Numerical computations with functions
defined on the sphere and disk
Alex Townsend
MIT
Grady Wright
Heather Wilber
NA and PDE Seminar, University of Delaware, 12th November 2015
Introduction
A trend in the computing era: adaptive algorithms
besselj(0, 3/2)
quadgk(f, a, b)
fft( x )
For example: (Code will soon be available as part of Chebfun)
f = spherefun( @(x,y,z) cos(1 + 2*pi*(x + y) + 5*sin(pi*z)) )
Goal: Adaptive algorithms for computing with functions on the sphere/disk.
Alex Townsend @ MIT
1/26
Introduction
Proxy computing
Proxy computing: Replace number/function by a substitute.
>> methods spherefun
abs, cdr, coeffs2, cos, diff, feval, grad, length, max, max2,
mean, min, min2, norm, plot, plus, pivots, poisson, real, restrict,
roots, rotate, sum2, surfacearea, trace, transpose, uplus, etc.
Noteworthy proxy software systems:
RKToolbox
Driscoll, Hale, Trefethen, and many
others
Alex Townsend @ MIT
Guttel and Berljafa
2/26
Introduction
Existing methods for the sphere
We desire:
1. Regularity preserving
2. “Uniform” resolution
3. Adaptive algorithms
Three ideas:
1. DFS method
2. Low rank approx.
3. Uniform grids Ñ FFT
Alex Townsend @ MIT
3/26
The double Fourier sphere (DFS) method
Part I: The double Fourier sphere method
Alex Townsend @ MIT
4/26
The double Fourier sphere (DFS) method
The globe
(a)
180θ{π
(c)
180θ{π
(b)
180λ{π
Alex Townsend @ MIT
180λ{π
5/26
The double Fourier sphere (DFS) method
Extended functions have an interesting symmetry
h
on sphere
f pλ, θq
r´π, πs ˆ r0, πs
f̃ pλ, θq
r´π, πs ˆ r´π, πs
180θ{π
g
f px, y, z q
flipph q
flippg q
Key papers: [Merilees, 1973], [Orszag,
1974], [Fornberg, 1995], [Cheong, 2000],
[Torres & Coutsias, 1999].
180λ{π
Steven Orszag
Alex Townsend @ MIT
6/26
The double Fourier sphere (DFS) method
Fourier series on spheres [Orszag, 1974]
Fourier series on spheres:
f̃ pλ, θq «
n
{2
ÿ
m
{2
ÿ
ajk e ijθ e ik λ
j “´n{2 k “´m{2
fi
»
fi »
fi »
f´
´f ´
g
h
f`
f`
Preserving symmetry:
fl
–
fl“–
fl ` –
even-odd decomp. f̃ “
´
´
`
`
´flippf q flippf q
flipph q flippg q
flippf q flippf q
[Yee, 1980]
looooooooooomooooooooooon loooooooooooomoooooooooooon
even, π-periodic
π
f ` “ pg `h q{2,
odd, π-antiperiodic
f ´ “ pg ´h q{2
θ
´π
´π
Alex Townsend @ MIT
Overresolving the poles
λ
π
7/26
Low rank approximation
Part II: Low rank approximation
Alex Townsend @ MIT
8/26
Low rank approximation
Skeleton grids [Goreinov, Tyrtyshnikov & Zamarashkin, 1997]
A tensor-product grid
A skeleton grid
“Skeleton grid” = criss-cross of
1D uniform grids.
Alleviates overresolving at
the poles
Computations can be
done with 1D algorithms
FFT applicable
Idea: Use a “skeleton grid”, constructed by a criss-cross of 1D uniform grids.
Related work: [Bebendorf 2000], [Carvajal, Chapman & Geddes, 2005], [Goreinov, Tyrtyshnikov & Zamarashkin,
1997], [Halko, Martinsson, & Tropp, 2011], [Hackbusch, 2009], [Mahoney & Drineas, 2009].
Alex Townsend @ MIT
9/26
Low rank approximation
Skeleton grids [Goreinov, Tyrtyshnikov & Zamarashkin, 1997]
A tensor-product grid
A skeleton grid
“Skeleton grid” = criss-cross of
1D uniform grids.
Alleviates overresolving at
the poles
Computations can be
done with 1D algorithms
FFT applicable
Idea: Use a “skeleton grid”, constructed by a criss-cross of 1D uniform grids.
