The Application of the Fast Fourier Transform toJacobi Polynomial expansions

by Akil C. Narayan and Jan S. Hesthaven

Abstract

We present an algorithm for transforming modal coefficients of one Jacobi Polynomial class

to the modal coefficients of another class. This transformation is invertible and is efficient

for certain pairs of Jacobi Polynomials classes. When one of the classes corresponds to the

Chebyshev case, the Fast Fourier Transform can be used to very quickly compute modal

coefficients for a wide variety Jacobi Polynomial expansions. Numerical results are pre-

sented that illustrate the computational and accuracy advantage of our method over stan-

dard quadrature methods.

1 Introduction

The classical Jacobi Polynomials Pn(α,β) are a family of orthogonal polynomials [13] that have

been used extensively in many applications for their ability to approximate general classes offunctions. They are a class of polynomials that encompasses the Chebyshev, Legendre, andGegenbauer/ultraspheric polynomials. In addition, they have a very close connection to theAssociated Legendre functions that are widely used in the Spherical Harmonic expansions.

Jacobi polynomial expansions have been used in a variety of applications, some of which arethe resolution of the Gibbs’ phenomenon [9], electrocardiogram data compression [14], and thesolution to differential equations [6]. Due to the range of applications for Jacobi polynomials, itis desirable to perform spectral expansion computations as quickly and accurately as possible.In this paper, we show that for a variety of classes of Jacobi polynomials, one can use the FFTto perform spectral transformations.

In section 2 we give an overview of Jacobi polynomials and the relevant properties needed forour discussion. Section 3 is devoted to a theoretical description of the method and includes mostof the major results. Section 4 is a special application of the results from section 3 to Cheby-shev-like systems where the FFT may be exploited. Finally, section 5 gives some numericalexamples.

2 Jacobi Polynomials

For a comprehensive treatement of Jacobi polynomials and their properties, [13], [7], and [1]prove to be excellent references. All properties and results in this section are taken from thesereferences. Jacobi polynomials are one of the two linearly independent solutions to the linear,second-order, singular Sturm-Liouville differential equation

−d

dr

[

(1− r)α+1 (1 + r)β+1) dρ

dr

]

−n(n + α + β + 1) (1− r)α(1+ r)βρ =0, r ∈ [− 1, 1]. (2.1)

1

The parameters α, β are restricted to take real values in the interval ( − 1, ∞). The monic

Jacobi polynomial of order n, as a solution to (2.1) is written as Pn(α,β)

(x). We define the spaceBN = span{xn: 0 ≤ n ≤ N }. For any α, β > − 1, they are orthogonal under a weighted L2 innerproduct:

∫

−1

1

Pm(α,β)

Pn(α,β)

(1− r)α (1 + r)β dr = hn(α,β)

δm,n, (2.2)

where δm,n is the Kronecker delta function, and hn(α,β)

is a normalization constant given in theappendix, equation (A.1). We take the integral (2.2) in the Lebesgue sense. We define theweight function

ω(α,β)(r)= (1− r)α (1 + r)β , (2.3)

and denote angled brackets 〈 · , · 〉(α,β)as the inner product of equation (2.2). I.e.,

〈f , g〉(α,β)=

∫

−1

1

f(r) g(r) ω(α,β) dr.

This inner product induces a norm ‖ · ‖(α,β) on the space L(α,β)2 4 {

f : [ − 1, 1] →R: f measur-

able, ‖f ‖(α,β) < ∞}

, where ‘measurable’ means with respect to Lebesgue measure. The Jacobi

polynomials of class (α, β) are complete and orthogonal in L(α,β)2 .

The monic Jacobi polynomials satisfy various relations and can be expressed in terms of theHypergeometric function, and also satisfy a generalized Rodrigues relation for all m≤n:

(

2n+ α + β

n

)

ω(α,β) Pn(α,β)

=(− 1)m

2m−n(n− 1)� (n−m +1)

dm

dxm

[

ω(α+m,β+m) Pn−m(α+m,β+m)

]

.

Although the formulae for the monic orthogonal polynomials are often easier to write downthan those for other normalizations, we shall prefer to work with the L(α,β)

2 -normalized polyno-mials. To this end, we define

P̃n(α,β)

=Pn

(α,β)

hn(α,β)

√ ,

which are orthonormal under the weight ω(α,β).

All orthogonal polynomials satisfy a three-term recurrence relation from which the Kernelpolynomials and the Christoffel-Darboux identity can be derived. These last two properties yieldthe following promotions and demotions of the Jacobi polynomial class parameters (α, β):

(1− r) P̃n(α,β)

= µn,0(α,β)

P̃n(α−1,β)

− µn,1(α,β)

P̃n+1(α−1,β)

, (2.4)

(1 + r)P̃n(α,β)

= µn,0(β,α)

P̃n(α,β−1)

+ µn,1(β,α)

P̃n+1(α,β−1)

, (2.5)

P̃n(α,β)

= νn,0(α,β)

P̃n(α+1,β)

− νn,−1(α,β)

P̃n−1(α+1,β)

, (2.6)

P̃n(α,β) = νn,0

(β,α)P̃n

(α,β+1) + νn,−1(β,α)

P̃n−1(α,β+1)

, (2.7)

where µn,0/1(α,β)

and νn,0/−1(α,β)

are constants for which we derive explicit formulae in appendix A.

