · A

B =VD VC · UA

UB = UA VC.

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Fractions of Recurrence Operators (Ore 1933)

A B = C

Addition:

A B +C

D = UA UB +VC

VD = UA+VC UB , Multiplication:

D C · A

B =VD VC · UA

UB = UA VC.

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Application to Chebyshev Recurrences

Definition

Letϕbe the “Chebyshev”morphism from the differential ring into the recurrence ring: ϕ(x) = 1/2 S+S⁻¹

andϕ _dx^d

= _−S+S²ⁿ−1.

Theorem

Let f be a function and L be a differential operator such that: L·f = 0 Anumeratorof a fraction of recurrence operatorϕ(L)is aChebyshev recurrence relationof f .

f =√

1−x²is cancelled by the differential operator: x+ (1−x²)_dx^d.

ϕ

x+ (1−x²)d dx

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Application to Chebyshev Recurrences

Definition

andϕ _dx^d

= _−S+S²ⁿ−1.

Theorem

Let f be a function and L be a differential operator such that: L·f = 0 A numerator of a fraction of recurrence operatorϕ(L)is a Chebyshev recurrence relation of f .

f =√

ϕ

x+ (1−x²)d dx

= S+S⁻¹

2 +

1−S²+ 2 +S⁻² 4

2n

−S+S⁻¹

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Application to Chebyshev Recurrences

Definition

andϕ _dx^d

= _−S+S²ⁿ−1.

Theorem

f =√

ϕ

x+ (1−x²)d dx

= S+S⁻¹

2 +

1−S²+ 2 +S⁻² 4

2n

−S+S⁻¹

= −S+S⁻¹

S+S⁻¹

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

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Application to Chebyshev Recurrences

Definition

andϕ _dx^d

= _−S+S²ⁿ−1.

Theorem

f =√

ϕ

x+ (1−x²)d dx

= S+S⁻¹

2 +

1−S²+ 2 +S⁻² 4

2n

−S+S⁻¹

= −S+S⁻¹

S+S⁻¹

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

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Application to Chebyshev Recurrences

Definition

andϕ _dx^d

= _−S+S²ⁿ−1.

Theorem

f =√

ϕ

x+ (1−x²)d dx

= −(n+ 3)S²+ 2n−(n−3)S⁻² 2 (−S+S⁻¹)

The Chebyshev coefficientsc_noff, satisfy the recurrence relation:

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Normalization

Definition

A fraction ^A_B is callednormalizedwhen the gcld of AandB is 1.

Example: Normalize fraction for√ 1−x² f =√

1−x²is cancelled by the differential operator : x+ (1−x²)_dx^d. we have:

ϕ

−x+ (−1 +x²)d dx

=−(n+ 3)S²+ 2n−(n−3)S⁻² 2 (−S+S⁻¹)

= −S+S⁻¹ (n+ 2)S−(n−2)S⁻¹

2(−S+S⁻¹) .

Smaller order

⇒(n+ 2)cn+1−(n−2)c_n−1= 0.

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Normalization

Definition

A fraction ^A_B is called normalized when the gcld ofAandB is 1.

ϕ

−x+ (−1 +x²)d dx

=−(n+ 3)S²+ 2n−(n−3)S⁻² 2 (−S+S⁻¹)

= −S+S⁻¹ (n+ 2)S−(n−2)S⁻¹

2(−S+S⁻¹) .

Smaller order

⇒(n+ 2)cn+1−(n−2)c_n−1= 0.

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Normalization

Definition

ϕ

−x+ (−1 +x²)d dx

=−(n+ 3)S²+ 2n−(n−3)S⁻² 2 (−S+S⁻¹)

= −S+S⁻¹

(n+ 2)S−(n−2)S⁻¹

2(−S+S⁻¹) .

Smaller order

⇒(n+ 2)cn+1−(n−2)c_n−1= 0.

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Normalization

Definition

ϕ

−x+ (−1 +x²)d dx

=−(n+ 3)S²+ 2n−(n−3)S⁻² 2 (−S+S⁻¹)

= −S+S⁻¹

(n+ 2)S−(n−2)S⁻¹

2(−S+S⁻¹) .

Smaller order

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III Algorithms

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Lewanowicz’s algorithm (1976)

Horner+Normalize at each step.

Example withf =√ 1−x²

(1−x²)d dx +x.

ϕ(1−x²) =−S²+ 2−S⁻²

4 =(S+S⁻¹)(−S+S⁻¹) 4

ϕ(1−x²)ϕ d

dx

= (n+ 1)S−(n−1)S⁻¹ 2

ϕ(1−x²)ϕ d

dx

+ϕ(x) =(n+ 2)S−(n−2)S⁻¹ 2

A recurrence verified by the Chebyshev coefficients off is: (n+ 2)cn+1−(n−2)c_n−1= 0

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Lewanowicz’s algorithm (1976)

(1−x²)d dx +x.

ϕ(1−x²) =−S²+ 2−S⁻²

4 =(S+S⁻¹)(−S+S⁻¹) 4

ϕ(1−x²)ϕ d

dx

= (n+ 1)S−(n−1)S⁻¹ 2

ϕ(1−x²)ϕ d

dx

+ϕ(x) =(n+ 2)S−(n−2)S⁻¹ 2

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Lewanowicz’s algorithm (1976)

(1−x²)d dx +x.