Related work: [Bebendorf 2000], [Carvajal, Chapman & Geddes, 2005], [Goreinov, Tyrtyshnikov & Zamarashkin,
1997], [Halko, Martinsson, & Tropp, 2011], [Hackbusch, 2009], [Mahoney & Drineas, 2009].
Alex Townsend @ MIT
9/26
Low rank approximation
Skeleton grids [Goreinov, Tyrtyshnikov & Zamarashkin, 1997]
A tensor-product grid
A skeleton grid
“Skeleton grid” = criss-cross of
1D uniform grids.
Alleviates overresolving at
the poles
Computations can be
done with 1D algorithms
FFT applicable
Idea: Use a “skeleton grid”, constructed by a criss-cross of 1D uniform grids.
Related work: [Bebendorf 2000], [Carvajal, Chapman & Geddes, 2005], [Goreinov, Tyrtyshnikov & Zamarashkin,
1997], [Halko, Martinsson, & Tropp, 2011], [Hackbusch, 2009], [Mahoney & Drineas, 2009].
Alex Townsend @ MIT
9/26
Low rank approximation
Skeleton grids [Goreinov, Tyrtyshnikov & Zamarashkin, 1997]
A tensor-product grid
A skeleton grid
“Skeleton grid” = criss-cross of
1D uniform grids.
Alleviates overresolving at
the poles
Computations can be
done with 1D algorithms
FFT applicable
Idea: Use a “skeleton grid”, constructed by a criss-cross of 1D uniform grids.
Related work: [Bebendorf 2000], [Carvajal, Chapman & Geddes, 2005], [Goreinov, Tyrtyshnikov & Zamarashkin,
1997], [Halko, Martinsson, & Tropp, 2011], [Hackbusch, 2009], [Mahoney & Drineas, 2009].
Alex Townsend @ MIT
9/26
Low rank approximation
Skeleton grids [Goreinov, Tyrtyshnikov & Zamarashkin, 1997]
A tensor-product grid
A skeleton grid
“Skeleton grid” = criss-cross of
1D uniform grids.
Alleviates overresolving at
the poles
Computations can be
done with 1D algorithms
FFT applicable
Idea: Use a “skeleton grid”, constructed by a criss-cross of 1D uniform grids.
Related work: [Bebendorf 2000], [Carvajal, Chapman & Geddes, 2005], [Goreinov, Tyrtyshnikov & Zamarashkin,
1997], [Halko, Martinsson, & Tropp, 2011], [Hackbusch, 2009], [Mahoney & Drineas, 2009].
Alex Townsend @ MIT
9/26
Low rank approximation
Skeleton grids [Goreinov, Tyrtyshnikov & Zamarashkin, 1997]
A tensor-product grid
A skeleton grid
“Skeleton grid” = criss-cross of
1D uniform grids.
Alleviates overresolving at
the poles
Computations can be
done with 1D algorithms
FFT applicable
Idea: Use a “skeleton grid”, constructed by a criss-cross of 1D uniform grids.
Related work: [Bebendorf 2000], [Carvajal, Chapman & Geddes, 2005], [Goreinov, Tyrtyshnikov & Zamarashkin,
1997], [Halko, Martinsson, & Tropp, 2011], [Hackbusch, 2009], [Mahoney & Drineas, 2009].
Alex Townsend @ MIT
9/26
Low rank approximation
Gaussian elimination on functions [T. & Trefethen 2013]
Standard point of view:
A
Alex Townsend @ MIT
PL
U
10/26
Low rank approximation
Gaussian elimination on functions [T. & Trefethen 2013]
Standard point of view:
Different point of view:
A ÐÝ A ´ A pj, :qA p:, k q{A pj, k q
f ÐÝ f ´ f px, :qf p:, y q{f px, y q
Each step of GE is a rank-1 update.
Alex Townsend @ MIT
A
PL
U
(GE step for matrices)
(GE step for functions)
10/26
Low rank approximation
Gaussian elimination on functions [T. & Trefethen 2013]
Standard point of view:
Different point of view:
A ÐÝ A ´ A pj, :qA p:, k q{A pj, k q
f ÐÝ f ´ f px, :qf p:, y q{f px, y q
Each step of GE is a rank-1 update.
Alex Townsend @ MIT
A
PL
U
(GE step for matrices)
(GE step for functions)
10/26
Low rank approximation
GE pivot locations
cosp10px 2 ` y qq ` sinp10px ` y 2 qq
Aip5px ` y 2 qqAip´5px 2 ` y 2 qq
1{p1 ` 100p 21 ´ x 2 ´ y 2 q2 q
Here we used complete pivoting, i.e., pick the absolute max. as the pivot.