The formulae (2.4)-(2.7) are the main ingredients for our results.

2 Section 2

Lastly we cover the spectral expansion of a function f(r) in Jacobi polynomials. For any f ∈L(α,β)

2 we define the modal coefficients

f̂n(α,β) =

⟨

f , P̃n(α,β)

⟩

(α,β).

We also have a Parseval-type relation:

‖f ‖(α,β)2 =

∑

n=0

∞[

f̂n(α,β)

]2.

Naturally, f̂n(a,b) is well-defined if a ≥ α and b ≥ β because of the inclusion L(α,β)

2 ⊆ L(a,b)2 . We

define the projection operator

PN(α,β)

f =∑

n=0

N−1

f̂n(α,β)

P̃n(α,β)

.

Due to completeness and orthogonality, this operator satisfies the relations

∥

∥

∥f −Pn

(α,β)f∥

∥

∥

(α,β)

� 0, n→∞

⟨

f −Pn(α,β)

f , φ⟩

(α,β)=0, φ∈Bn

3 Jacobi-Jacobi transformations

The purpose of the section is to form a relationship between the expansions PN(α,β)

f and PN(γ,δ)

f

for α � γ and/or β � δ. More than that, we will determine a relationship that will allow us totravel between the two expansions very quickly and with very little effort. Of course, the speedof this transformation comes with a natural price: this can only be done for certain values of|δ − β | and |γ −α|.

We begin by proving a lemma that is an inductive application of equations (2.4) and (2.5):

Lemma 3.1.

For any A, B ∈N and α, β >− 1, the following promotion relations hold:

(1− r)AP̃n(α+A,β)

=∑

m=0

A

Mm,n(α,A,β)

P̃n+m(α,β)

(3.1)

(1 + r)BP̃n(α,β+B)

=∑

m=0

B

(− 1)mMm,n(β,B,α)

P̃n+m(α,β)

, (3.2)

where the Mm,n(α,β)

are constants. Similarly, we can demote the class parameters α + A and β + B

down to α and β as well:

P̃n(α,β)

=∑

m=0

A

Nm,n(α,A,β)

P̃n−m(α+A,β)

(3.3)

P̃n(α,b)

=∑

m=0

B

(− 1)mNm,n(β,B,α)

P̃n−m(α,β+B)

(3.4)

Jacobi-Jacobi transformations 3

Proof. The proofs of (3.1) and (3.2) are accomplished via the demotion relations (2.4) and(2.5): repeated application of (2.4) to the left-hand side of (3.1), and repeated application of(2.5) to the left-hand side of (3.2) yields the desired result.

The relations (3.3) and (3.4) are proven in exactly the same fashion using the promotionrelations (2.6) and (2.7). �

Note that we did not give the formulae for the constants Mm,n(α,A,β) and Nm,n

(α,A,β) in lemma

3.1. It would be possible for us to derive such formulae in terms of the µn,i(α,β) and νn,i

(α,β), but it

is relatively complicated and of little value for us. The main use of lemma 3.1 is in the proof ofthe following theorem:

Theorem 3.2.

For A, B ∈N0 and α, β >− 1, let f ∈L(α,β)2 . Then

f̂n(α+A,β+B) =

∑

m=0

A+B

Λm,n(α,A,β,B)

f̂n+m(α,β)

, n≥ 0 (3.5)

for some constants Λm,n(α,A,β,B).

Proof. We first use lemma 3.1 twice on (1 − r)A(1 + r)BP̃n(α+A,β+B)

to show that there existconstants Λm,n

(α,A,β,B) such that

ω(A,B)P̃n(α+A,β+B)

=∑

m=0

A+B

Λm,n(α,A,β,B)

P̃n+m(α,β)

.

Following this we note that f̂n(α,β) is well-defined because f has membership in L(α,β)

2 , and

f̂n(α+A,β+B) is well-defined because of the inclusion of L(α,β)

2 in L(α+A,β+B)2 . Finally, we have

f̂n(α+A,β+B) =

⟨

f , P̃n(α+A,β+B)

⟩

(α+A,β+B)

=⟨

f , (1− r)A (1 + r)B P̃n(α+A,β+B)

⟩

(α,β)

=

⟨

f ,∑

m=0

A+B

Λm,n(α,A,β,B)

P̃n+m(α,β)

⟩

(α,β)

=∑

m=0

A+B

Λm,n(α,A,β,B)

⟨

f , P̃n+m(α,β)

⟩

(α,β)

=∑

m=0

A+B

Λm,n(α,A,β,B)

f̂n+m(α,β)

.

