A fast Chebyshev–Legendre transform using
an asymptotic formula
Alex Townsend
MIT
(Joint work with Nick Hale)
SIAM OPSFA, 5th June 2015
Introduction
What is the Chebyshev–Legendre transform?
Suppose we have a degree N ´ 1 polynomial. It can be expressed in the
Chebyshev or Legendre polynomial basis:
pN ´1 px q “
Nÿ
´1
leg
ck Pk px q
ðñ
pN ´1 px q “
k “0
Nÿ
´1
ckcheb Tk px q
k “0
The Chebyshev–Legendre transform:
forward transform
leg
leg
c0 , . . . , cN ´1
ÝÝÝÝÝÝÝÝÝÝÝÝÝÝá
OpN plog N q2 { log log N q
âÝÝÝÝÝÝÝÝÝÝÝÝÝ
c0cheb , . . . , cNcheb
.
´1
inverse transform
Applications in:
I
I
I
I
Convolution [Hale & T., 14]
Legendre-tau spectral methods
QR of a quasimatrix [Trefethen, 08]
best least squares approximation
Alex Townsend @ MIT
1/18
Introduction
What is the Chebyshev–Legendre transform?
Suppose we have a degree N ´ 1 polynomial. It can be expressed in the
Chebyshev or Legendre polynomial basis:
pN ´1 px q “
Nÿ
´1
leg
ck Pk px q
ðñ
pN ´1 px q “
k “0
Nÿ
´1
ckcheb Tk px q
k “0
The Chebyshev–Legendre transform:
forward transform
leg
leg
c0 , . . . , cN ´1
ÝÝÝÝÝÝÝÝÝÝÝÝÝÝá
OpN plog N q2 { log log N q
âÝÝÝÝÝÝÝÝÝÝÝÝÝ
c0cheb , . . . , cNcheb
.
´1
inverse transform
Applications in:
I
I
I
I
Convolution [Hale & T., 14]
Legendre-tau spectral methods
QR of a quasimatrix [Trefethen, 08]
best least squares approximation
Alex Townsend @ MIT
1/18
Introduction
What is the Chebyshev–Legendre transform?
Suppose we have a degree N ´ 1 polynomial. It can be expressed in the
Chebyshev or Legendre polynomial basis:
pN ´1 px q “
Nÿ
´1
leg
ck Pk px q
ðñ
pN ´1 px q “
k “0
Nÿ
´1
ckcheb Tk px q
k “0
The Chebyshev–Legendre transform:
forward transform
leg
leg
c0 , . . . , cN ´1
ÝÝÝÝÝÝÝÝÝÝÝÝÝÝá
OpN plog N q2 { log log N q
âÝÝÝÝÝÝÝÝÝÝÝÝÝ
c0cheb , . . . , cNcheb
.
´1
inverse transform
Applications in:
I
I
I
I
Convolution [Hale & T., 14]
Legendre-tau spectral methods
QR of a quasimatrix [Trefethen, 08]
best least squares approximation
Alex Townsend @ MIT
1/18
Introduction
Related work
discrete cosine transform
O pN log N q
Values at
Chebyshev
points
Potts et al.
`
˘
O N plog N q2
Mori et al.
O pN log N q
Values in
the complex
plane
Legendre
coefficients
Iserles
O pN log N q
Alpert & Rokhlin
O pN q
`Tygert
˘
O N plog N q2
Chebyshev
coefficients
Values at
Legendre
points
New features of our algorithms
FFT-based approach
Analysis-based
No precomputation
Alex Townsend @ MIT
2/18
Introduction
The fast Fourier transform
The FFT computes the DFT in OpN log N q operations:
ř
Xk “ nN“´01 xn ¨ e ´2πink {N ,
0 ď k ď N ´ 1.
sinp
´2π
t
q
Imag axis
e
´2πit
Top 10
algorithm
;
cos
p´2
James Cooley
t
Alex Townsend @ MIT
;
πt q
John Tukey
xis
al a
Re
3/18
Introduction
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
4/18
Introduction
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
4/18
Introduction
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
4/18
Introduction
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
4/18
Introduction
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
4/18
Introduction
Many special functions are trigonometric-like
Trigonometric functions
cospωx q,
sinpωx q
Chebyshev polynomials
Tn px q
Legendre polynomials
Pn px q
Bessel functions
Jν pz q
Airy functions
Ai px q
Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc.
Alex Townsend @ MIT
4/18
Introduction
An asymptotic expansion of Legendre polynomials
Legendre polynomials
Pn pcos θq “ Cn
Mÿ
´1
hm,n
m “0
c
Cn “
4 Γpn ` 1q
,
π Γpn ` 3{2q
˘
`
cos pm ` n ` 12 qθ ´ pm ` 21 q π2
p2 sin θqm`1{2
#
hm,n “
1,
pj ´1{2q2
j “1 j pn`j `1{2q ,
śm
` Rn,M pθq
m “ 0,
m ą 0.