Alex Townsend @ MIT
11/26
Low rank approximation
GE with any standard pivoting does not preserve symmetry
π
f̃ pλ, θq «
pλ˚ , θ˚ q
f̃ pλ˚ , θqf̃ pλ, θ˚ q
f̃ pλ˚ , θ˚ q
“ d1 c1 pθqr1 pλq
θ
´π
´π
λ
π
Problem: GE with any standard pivoting does not preserve symmetry.
Alex Townsend @ MIT
12/26
Low rank approximation
GE with any standard pivoting does not preserve symmetry
π
f̃ pλ, θq «
pλ˚ , θ˚ q
f̃ pλ˚ , θqf̃ pλ, θ˚ q
f̃ pλ˚ , θ˚ q
“ d1 c1 pθqr1 pλq
θ
´π
´π
λ
π
Problem: GE with any standard pivoting does not preserve symmetry.
Alex Townsend @ MIT
12/26
Low rank approximation
GE with any standard pivoting does not preserve symmetry
π
f̃ pλ, θq «
pλ˚ , θ˚ q
f̃ pλ˚ , θqf̃ pλ, θ˚ q
f̃ pλ˚ , θ˚ q
“ d1 c1 pθqr1 pλq
θ
´π
´π
λ
π
Problem: GE with any standard pivoting does not preserve symmetry.
Alex Townsend @ MIT
12/26
Low rank approximation
GE with any standard pivoting does not preserve symmetry
π
f̃ pλ, θq «
pλ˚ , θ˚ q
f̃ pλ˚ , θqf̃ pλ, θ˚ q
f̃ pλ˚ , θ˚ q
“ d1 c1 pθqr1 pλq
θ
´π
´π
λ
π
Problem: GE with any standard pivoting does not preserve symmetry.
Alex Townsend @ MIT
12/26
Low rank approximation
Structure preserving GE, motivated by [Bunch & Parlett 1971]
π
«
pλ ´ π, θ q
˚
˚
”
ı
f̃ pλ, θq « f̃ pλ ´ π, θq f̃ pλ , θq M ´1
pλ , θ q
˚
˚
θ
˚
˚
f̃ pλ, θ˚ q
ff
f̃ pλ, ´θ˚ q
Rank 2
approx.
pλ˚´ π, ´θ˚q pλ˚ , ´θ˚ q
´π
´π
λ
π
Problem: GE with 2 ˆ 2 pivots
does preserve symmetry.
No pole
singularity
M ´1 is replaced by an -pseudoinverse.
Alex Townsend @ MIT
13/26
Low rank approximation
Structure preserving GE, motivated by [Bunch & Parlett 1971]
π
«
pλ ´ π, θ q
˚
˚
”
ı
f̃ pλ, θq « f̃ pλ ´ π, θq f̃ pλ , θq M ´1
pλ , θ q
˚
˚
θ
˚
˚
f̃ pλ, θ˚ q
ff
f̃ pλ, ´θ˚ q
Rank 2
approx.
pλ˚´ π, ´θ˚q pλ˚ , ´θ˚ q
´π
´π
λ
π
Problem: GE with 2 ˆ 2 pivots
does preserve symmetry.
No pole
singularity
M ´1 is replaced by an -pseudoinverse.
Alex Townsend @ MIT
13/26
Low rank approximation
No pole singularities
Rank 2
Rank 4
Rank 6
Rank 2
Rank 4
Rank 6
Standard GE
Structure
preserving GE
Alex Townsend @ MIT
14/26
Low rank approximation
Skeleton grid review
cospxyz q
cospxz ´ sin y q
f̃ pλ, θq « d1 c1 pθqr1 pλq
` d2 c2 pθqr2 pλq ` . . .
` dK cK pθqrK pλq
Observation:
Mathematically,
most
functions are of infinite
rank but can be of low
numerical rank.
[Chapter 3, T. 2014]
Rank 5
Alex Townsend @ MIT
Rank 17
15/26
Low rank approximation
Approximation theory for our GE procedure
Observation: SVD is optimal, but GE
is near-optimal.
f px, y, z q “ 1{p1 ` γx 2 y 2 z 2 q
Best (SVD)
GE
γ“
L2 error
γ“
10
00
10
Rank of approximant
Alex Townsend @ MIT
Exact recovery: If f pλ, θq has a finite
spherical harmonic expansion, then GE
exactly recovers f .
Convergence: If f̃ pλ, θq is analytic in
both variables separably in a sufficiently
large region, then our GE procedure
constructs a sequence of low rank approximants that converge geometrically
to f̃ .