�

In other words, theorem 3.2 says that we can express modes of higher-class Jacobi expan-sions in terms of linear combinations of lower-class Jacobi expansions. Roughly speaking, inorder to convert an expansion of class (α0, β0) to one of class (α1, β1), we require about O(α1 −α0 + β1− β0) computations, as long as α1−α0 and β1− β0 are both natural numbers.

4 Section 3

The above result is likely known by many authors in this field due to the voluminous litera-ture on Jacobi polynomials. However, we have not seen it written down explicitly, and so wepresent the above theorem. As in lemma 3.1, we do not present explicit formulae for the con-

stants Λm,n(α,A,β,B). Instead, we present the following algorithm for transforming N (α, β) modes

into N (α +A, β +B) modes.

3.1 Computing the transformation

We collect the modes{

f̂n(α,β)

}

ninto the vector f̂ (α,β). Theorem 3.2 makes it clear that

there exists an N × (N +A +B) matrix R̂, dependent on α, β, A, and B such that

f̂ (α+A,β+B) = R̂f̂ (α,β). (3.6)

This relation transports N + A + B modes from an expansion in the polynomials P̃ (α,β) to N

modes from an expansion in P̃ (α+A,β+B). We do not (yet) know the explicit entries of thematrix R̂ , but we do know that R̂ is a sparse matrix, banded upper-triangular, with about (A +

B)(N + A + B) non-zero entries. In order to construct the matrix R̂, we essentially write outthe proof of lemma 3.1 via induction. To this end, we define the following set of matrices: let

U (α,β) and V (α,β) be sparse bidiagonal matrices with entries defined by

Un,n(α,β)

= µn,0(α,β)

Un,n+1(α,β)

= − µn,1(α,β)

Vn,n(α,β)

= µn,0(β,α)

Vn,n+1(α,β) = − µn,1

(β,α)

n = 1, 2,� , N + A+ B − 1

UN+A+B,N+A+B(α,β)

= µn,0(α,β)

VN+A+B,N+A+B(α,β) = µn,0

(β,α)

The matrix U (α,β) transforms the modes of an (α − 1, β) expansion to those of an (α, β) expan-sion whenever α > 0. Simiarly, V (α,β) transforms the modes of an (α, β − 1) expansion intothose of an (α, β) expansion whenever β > 0. To define the entries of U and V we have used thedemotion relations (2.4) and (2.5). Note, however, that the last mode N + A + B will be incor-rect because we require information about mode N + A + B + 1 (which we don’t have) to deter-mine it. Thus, the last output mode will be corrupted. However, as we will see later, we canactually characterize this corruption.

We can now define a square (N + A+ B)× (N + A+ B) matrix R as

R =∏

b=1

B

V (α+A,β+b)∏

a=1

A

U (α+a,β) (3.7)

The matrix R has some nice properties. As the product of banded bidiagonal matrices, we alsoknow that R has non-zero entries on at most (A+ B + 1) diagonals, and is upper-triangular.

Proposition 3.3.

Jacobi-Jacobi transformations 5

The matrix R defined by equation ( 3.7) is invertible and positive-definite.

Proof. R is upper-triangular, and thus the diagonal entries are the eigenvalues. By utilizingequation (3.7) and expressions (A.2) in the appendix, we have, for all n = 1, 2,� , N + A+ B

Rn,n =∏

b=1

B

µn,0(β+b,α+A)

∏

a=1

A

µn,0(α+a,β)

> 0

�

Proposition 3.3 tells us that the matrix R is invertible, which means we can travel back andforth between the modes f̂n

(α+A,β+B) and f̂n(α,β). In practice, this can be accomplished via

back-substitution because R is upper-triangular. The back substitution has a sequential compu-tational cost O(N(A + B)), similar to the cost of the formation or application of the matrix. Wetangentially note that it is not entirely necessary to form the matrix R; one can merely store the

coefficients µn,i(α,β)

required for the formation of R and view the application (inversion) of R as

A +B sequential applications (inversions) of invertible bidiagonal matrices.

Returning to the goal presented at the beginning of this section, we can define the non-

square matrix R̂ as the first N rows of R, and this is exactly the matrix we were looking for.Thus, the formation of R̂ (or R) requires (A + B) multiplications between sparse bandedmatrices.

However, in contrast to the relation presented in (3.6), we would like an invertible transfor-mation. The reason is that we will eventually use this transform as an ingredient in a

modal/nodal transformation, so we’d like an invertible transformation for this. The matrix R̂ isnot invertible (it’s not even square), but of course R is invertible, so we shall frequently use thatmatrix. (As mentioned before, one should be aware that the last A + B modes are not the truemodes, but some version of them.)

We note that the constants Λm,n(α,A,β,B) in theorem (3.2) are the entries of the matrix R.

Thus, we have developed an inexpensive algorithm to compute these constants, even if we don’tgive explicit formulae for them.

3.2 Properties of the transformation

In the following, we will replace all instances of N + A + B with just N . Indeed this is atrivial thing to do by simply augmenting the value of N accordingly.