Thomas Stieltjes
Boundary Region: Bessel−like
Interior Region: Trig−like
Alex Townsend @ MIT
5/18
Introduction
Numerical pitfalls of an asymptotic expansionist
`
˘
Mÿ
´1
cos pm ` n ` 12 qθ ´ pm ` 12 q π2
Pn pcos θq “ Cn
hm,n
` Rn,M pθq
p2 sin θqm`1{2
m “0
|Rn,M pθq|
M“1
M“3
n “ 1,000
M “ 5, 7
x
Fix M. Where is the asymptotic expansion accurate?
Alex Townsend @ MIT
6/18
Computing the Chebyshev–Legendre transform
The transform comes in two parts
The Chebyshev–Legendre transform naturally splits into two parts:
This bit
leg
leg
c0 , . . . , cN ´1
ÝÝÝÝÝÝÝá
âÝÝÝÝÝÝÝ
IDCT
q
pN ´1 px0cheb q, . . . , pN ´1 pxNcheb
´1
ÝÝÝÝÝÝá
DCT
c0cheb , . . . , cNcheb
´1
âÝÝÝÝÝÝ
Task: Compute the following matrix-vector product in quasilinear time?
¨
˛ ¨ leg ˛
c0
P0 pcos θ0 q . . . PN ´1 pcos θ0 q
˚
˚
‹˚ . ‹
kπ
..
..
...
‹ ˚ .. ‹
PN c “ ˚
‹ , θk “
.
.
˝
‚˝
‚
N´1
leg
P0 pcos θN ´1 q . . . PN ´1 pcos θN ´1 q
c N ´1
Alex Townsend @ MIT
7/18
Computing the Chebyshev–Legendre transform
The transform comes in two parts
The Chebyshev–Legendre transform naturally splits into two parts:
This bit
leg
leg
c0 , . . . , cN ´1
ÝÝÝÝÝÝÝá
âÝÝÝÝÝÝÝ
IDCT
q
pN ´1 px0cheb q, . . . , pN ´1 pxNcheb
´1
ÝÝÝÝÝÝá
DCT
c0cheb , . . . , cNcheb
´1
âÝÝÝÝÝÝ
Task: Compute the following matrix-vector product in quasilinear time?
¨
˛ ¨ leg ˛
c0
P0 pcos θ0 q . . . PN ´1 pcos θ0 q
˚
˚
‹˚ . ‹
kπ
..
..
...
‹ ˚ .. ‹
PN c “ ˚
‹ , θk “
.
.
˝
‚˝
‚
N´1
leg
P0 pcos θN ´1 q . . . PN ´1 pcos θN ´1 q
c N ´1
Alex Townsend @ MIT
7/18
Computing the Chebyshev–Legendre transform
Asymptotic expansions as a matrix decomposition
The asymptotic expansion
Pn pcos θk q “ Cn
Mÿ
´1
hm,n
m “0
˘
`
cos pm ` n ` 21 qθk ´ pm ` 12 q π2
p2 sin θk qm`1{2
` Rn,M pθk q
gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):
»
fi
˛
¨
0
0
0
Mÿ
´1
˝Du CN DCh ` Dv –0 SN ´2 0fl DCh ‚ ` RN,M .
PN “
m
m
m
m
m “0
0
0
0
PN
Alex Townsend @ MIT
“
PASY
N
`
RN,M
8/18
Computing the Chebyshev–Legendre transform
Asymptotic expansions as a matrix decomposition
The asymptotic expansion
Pn pcos θk q “ Cn
Mÿ
´1
hm,n
m “0
˘
`
cos pm ` n ` 21 qθk ´ pm ` 12 q π2
p2 sin θk qm`1{2
` Rn,M pθk q
gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):
»
fi
˛
¨
0
0
0
Mÿ
´1
˝Du CN DCh ` Dv –0 SN ´2 0fl DCh ‚ ` RN,M .
PN “
m
m
m
m
m “0
0
0
0
PN
Alex Townsend @ MIT
“
PASY
N
`
RN,M
8/18
Computing the Chebyshev–Legendre transform
Asymptotic expansions as a matrix decomposition
The asymptotic expansion
Pn pcos θk q “ Cn
Mÿ
´1
hm,n
m “0
˘
`
cos pm ` n ` 21 qθk ´ pm ` 12 q π2
p2 sin θk qm`1{2
` Rn,M pθk q
gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):
»
fi
˛
¨
0
0
0
Mÿ
´1
˝Du CN DCh ` Dv –0 SN ´2 0fl DCh ‚ ` RN,M .
