A fast Chebyshev–Legendre transform using an asymptotic formula ...

A fast Chebyshev–Legendre transform usingan asymptotic formula

Alex TownsendMIT

(Joint work with Nick Hale)

SIAM OPSFA, 5th June 2015

IntroductionWhat is the Chebyshev–Legendre transform?

Suppose we have a degree N ´ 1 polynomial. It can be expressed in theChebyshev or Legendre polynomial basis:

pN´1pxq “N´1ÿ

c legk Pk pxq ðñ pN´1pxq “

N´1ÿ

cchebk Tk pxq

The Chebyshev–Legendre transform:forward transform

ÝÝÝÝÝÝÝÝÝÝÝÝÝÝá

c leg0 , . . . , c leg

N´1 OpNplog Nq2{ log log Nq ccheb0 , . . . , ccheb

N´1 .âÝÝÝÝÝÝÝÝÝÝÝÝÝ

inverse transform

Applications in:I Convolution [Hale & T., 14]I Legendre-tau spectral methodsI QR of a quasimatrix [Trefethen, 08]I best least squares approximation

Alex Townsend @ MIT 1/18

pN´1pxq “N´1ÿ

N´1ÿ

cchebk Tk pxq

c leg0 , . . . , c leg

inverse transform

pN´1pxq “N´1ÿ

N´1ÿ

cchebk Tk pxq

c leg0 , . . . , c leg

inverse transform

IntroductionRelated work

Values atChebyshev

points

Legendrecoefficients

Chebyshevcoefficients

Values inthe complex

Values atLegendre

points

Potts et al.O`

Nplog Nq2˘

Mori et al.O pN log Nq

Alpert & Rokhlin

discrete cosine transform

O pN log Nq

IserlesO pN log Nq

TygertO`

Nplog Nq2˘

New features of our algorithmsFFT-based approachAnalysis-basedNo precomputation

IntroductionThe fast Fourier transform

The FFT computes the DFT in OpN log Nq operations:

Xk “řN´1

n“0 xn ¨ e´2πink{N, 0 ď k ď N ´ 1.Top 10

algorithm

James Cooley;

sinp´2πtqe´2πit

cosp´2πtqReal axis

John Tukey;

IntroductionMany special functions are trigonometric-like

Trigonometric functionscospωxq, sinpωxq

Chebyshev polynomialsTnpxq

Legendre polynomialsPnpxq

Bessel functionsJνpzq

Airy functionsAipxq

Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.

Airy functionsAipxq

IntroductionAn asymptotic expansion of Legendre polynomials

Legendre polynomials

Pnpcosθq “ Cn

M´1ÿ

hm,ncos

pm ` n ` 12qθ ´ pm `

12qπ2

p2 sinθqm`1{2`Rn,Mpθq

Cn “

Γpn ` 1qΓpn ` 3{2q

, hm,n “

1, m “ 0,śm

j“1pj´1{2q2

jpn`j`1{2q , m ą 0. Thomas Stieltjes

Interior Region: Trig−like

Boundary Region: Bessel−like

cos(5π/6) cos(π/6)Alex Townsend @ MIT 5/18

IntroductionNumerical pitfalls of an asymptotic expansionist

Pnpcosθq “ Cn

M´1ÿ

hm,ncos

pm ` n ` 12qθ ´ pm `

12qπ2

p2 sinθqm`1{2` Rn,Mpθq

M “ 1

M “ 3

M “ 5,7

n “ 1,000

Mpθq|

Fix M. Where is the asymptotic expansion accurate?Alex Townsend @ MIT 6/18

Computing the Chebyshev–Legendre transformThe transform comes in two parts

The Chebyshev–Legendre transform naturally splits into two parts:

This bitÝÝÝÝÝÝÝá

IDCTÝÝÝÝÝÝá

c leg0 , . . . , c leg

N´1 pN´1pxcheb0 q, . . . ,pN´1pxcheb

N´1 q ccheb0 , . . . , ccheb

N´1âÝÝÝÝÝÝÝ

DCTâÝÝÝÝÝÝ

Task: Compute the following matrix-vector product in quasilinear time?

PNc “

P0pcosθ0q . . . PN´1pcosθ0q

.... . .

P0pcosθN´1q . . . PN´1pcosθN´1q

c leg0...

c legN´1

, θk “kπ

N ´ 1

Computing the Chebyshev–Legendre transformThe transform comes in two parts

The Chebyshev–Legendre transform naturally splits into two parts:

This bitÝÝÝÝÝÝÝá

IDCTÝÝÝÝÝÝá

c leg0 , . . . , c leg

N´1 pN´1pxcheb0 q, . . . ,pN´1pxcheb

N´1 q ccheb0 , . . . , ccheb

N´1âÝÝÝÝÝÝÝ

DCTâÝÝÝÝÝÝ

Task: Compute the following matrix-vector product in quasilinear time?

PNc “

P0pcosθ0q . . . PN´1pcosθ0q

.... . .