Given that the last A + B modes{

f̂n(α+A,β+B)

}

n=N−A−B

N−1are not the true modes, we might

wonder how the expansion

f (α+A,β+B)(r)=∑

n=0

N−1

f̂n(α+A,β+B)

P̃n(α+A,β+B)

(r) (3.8)

compares to the original expansion

f (α,β)(r)=∑

n=0

N−1

f̂n(α,β)

P̃n(α,β)

(r), (3.9)

6 Section 3

assuming that the modes f̂n(α,β) and f̂n

(α+A,β+B) are related by (3.7). It is not trivial, but alsonot surprising, that these expansions are exactly the same function:

Proposition 3.4.

Let the modes{

f̂n(α,β)

}

n=0

N−1be given, and let the modes

{

f̂n(α+A,β+B)

}

n=0

N−1be derived via

the matrix R. Then the expansions f (α,β) and f (α+A,β+B) given by equations ( 3.9) and ( 3.8),respectively, are equal.

Proof. Without loss, we take f̂n(α,β) = f̂n

(α+A,β+B) = 0 for all n ≥ N . We first determine the

expansion coefficients of the given function f (α,β) ∈ BN in the space L(α+A,β+B)2 with the func-

tions P̃n(α+A,β+B)

:

⟨

f (α,β), P̃n(α+A,β+B)

⟩

(α+A,β+B)=

∑

l=0

N−1

f̂l(α,β)

⟨

P̃l(α,β)

, ω(A,B)P̃n(α+A,β+B)

⟩

(α,β)

=∑

l=0

N−1

f̂l(α,β)

⟨

P̃l(α,β)

,∑

m=0

A+B

Λm,n(α,A,β,B)

P̃n+m(α,β)

⟩

(α,β)

=∑

l=0

N−1

f̂l(α,β)

∑

m=0

A+B

Λm,n(α,A,β,B)

δl,n+m

=∑

m=0

A+B

Λm,n(α,A,β,B)

f̂n+m(α,β)

.

Therefore, since f (α,β)∈BN and by the definition of f̂ (α+A,β+B) in relation (3.5):

f (α,β) = PN(α+A,β+B)

f (α,β)

=∑

n=0

N−1⟨

f (α,β), P̃ (α+A,β+B)⟩

(α+A,β+B)P̃n

(α+A,β+B)

=∑

n=0

N−1(

∑

m=0

A+B

Λm,n(α,A,β,B)

f̂n+m(α,β)

)

P̃n(α+A,β+B)

=∑

n=0

N−1

f̂n(α+A,β+B)

P̃n(α+A,β+B)

= f (α+A,β+B).

�

4 Exploiting the FFT

Exploiting the FFT 7

Up until this point we have not discussed any real implementation issues: it has all beentheory. In this section we explore the use of the FFT for performing Jacobi polynomial spectraltransformations. To facilitate discussions, we must introduce more notation.

4.1 Quadrature and interpolation

All the results in this section are well-known and we point to the given references for details.

We shall call an N -point quadrature rule Gaussian under a particular weight function ω(α,β)

if it exactly integrates any polynomial in the space B2N under the weight ω(α,β). Such a quadra-ture rule always exists and is unique if the weight function is non-negative. A quadrature rule is

defined by N nodes and weights {rn, wn}n=1N and we shall write the Gaussian quadrature rule

evaluation under weight ω(α,β) as

∫

−1

1

f(r) ω(α,β) dr ≃ QN(α,β)[f ] 4 ∑

n=1

N

f(rq)wq.

With this notation, we can write the definition of a Gaussian quadrature rule as one that satis-fies

QN(α,β)

[rn] =

∫

−1

1

rn ω(α,β) dr, n≤ 2N − 1.

We recall a fundamental result due to Golub and Welsch [8]: the determination of the nodesand weights {rn, wn} can be accomplished via the computation of the eigenvalues and eigenvec-tors of a symmetric tridiagonal matrix. A more efficient way is to just compute the eigenvalues(which are the nodes rn) and then use known formulae [3] to compute the weights wn. Thisbrings the cost of computation down to O

(

N2)

operations. We refer to the nodes and weights

corresponding to the Gaussian quadrature rule QN(α,β)

[ · ] as{

rn(α,β)

, wn(α,β)

}

.

In many computations one cannot exactly compute the modal coefficients f̂n(α,β) because we

cannot compute the integral exactly, or we can only evaluate f but do not know an analyticform for it. Instead, we can use quadrature rules to approximate the modal coefficients:

f̂n(α,β)

≃ f̃n(α,β)4 QN

(α,β)[

f P̃n(α,β)

]

.

We can then form the following approximation to PN(α,β)

f :

IN(α,β)4 ∑

n=0

N−1

f̃n(α,β)

P̃n(α,β)

.