More details in [T. , Wilber, Wright, 2015].
16/26
Operations and differential equations
Part III: Operations and differential equations
Alex Townsend @ MIT
17/26
Operations and differential equations
Tensor product operations
Idea: Two-dimensional operations can exploit 1D Fourier technology
f̃ pλ, θq
K
ÿ
dk lo
cokmo
pθon
q
«
k “1
2π-periodic
sum2( f ):
ż
f px, y, z qdxdydz «
S2
K
ÿ
k “1
żπ
dk
0
rkomo
pλoqn
lo
2π-periodic
ck pθq sin θdθ
żπ
´π
rk pλqdλ
diff( f ):
» fi
˙ˆ
˙ ÿ
˙
1
K ˆ
K ˆ
t
ÿ
ck pθq
B rk pλq
B ck pθq
Bf
–
fl
pcos λrk pλqq
“ 0 ¨∇S2 f « ´
sin λ
`
cos θ
Bx
sin
θ
B
λ
B
θ
k “1
k “1
0
Others include:
Very different to [Cheong 2000] and [Yee 1981].
f(x,y,z), curl(f), divergence(F), gradient(f), lu(f), sample(f)
Alex Townsend @ MIT
18/26
Operations and differential equations
Tensor product operations
Idea: Two-dimensional operations can exploit 1D Fourier technology
f̃ pλ, θq
K
ÿ
dk lo
cokmo
pθon
q
«
k “1
2π-periodic
sum2( f ):
ż
f px, y, z qdxdydz «
S2
K
ÿ
k “1
żπ
dk
0
rkomo
pλoqn
lo
2π-periodic
ck pθq sin θdθ
żπ
´π
rk pλqdλ
diff( f ):
» fi
˙ˆ
˙ ÿ
˙
1
K ˆ
K ˆ
t
ÿ
ck pθq
B rk pλq
B ck pθq
Bf
–
fl
pcos λrk pλqq
“ 0 ¨∇S2 f « ´
sin λ
`
cos θ
Bx
sin
θ
B
λ
B
θ
k “1
k “1
0
Others include:
Very different to [Cheong 2000] and [Yee 1981].
f(x,y,z), curl(f), divergence(F), gradient(f), lu(f), sample(f)
Alex Townsend @ MIT
18/26
Operations and differential equations
Tensor product operations
Idea: Two-dimensional operations can exploit 1D Fourier technology
f̃ pλ, θq
K
ÿ
dk lo
cokmo
pθon
q
«
k “1
2π-periodic
sum2( f ):
ż
f px, y, z qdxdydz «
S2
K
ÿ
k “1
żπ
dk
0
rkomo
pλoqn
lo
2π-periodic
ck pθq sin θdθ
żπ
´π
rk pλqdλ
diff( f ):
» fi
˙ˆ
˙ ÿ
˙
1
K ˆ
K ˆ
t
ÿ
ck pθq
B rk pλq
B ck pθq
Bf
–
fl
pcos λrk pλqq
“ 0 ¨∇S2 f « ´
sin λ
`
cos θ
Bx
sin
θ
B
λ
B
θ
k “1
k “1
0
Others include:
Very different to [Cheong 2000] and [Yee 1981].
f(x,y,z), curl(f), divergence(F), gradient(f), lu(f), sample(f)
Alex Townsend @ MIT
18/26
Operations and differential equations
Tensor product operations
Idea: Two-dimensional operations can exploit 1D Fourier technology
f̃ pλ, θq
K
ÿ
dk lo
cokmo
pθon
q
«
k “1
2π-periodic
sum2( f ):
ż
f px, y, z qdxdydz «
S2
K
ÿ
k “1
żπ
dk
0
rkomo
pλoqn
lo
2π-periodic
ck pθq sin θdθ
żπ
´π
rk pλqdλ
diff( f ):
» fi
˙ˆ
˙ ÿ
˙
1
K ˆ
K ˆ
t
ÿ
ck pθq
B rk pλq
B ck pθq
Bf
–
fl
pcos λrk pλqq
“ 0 ¨∇S2 f « ´
sin λ
`
cos θ
Bx
sin
θ
B
λ
B
θ
k “1
k “1
0
Others include:
Very different to [Cheong 2000] and [Yee 1981].
f(x,y,z), curl(f), divergence(F), gradient(f), lu(f), sample(f)
Alex Townsend @ MIT
18/26
Operations and differential equations
Vector calculus operations
Great for vector calculus: div, grad, and curl are surprisingly efficient.