P0pcosθN´1q . . . PN´1pcosθN´1q

c leg0...

c legN´1

, θk “kπ

N ´ 1

Computing the Chebyshev–Legendre transformAsymptotic expansions as a matrix decomposition

The asymptotic expansion

Pnpcosθk q “ Cn

M´1ÿ

hm,ncos

pm ` n ` 12qθk ´ pm `

12qπ2

p2 sinθk qm`1{2

` Rn,Mpθk q

gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):

PN “

M´1ÿ

˝DumCNDChm ` Dvm

0 0 00 SN´2 00 0 0

flDChm

‚` RN,M.

“ `PN PASYN RN,M

Pnpcosθk q “ Cn

M´1ÿ

hm,ncos

pm ` n ` 12qθk ´ pm `

12qπ2

p2 sinθk qm`1{2

` Rn,Mpθk q

PN “

M´1ÿ

˝DumCNDChm ` Dvm

0 0 00 SN´2 00 0 0

flDChm

‚` RN,M.

“ `PN PASYN RN,M

Pnpcosθk q “ Cn

M´1ÿ

hm,ncos

pm ` n ` 12qθk ´ pm `

12qπ2

p2 sinθk qm`1{2

` Rn,Mpθk q

PN “

M´1ÿ

˝DumCNDChm ` Dvm

0 0 00 SN´2 00 0 0

flDChm

‚` RN,M.

“ `PN PASYN RN,M

Computing the Chebyshev–Legendre transformBe careful and stay safe

Error curve: |Rn,Mpθk q| “ ε

RN,M “

M“15M“7

M“ 6

|Rn,Mpθk q| ď2CnhM,n

p2 sinθk qM`1{2

Fix M. Where is the asymptotic expan-sion accurate?

Computing the Chebyshev–Legendre transformPartitioning and balancing competing costs

PN “ P(1)NP(2)

NP(3)N

PEVALN

αNα2Nα3N N

TheoremThe matrix-vector product f “ PNc canbe computed in OpNplog Nq2{ loglog Nqoperations.

PEVALN c

P(j)N c

OpN log Nq

α too small

PN “ P(1)NP(2)

NP(3)N

PEVALN

αNα2Nα3N N

PEVALN c

P(j)N c

OpN log Nq

α too small

PN “ P(1)NP(2)

NP(3)N

PEVALN

αNα2Nα3N N

PEVALN c

P(j)N c

OpN log Nq

α too small

PN “ P(1)NP(2)

NP(3)N

PEVALN

αNα2Nα3N N

PEVALN c

P(j)N c

OpN log Nq

OpN2 log Nq

α too large

PN “ P(1)NP(2)

NP(3)N

PEVALN

αNα2Nα3N N

PEVALN cP(j)

OpNplog Nq2{ loglog NqOpNplog Nq2{ loglog Nq

α “ Op1{ log Nq

Computing the Chebyshev–Legendre transformNumerical results

log Nq2 { lo

glog Nq

No precomputation.

Mollification of rough signals

´1Pnpxqe´iωxdx “ im

2π´ω

Jm`1{2p´ωq

Computing the Chebyshev–Legendre transformThe inverse Chebyshev–Legendre transform

The integral formula for Legendre coefficients gives the following relation:

c legN “

IN`1 |0N‰

Ds2NP2Npxcheb

2N qTDw2N

T2Npxcheb2N q

IN`10N

cchebN ,

PEVAL2N

P(1)2N

P(2)2N

P(3)2N

2α2N2α3N

i11 i12 i13

P2Npxcheb2N qT

log Nq2 { lo

glog Nq

Future workFast spherical harmonic transform

Spherical harmonic transform:

fpθ, φq “N´1ÿ

m“´l

αml P |m|l pcosθqeimφ

[Mohlenkamp, 1999], [Rokhlin & Tygert, 2006], [Tygert, 2008]

A new generation of fast transforms with no precomputation.

Thank you

References

D. H. Bailey, K. Jeyabalan, and X. S. Li, A comparison of three high-precision quadrature schemes, 2005.

P. Baratella and I. Gatteschi, The bounds for the error term of an asymptotic approximation of Jacobipolynomials, Lecture notes in Mathematics, 1988.

Z. Battels and L. N. Trefethen, An extension of Matlab to continuous functions and operators, SISC, 2004.

Iteration-free computation of Gauss–Legendre quadrature nodes and weights, SISC, 2014.

G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature rules, Math. Comput., 1969.

N. Hale and A. Townsend, Fast and accurate computation of Gauss–Legendre and Gauss–Jacobi quadraturenodes and weights, SISC, 2013.

N. Hale and A. Townsend, A fast, simple, and stable Chebyshev–Legendre transform using an asymptoticformula, SISC, 2014.

Y. Nakatsukasa, V. Noferini, and A. Townsend, Computing the common zeros of two bivariate functions viaBezout resultants, Numerische Mathematik, 2014.

References

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.

A. Townsend and L. N. Trefethen, An extension of Chebfun to two dimensions, SISC, 2013.

A. Townsend, T. Trogdon, and S. Olver, Fast computation of Gauss quadrature nodes and weights on thewhole real line, to appear in IMA Numer. Anal., 2015.

A. Townsend, The race to compute high-order Gauss quadrature, SIAM News, 2015.

A. Townsend, A fast analysis-based discrete Hankel transform using asymptotics expansions, submitted, 2015.

F. G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura. Appl., 1950.

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