Due to the exactness of the Gauss quadrature rule, IN(α,β)

f = PN(α,β)

f for any f ∈ BN. Because

of the Christoffel-Darboux identity, the expansion IN(α,β)

f is the unique (N − 1)st degree poly-

nomial interpolant of f at the nodes{

rn(α,β)

}

n=1

N

. For f � BN, the difference between the inter-

polation IN(α,β)

f and the projection PN(α,β)

f is called the aliasing error [11], and arises due tothe error in the quadrature rule.

Let A, B ∈ N0. We define the following discrete approximations to the N modal coefficients{

f̂n(α+A,β+B)

}

n=0

N−1:

8 Section 4

f̃n;GQ(α+A,β+B) = QN

(α+A,β+B)[

f P̃n(α+A,β+B)

]

(4.1)

f̃n;R(α+A,β+B) =

∑

m=0

A+B

Λm,n(α,A,β,B)

f̃n+m;GQ(α,β) (4.2)

f̃n;DQ(α+A,β+B) = QN

(α,β)[

f P̃n(α+A,β+B)

w(A,B)]

(4.3)

The modes f̃n;GQ are obtained using the Gaussian quadrature native to the expansion class (α +

A, β + B). The modes f̃n;R are obtained by performing the matrix multiplication in equation(3.6); i.e. the matrix R is used to obtain the modes by promoting the lower-class discerte modes

from class (α, β) to class (α + A, β + B). The last class of modes f̃n;DQ are obtained by using aGaussian quadrature under the weight ω(α,β) to simulate quadrature under the weight

ω(α+A,β+B). We call this last expansion one via ‘demoted quadrature’, which motivates the sub-script DQ.

With these three modal definitions, we can define the three expansions fi(α+A,β+B)

=∑

n=0N−1

f̃n;i(α+A,β+B)

P̃n(α+A,β+B), for i = GQ,R,DQ. Since GQ represents a strict Gaussian

quadrature, we know that fGQ(α+A,β+B)

= IN(α+A,β+B)

f . As a corollary of proposition 3.4, wehave the following:

Corollary 4.1. fR(α+A,β+B)

= IN(α,β)

f.

The first two expansions GQ and R are exact for any f ∈ BN. The last expansion, DQ, isonly exact if f ∈ BN−A−B. However, it is important to notice that all of the three expansionsare different for f � BN because the aliasing error for each method is different.

4.2 Chebyshev Interpolants

We consider the following problem: Given a function f(r) for r ∈ [ − 1, 1], a Jacobi class (α,

β) =(

−1

2+ A,−

1

2+ B

)

, and a maximum modal number N , use an N -point quadrature rule to

compute an approximation to the first N modes. A standard practice is to compute the Gauss-

Jacobi interpolant fn(α,β) using the Gaussian quadrature native to class (α, β). This amounts to

performing the operation in equation (4.1) N times, which is an O(

N2)

matrix multiplication.

However, given the results in section 3, we can instead compute the modes for f in class(

−1

2,−

1

2

)

and then translate them to class (α, β) using the matrix R. Computing the modes for f

in class(

−1

2, −

1

2

)

(i.e. the Chebyshev modes) can be accomplished with a fast Fourier Trans-

form (FFT) [11]. Therefore, for general classes(

−1

2+ A, −

1

2+ B

)

with any natural numbers

A, B we can compute the modes using an FFT coupled with a sparse matrix-multiply.

In the determination of the expansion coefficients f̃n;GQ and f̃n;R, we can characterize fourtypes of computations to be peformed:

Overhead Time (GQ)

• Computation of the quadrature rule. This requires evaluation of the values

P̃n−1(α,β)

(

rm(−1/2,−1/2)

)

for m, n =1,� , N . The total cost for this is about O(

N2)

Exploiting the FFT 9

Online Time (GQ)

• Performing the quadrature rule evaluation in equation (4.1). This is basically a matrix-vector multiplication requiring O

(

N2)

computations.

Overhead Time (R)

• Precomputations for the FFT. This involves a small prime factorization of N and storageof function evaluations.

• Computing the shift to transform standard Fourier modes, the output of the FFT, to theChebyshev modes that we desire. This is O(N).

• Calculating the entries in the matrix R. This is an O(N(A+ B)) operation.

Online Time (R)

• Performing the FFT to recover the Fourier modes and transforming them to the Cheby-shev modes. O(N logN )

• Multiplication by the sparse matrix R. This requires O(N(A+ B)) operations.

To summarize, the GQ method has O(

N2)

complexity for both the overhead and online

times. The R method is about O(N logN) for both, as long as A + B is not very large. The GQ

method produces the modes f̃n;GQ(α,β), and the R method produces the modes f̃n;R

(α,β).

We will use these characterizations of the online and overhead times for our results in thenext section.

5 Numerical examples

In this section we will test the theoretical methods developed in section 3 applied to Chebyshev-

Jacobi polynomial expansions(

−1

2+ A, −

1

2+ B

)

as described in section 4.2. We take the test

function

f(r)= exp

(

− 5(

r −π

6

)2)

+ sin(r),

which is analytic but does not exhibit any symmetry on the interval r ∈ [− 1, 1]. Using the algo-

rithms presented in section 4, we can compute two spectral expansions of f : fGQ(α,β)

and fR(α,β).