ϕ(1−x²) =−S²+ 2−S⁻²

4 =(S+S⁻¹)(−S+S⁻¹) 4

ϕ(1−x²)ϕ d

dx

=(S+S⁻¹)(−S+S⁻¹) 4

2n

−S+S⁻¹

= (n+ 1)S−(n−1)S⁻¹ 2

ϕ(1−x²)ϕ d

dx

+ϕ(x) =(n+ 2)S−(n−2)S⁻¹ 2

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Lewanowicz’s algorithm (1976)

(1−x²)d dx +x.

ϕ(1−x²) =−S²+ 2−S⁻²

4 =(S+S⁻¹)(−S+S⁻¹) 4

ϕ(1−x²)ϕ d

dx

= (n+ 1)S−(n−1)S⁻¹ 2

ϕ(1−x²)ϕ d

dx

+ϕ(x) = (n+ 1)S−(n−1)S⁻¹

2 +S+S⁻¹

2

=(n+ 2)S−(n−2)S⁻¹ 2

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Lewanowicz’s algorithm (1976)

(1−x²)d dx +x.

ϕ(1−x²) =−S²+ 2−S⁻²

4 =(S+S⁻¹)(−S+S⁻¹) 4

ϕ(1−x²)ϕ d

dx

= (n+ 1)S−(n−1)S⁻¹ 2

ϕ(1−x²)ϕ d

dx

+ϕ(x) =(n+ 2)S−(n−2)S⁻¹ 2

A recurrence verified by the Chebyshev coefficients off is:

(n+ 2)c −(n−2)c = 0

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Algorithms of Paszkowski (1975) and Rebillard (1998)

Observation: ifD=ϕ(_dx^d) =_−S+S²ⁿ−1 thenD⁻¹is a polynomial.

INPUT :

L=

k

X

i=0

p_i(x) d

dx ⁱ

OUTPUT : A numerator of ϕ(L)

Computation with polynomials only.

Paszkowski

Computeqi(x) such that

k

X

i=0

pi(x) d

dx i

=

k

X

i=0

d dx

i

qi(x).

k

X

i=0

pi(X)Dⁱ=

k

P

i=0

D^−k+iq_i(X) D^−k .

Rebillard

Xk :=D^−kXD^k.

k

X

i=0

pi(X)Dⁱ =

k

P

i=0

p_i(X_k)D^−k+i

D^−k .

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Algorithms of Paszkowski (1975) and Rebillard (1998)

INPUT :

L=

k

X

i=0

p_i(x) d

dx ⁱ

Paszkowski

k

X

i=0

pi(x) d

dx i

=

k

X

i=0

d dx

i

qi(x).

k

PD^−k+iq_i(X)

Rebillard

Xk :=D^−kXD^k.

k

X

i=0

pi(X)Dⁱ=

k

P

i=0

p_i(X_k)D^−k⁺ⁱ

D^−k .

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Algorithms of Paszkowski (1975) and Rebillard (1998)

INPUT :

L=

k

X

i=0

p_i(x) d

dx ⁱ

Paszkowski

k

X

i=0

pi(x) d

dx i

=

k

X

i=0

d dx

i

qi(x).

k

PD^−k+iq_i(X)

Rebillard

Xk :=D^−kXD^k.

k

Xpi(X)Dⁱ=

k

P

i=0

p_i(X_k)D^−k⁺ⁱ

D^−k .

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Our algorithm: Divide and conquer

D⁻ⁱ is of bidegree (2i,2i).

New, fast algorithm

Step 1: Computeqi(x) such that

k

X

i=0

p_i(x) d

dx i

=

k

X

i=0

d dx

i

q_i(x).

Step 2 : Divide and conquer

k

X

i=0

D^−k+iq_i(X) =

D⁻^k²

k 2

XD⁻^k²⁺ⁱqi(X)+

k

X D^−k⁺ⁱqi(X).

Theorem

If the degrees of p_i are at most k, New: O(k^ω)arithmetic operations.

Paszkowski and Lewanowicz algorithms : O(k⁴)arithmetic operations.

Rebillard : O(k⁵)arithmetic operations.

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Our algorithm: Divide and conquer

D⁻ⁱ is of bidegree (2i,2i).

New, fast algorithm

Step 1: Computeqi(x) such that

k

X

i=0

p_i(x) d

dx i

=

k

X

i=0

d dx

i

q_i(x).

Step 2 : Divide and conquer

k

X

i=0

D^−k+iq_i(X) =

D⁻^k²

k 2

XD⁻^k²⁺ⁱqi(X)+

k

X D^−k⁺ⁱqi(X).

Theorem

If the degrees of p_i are at most k, New: O(k^ω)arithmetic operations.

Paszkowski and Lewanowicz algorithms : O(k⁴)arithmetic operations.

Rebillard : O(k⁵)arithmetic operations.

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IV Conclusion and Future Works

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Other Orthogonal Polynomial Families

Same relation to multiplication byx and differentiation for the Gegenbauer and Jacobi Polynomials: same algorithm.

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Other Orthogonal Polynomial Families

Same relation to multiplication byx and differentiation for the Gegenbauer and Jacobi Polynomials: same algorithm.

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Conclusion and Future works

Contributions:

Use of fractions of recurrence operators.

New algorithm.

Maple code.

Available in the Dynamic Dictionary of Mathematical Functions.

Perspectives:

Recurrence in other bases (Jacobi, Legendre and Laguerre polynomials, Bessel functions)