PN “
m
m
m
m
m “0
0
0
0
PN
Alex Townsend @ MIT
“
PASY
N
`
RN,M
8/18
Computing the Chebyshev–Legendre transform
Be careful and stay safe
Error curve: |Rn,M pθk q| “ Fix M. Where is the asymptotic expansion accurate?
RN,M “
M “5
M “7
M “ 15
|Rn,M pθk q| ď
Alex Townsend @ MIT
M“ 6
2Cn hM,n
p2 sin θk qM `1{2
9/18
Computing the Chebyshev–Legendre transform
Partitioning and balancing competing costs
α3 N α2 N
αN
N
PEVAL
N
PN “
(3)
PN
Theorem
(2)
PN
The matrix-vector product f “ PN c can
be computed in OpN plog N q2 { loglog N q
operations.
(1)
PN
α too small
PEVAL
c
N
OpN 2 q
(j)
PN c
OpN log N q
Alex Townsend @ MIT
10/18
Computing the Chebyshev–Legendre transform
Partitioning and balancing competing costs
α3 N α2 N
αN
N
PEVAL
N
PN “
(3)
PN
Theorem
(2)
PN
The matrix-vector product f “ PN c can
be computed in OpN plog N q2 { loglog N q
operations.
(1)
PN
α too small
PEVAL
c
N
OpN 2 q
(j)
PN c
OpN log N q
Alex Townsend @ MIT
10/18
Computing the Chebyshev–Legendre transform
Partitioning and balancing competing costs
α3 N α2 N
αN
N
PEVAL
N
PN “
(3)
PN
Theorem
(2)
PN
The matrix-vector product f “ PN c can
be computed in OpN plog N q2 { loglog N q
operations.
(1)
PN
α too small
PEVAL
c
N
OpN 2 q
(j)
PN c
OpN log N q
Alex Townsend @ MIT
10/18
Computing the Chebyshev–Legendre transform
Partitioning and balancing competing costs
α3 N α2 N
αN
N
PEVAL
N
PN “
(3)
PN
Theorem
(2)
PN
The matrix-vector product f “ PN c can
be computed in OpN plog N q2 { loglog N q
operations.
(1)
PN
(j)
PN c
α too large
OpN 2 log N q
PEVAL
c
N
OpN log N q
Alex Townsend @ MIT
11/18
Computing the Chebyshev–Legendre transform
Partitioning and balancing competing costs
α3 N α2 N
αN
N
PEVAL
N
PN “
(3)
PN
Theorem
(2)
PN
The matrix-vector product f “ PN c can
be computed in OpN plog N q2 { loglog N q
operations.
(1)
PN
α “ Op1{ log Nq
(j)
PN c
OpN plog N q2 { loglog N q
Alex Townsend @ MIT
PEVAL
c
N
OpN plog N q2 { loglog N q
12/18
Computing the Chebyshev–Legendre transform
Numerical results
Op
N2
q
Mollification of rough signals
g
plo
N
Op
gN
o
l
og
2 {l
q
N
q
d
ż1
Pn px qe ´iωx dx “ im
No precomputation.
´1
Alex Townsend @ MIT
2π
Jm`1{2 p´ωq
´ω
13/18
Computing the Chebyshev–Legendre transform
The inverse Chebyshev–Legendre transform
The integral formula for Legendre coefficients gives the following relation:
„

“
‰
IN `1 cheb
leg
cheb T
cheb
c N “ IN `1 | 0N Ds 2N P2N px 2N q Dw 2N T2N px 2N q
cN ,
0N
2αN
PEVAL
2N
T
(3) T
P2N
(2) T
P2N
(1) T
P2N
2N
Alex Townsend @ MIT
qT
P2N px cheb
2N
Op
N2
q
2α3 N
2α2 N
i31
Execution time
i11 i21
g
plo
N
Op
g
glo
Nq
o
2 {l
q
N
N
14/18
Future work
Fast spherical harmonic transform
Spherical harmonic transform:
f pθ, φq “
Nÿ
´1
l
ÿ
|m|
imφ
αm
l Pl pcos θqe
l “0 m“´l
[Mohlenkamp, 1999], [Rokhlin & Tygert, 2006], [Tygert, 2008]
A new generation of fast transforms with no precomputation.
Alex Townsend @ MIT
15/18
Thank you
Thank you
Alex Townsend @ MIT
16/18
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 Jacobi
polynomials, 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 quadrature
nodes and weights, SISC, 2013.
N. Hale and A. Townsend, A fast, simple, and stable Chebyshev–Legendre transform using an asymptotic
formula, SISC, 2014.
Y. Nakatsukasa, V. Noferini, and A. Townsend, Computing the common zeros of two bivariate functions via
Bezout resultants, Numerische Mathematik, 2014.
Alex Townsend @ MIT
17/18
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 the
whole 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.
Alex Townsend @ MIT
18/18