We split the computer work required into the ‘overhead’, and ‘online’ divisions, which aredefined in section 4.2. By performing the algorithms delineated in section 4.2, we can measurethe time required to perform spectral expansions using the GQ and R methods. The computa-tions for online time are performed using C libraries, and those for overhead time are performedusing Python wrapped around C linear algebra subroutines.

In figure 5.1 we show the online times for the GQ and R methods for A and B taking values5 and 10. We see that the online time for the R method with the FFT is relatively insensitive toN , growing only linearly with N . However, the GQ (quadrature) method grows with N2 and ismore expensive than the R method. The time spent for the R method is split almost evenlybetween the FFT and application of the matrix R. Application of R becomes the dominantfactor when A and B become large.

10 Section 5

Figure 5.1. Plots of the online times for determining N modal coefficients using N nodal evaluations.

FFT+R-matrix calculations are plotted (solid line) vs. quadrature calculations (dashed line) for various

expansion classes (α, β). The kink around N = 128 represents the limitations of the machine used to per-

form computations and does not reflect the algorithm complexity.

It is worth noting that the timings in figure 5.1 should be taken with a grain of salt: compu-tational timings for the FFT vs quadrature for small N are extremely sensitive to the method ofcoding, the particular libraries used, the compiler, and the computer hardware. However, it isclear that for large N the FFT is undoubtedly faster than a direct application of quadrature,and that this speed advantage remains despite influence by the multiplication by R.

In figure 5.2 we show the same results as in figure 5.1, but we focus on how the parametersA and B affect the online time required for the R method. As expected, the slope of the linearrelation between computational time and N is directly related to A + B. In the cases (α, β) =(

9

2,

9

2

)

and (α, β) =(

−1

2,19

2

)

, we have A + B = 10, and the online time required is almost iden-

tical.

Another topic to consider is the overhead time required to compute the matrix R for the Rmethod or the quadrature rule for the GQ method. In figure 5.3 we show these results. We seethat even for small N the overhead time required for the R method is less than that required forthe quadrature method. Note that we have even given the GQ method a bit of an advantage: insection 4.2, we defined the overhead time for the GQ method, and we did not include the timerequired to form the Gaussian quadrature rule nodes and weights rn

(α,β) and wn(α,β), respectively.

This requires an additional O(

N2)

operations (via the Golub-Welsch algorithm); we did not

include it because in practice one may not compute fGQ, but instead the demoted quadratureexpansion fDQ from equation (4.3) because the Chebyshev-Gauss quadrature rule has an ana-lytic form and is very easy to compute [11]. However, the DQ quadrature becomes more andmore inaccurate as A and B are increased. The ‘quadrature’ overhead time plotted in figure 5.3is actually that for the DQ method, and not the GQ method.

Numerical examples 11

Figure 5.2. Plot of the online computational time required for the R method vs. N . These are the same

results as in figure 5.1.

Figure 5.3. Plot of the overhead times vs. number of nodal/modal degrees of freedom N . The over-

heads are calculated for the fGQ (quadrature) and fR (FFT) methods.

One final topic we may wish to explore is the accuracy of the expansions. We know from

proposition 3.4 that fGQ = IN(α,β)

(f), and from corollary 4.1 that fR = IN(−1/2,−1/2)

(f). It is also

well-known that the nodes{

rn(α,β)

}

n=1

N

approach the equidistant nodal set as α and β are

increased. Furthermore, the equidistant nodal set for polynomial interpolation has an exponen-tially growing Lebesgue constant [12], whose magnitude bounds the maximum pointwise error of

12 Section 5

an interpolant. Because large (α, β) will yield an interpolant fGQ on near-equidistant nodes, wehave good reason to suspect that fGQ will be a bad pointwise estimator for the function f . Bycontrast, it is also known that the Chebyshev-Gauss nodes have a nearly-optimal Lebesgue con-stant [10], of order log(N). Therefore, we expect that fR to be an orders-of-magnitude betterpointwise interpolant than fGQ for large α, β.

This suspicion is validated in figure 5.4. We see that for α = β = −1

2, when fGQ and fR are

the same function, the errors are of order machine precision. When we increase α and β, we seethat fGQ develops very bad errors at the boundaries, which is consistent with our observationthat this function is an interpolant on nearly-equidistant nodes. However, we also see that thefunction fR maintains its pointwise error near machine precision. This is not surprising since we

know that for all (α, β) =(

−1

2+ A, −

1

2+ B

)

, the function fR is the the Chebyshev-Gauss

interpolant, which has a well-behaved Lebesgue constant.

We remark that all the functions fR in figure 5.4 are the same function, so we expect all thesolid lines in the plot to be identical. We see that this is not the case, which we can attribute toroundoff errors. If one were to expand a function whose spectral coefficients did not decay expo-nentially (and thus whose error is not near machine roundoff), one would see that all the errorpatterns for fR are identical, in agreement with proposition 3.4 and corollary 4.1.

Figure 5.4. Plots of the pointwise errors |f − fGQ | (dashed line) and |f − fR| (solid line) with an N =

40, class (α, β) modal reconstruction of the FFT approximation fR and the quadrature approximation

fGQ . The grid is a 104-point Chebyshev-Gauss set of nodes.

The very large errors in figure 5.4 for the quadrature expansion fGQ for large α, β are due to

the classical Runge phenomenon: since fGQ = IN(α,β)

f and fR = IN(−1/2,−1/2)

f , we need only notethat the interpolant fGQ is taken over the Jacobi-Gauss (α, β) quadrature points, which tend toequidistant nodes [11] as α, β →∞. It is well-known that polynomial approximation via interpo-lation on equidistant nodes is very ill-conditioned; this results in the bad errors for fGQ in figure5.4.

Numerical examples 13

Note that a very similar phenomenon has been found in the context of Gegenbauer recon-struction: for functions whose singularities are close to the real axis, the projection of these func-tions exhibits bad pointwise error near the boundaries for large α, β. This has been termedthe ‘generalized Runge phenomenon’ by Boyd [5]. It is not the same as the classical Runge phe-nomenon because the classical phenomenon is due to interpolation, and the generalized phe-nomenon is due to polynomial projections. However, Boyd shows that expansions in symmetricJacobi polynomials for large α is similar to computing a power series, which in general is ill-behaved.

We have already shown that the R method is superior to the GQ method for the suppressionof the Runge phenomenon for interpolation; this is a simple consequence of analyzing the inter-polation nodes for each method as explained above. However, the R method is also able to sup-press the generalized Runge phenomenon for projection – with a caveat. Assume we have access

to the exact modes f̂n(α,β)

, and are seeking the modes f̂n(α+A,β+B)

. The degree of improvement

depends on which version of the matrix R is used. If we use the N × (N + A + B) matrix R̂

defined in equation (3.6), then fR̂ is identical to fGQ and there is no improvement. However, ifwe use the square matrix R, then the generalized Runge phenomenon is mitigated since we

merely transform the more well-behaved projection f̂n(α,β) into modes of Jacobi class (α + A, β +

B).

The caveat is that the last A + B modes of the projection are incorrect: they can be viewedas a filtered version of the true modes. This filtered function fR has much better error patternsat the boundaries, similar to what is shown in figure 5.4. In other words, this method augmentsthe modes to ‘defeat’ the generalized Runge phenomenon as explored in [5]. However, as it is afiltered version of the true function, some information is lost. In particular, if we attempt to per-form Gegenbauer reconstruction [9], we are actually trying to recover f from some Gibbs’-oscil-lation-polluted version f̃ . The R method, in suppressing the generalized Runge phenomenon

produces a function that very closely approximates f̃ – this is not what we want. Indeed, wereally want the true Gegenbauer approximation that is far enough away from f̃ and is closer tothe hidden function f . We have performed numerical experiments to confirm that using the Rmethod in a projection sense, which pollutes the last A + B modes with respect to the truemodes, actually prevents the Gegenbauer reconstruction method from succeeding! By paddingA + B modes to the expansion, we can recover the true Gegenbauer modes required to suppress

the Gibbs’ oscillation in f̃ , but then this function, possessing the true Gegenbauer modes, doesexhibit the generalized Runge phenomenon.

6 Summary

We have presented theory relating the modal coefficients of a Jacobi polynomial expansion ofclass (α, β) to the modal coefficients of class (α + A, β + B) for A, B ∈N0. Such a transforma-tion is possible due to certain equations for promoting and demoting Jacobi class parameters α

and β by integral values. This transformation can also be efficiently implemented, transforming

N modes in O(N(A + B)) computations. In the case where α = β =−1

2, one can apply the FFT

to compute modal expansions of class(

−1

2+ A,−

1

2+ B

)

in O(N logN) time. For large N and

large A and B the FFT algorithm proves to be a more efficient method for computing modalcoefficients than direct quadrature.

14 Section 6

In addition to the advantage of speed, we can also claim a victory in accuracy. We haveproven that the reconstructed function can be characterized as the Chebyshev-Gauss inter-polant. For large A, B, this implies that our reconstructed function is much better behaved than

the Jacobi-Gauss(

−1

2+ A, −

1

2+ B

)

interpolant. From one point of view, the Chebyshev-

Gauss interpolant is approximately a filtered version of the Jacobi-Gauss interpolant, wherein achange in the final A + B modes is enough to mold the latter interpolant into the former. Onthe same topic of accuracy, it is known that using the FFT rather than direct use of directquadrature is more accurate because fewer calculations overall leads to less roundoff error. Thisis the reason why fR is more accurate than fGQ in figure 5.4 for α = β =−

1

2.

Although our numerical examples have concentrated on the Jacobi-Gauss quadrature, themethod is equally applicable to the Radau and Lobatto quadratures as well, since Chebyshev-Gauss-Radau and Chebyshev-Gauss-Lobatto quadrature rules are FFT-implementable as well.In addition, fast algorithms for Legendre polynomial expansions also exist [2], which means thatthis algorithm also yields fast spectral transformations for polynomial expansions in any class ofthe form (A, B) as well when combined with the algorithm in [2]. We also note that Boyd pro-vides the possibility of using the fast multipole method (FMM) for evaluating modal coefficientson Gaussian quadrature nodes [4]. This enables the possibility of an O(N log N) algorithm forthe GQ method as well, but as mentioned in [4], while the FMM has the same asymptotic com-plexity as the FFT, it unfortunately has a much larger constant of proportionality than theFFT, so the FFT is much superior if one can use it.

There is still room for improvement for our method: we have used packaged FFT routines,but for our purpose of calculating Chebyshev modes, the method is more efficient if instead weuse fast cosine transform routines. In addition, we have assumed for given A and B that exactly

(A + B + 1)(

N −(A + B)

2

)

elements of R are nonzero (that is, the first A + B + 1 superdiago-

nals). However, if α = β and A = B, then the odd superdiagonals are actually zero. This meansthat an online application of R for Gegenbauer/Ultraspheric expansions can be optimized by afactor of roughly 2, which we have not done in this paper. Because application of R is usuallyon the same order of magnitude as application of the FFT, this is can result in a significantspeedup of the method.

Finally, the inverse transform from modes of order(

−1

2+ A, −

1

2+ B

)

to Chebyshev-

Gauss(-Radau/Lobatto) nodes is also possible via the FFT: one must back-substitute to solvethe system

f̃ (α+A,β+B) = Rf̃ (α,β),

for the modes f̃ (α,β), which is a sequential operation with nearly the same number of operationsas the forward application of R. (Recall by proposition 3.3 that R is invertible.) Then if α = β =−

1

2, one can use the FFT to recover the nodal evaluations.

We provide the Python code used to generate the figures athttp://www.dam.brown.edu/people/anaray/research.html.

The authors acknowledge partial support of this work by AFOSR award FA9550-07-1-0422.

Appendix A Jacobi polynomial properties

This appendix is devoted to providing the formulae necessary to carry out the algorithms pre-sented in this paper. The Jacobi polynomials satisfy recurrence relations (2.4)-(2.7) for the con-

stants µn,0/1(α,β)

and νn,0/1(α,β)

. To derive these constants, we note that the usual scaling found for the

Jacobi polynomials is that adhering to the criterion

P̂n(α,β)

(r = 1)=

(

n +α

n

)

,

Jacobi polynomial properties 15

where we have introduced the scaled Jacobi polynomials P̂n(α,β)

, which are the scaled polyno-mials that usually appear in the literature [1]. From various sources, e.g. [13], we have thatthese polynomials satisfy promotion and demotion formulae in a form similar to (2.4)-(2.7), butwith different constants. With these formulae from the literature, we have that the monicJacobi polynomials Pn

(α,β)satisfy the same promotion and demotion formulae with the con-

stants:

(1− r)Pn(α,β)

=2(n +α)(n +α + β)

(2n +α + β)(2n+ α + β + 1)Pn

(α−1,β)−Pn+1

(α−1,β)

(1 + r)Pn(α,β)

=2(n + β)(n +α + β)

(2n +α + β)(2n+ α + β + 1)Pn

(α,β−1)+ Pn+1

(α,β−1)

Pn(α,β)

= Pn(α+1,β)

−2n(n + β)

(2n + α + β)(2n +α + β + 1)Pn−1

(α+1,β)

Pn(α,β) = Pn

(α,β+1) +2n(n + α)

(2n +α + β)(2n +α + β + 1)Pn−1

(α,β+1)

The monic polynomials have norm∥

∥

∥Pn

(α,β)∥

∥

∥

2= hn

(α,β), which is given by

hn(α,β)

=22n+α+β+1(n!) Γ(n + α+β + 1) Γ(n + α + 1) Γ(n + β + 1)

Γ(2n +α + β + 1) Γ(2n +α + β + 2), (A.1)

where Γ( · ) represents the Gamma function. The above formulae allow us to translate the pro-motion/demotion formulae for the monic polynomials into those for the normalized Jacobi poly-nomials, as given in equations (2.4)-(2.7). After some algebra, we determine that the constantsin those equations are given by

µn,0(α,β) =

2(n+ α)(n +α + β)

(2n+ α + β)(2n + α + β +1)

√

(A.2)

µn,1(α,β) =

2(n +1)(n + β +1)

(2n+ α + β + 1)(2n +α + β + 2)

√

(A.3)

νn,0(α,β) =

2(n + α +1)(n +α + β + 1)

(2n+ α + β + 1)(2n +α + β + 2)

√

(A.4)

νn,−1(α,β) =

2n(n + β)

(2n+ α + β)(2n + α + β +1)

√

(A.5)

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