Why? Based on 1D algorithms for Fourier series
How? All standard vector calculus operations on
the sphere can be expressed in terms of
Bt f
,
Bx
Bt f
,
By
Bt f
.
Bz
% Rossby-Haurtwitz benchmark problem
psi = spherefun(@(la,th) cos(th) + sin(th).ˆ4.*cos(th).*cos(4*la));
u = curl( psi ); % vector-valued spherefun
plot( vort( u ) ), hold on, quiver( u )
Alex Townsend @ MIT
19/26
Operations and differential equations
Composition operators: e.g. f + g, f.*g, and exp(f)
cosp5pxy ` z qq
ˆ
Rank 15
Ai p5px ´ y ` z qq
Rank 37
“
h
Rank 43 ! 15 ˆ 37 “ 645
Option: (Call the constructor)
h = spherefun( @(la, th) f(la,th).*g(la,th) ); % h = f .* g
Composition operations call the constructor. (f+g is THE exception.)
Alex Townsend @ MIT
20/26
Operations and differential equations
Fourier spectral method for Poisson’s equation
Poisson equation on the sphere
ż
∇ u “ f,
2
px, y, z q P S ,
2
upx, y, z qdxdydz “ 0.
S2
Poisson equation in DFS method
psin θq2
B ũ B 2 ũ
B 2 ũ
2
`
cos
θ
sin
θ
` 2 “ psin θq f̃ ,
2
Bθ Bλ
B θ
Ansatz:
ũpλ, θq «
n
{2
ÿ
pλ, θq P r´π, πs ˆ r´π, πs.
m
{2
ÿ
ujk e ijθ e ik λ
j “´n{2 k “´m{2
Because B 2 pe ik λ q{B λ2 “ ´k 2 e ik λ , Poisson’s equation decouples.
Alex Townsend @ MIT
21/26
Operations and differential equations
Fourier spectral method for Poisson’s equation
Poisson equation on the sphere
ż
∇ u “ f,
2
px, y, z q P S ,
2
upx, y, z qdxdydz “ 0.
S2
Poisson equation in DFS method
psin θq2
B ũ B 2 ũ
B 2 ũ
2
`
cos
θ
sin
θ
` 2 “ psin θq f̃ ,
2
Bθ Bλ
B θ
Ansatz:
ũpλ, θq «
n
{2
ÿ
pλ, θq P r´π, πs ˆ r´π, πs.
m
{2
ÿ
ujk e ijθ e ik λ
j “´n{2 k “´m{2
Because B 2 pe ik λ q{B λ2 “ ´k 2 e ik λ , Poisson’s equation decouples.
Alex Townsend @ MIT
21/26
Operations and differential equations
Fourier spectral method for Poisson’s equation
Poisson equation on the sphere
ż
∇ u “ f,
2
px, y, z q P S ,
2
upx, y, z qdxdydz “ 0.
S2
Poisson equation in DFS method
psin θq2
B ũ B 2 ũ
B 2 ũ
2
`
cos
θ
sin
θ
` 2 “ psin θq f̃ ,
2
Bθ Bλ
B θ
Ansatz:
ũpλ, θq «
n
{2
ÿ
pλ, θq P r´π, πs ˆ r´π, πs.
m
{2
ÿ
ujk e ijθ e ik λ
j “´n{2 k “´m{2
Because B 2 pe ik λ q{B λ2 “ ´k 2 e ik λ , Poisson’s equation decouples.
Alex Townsend @ MIT
21/26
Operations and differential equations
Fourier spectral method for Poisson’s equation (cont.)
ż
2
2
∇ u “ sinp50xyz q, px, y, z q P S ,
upx, y, z qdxdydz “ 0.
S2
10
101
Op
n2
q
Computational time (sec)
upx, y, z q
2
100
10-1
10-2
10-3
102
103
104
105
n2 {2
106
107
108
f = spherefun(@(x,y,z) sin(50*x.*y.*z)); u = Poisson(f,0,150,150);
When are spherical harmonic expansions needed?
Alex Townsend @ MIT
22/26
Operations and differential equations
Fourier spectral method for Poisson’s equation (cont.)
ż
2
2
∇ u “ sinp50xyz q, px, y, z q P S ,
upx, y, z qdxdydz “ 0.
S2
10
101
Op
n2
q
Computational time (sec)
upx, y, z q
2
100
10-1
10-2
10-3
102
103
104
105
n2 {2
106
107
108
f = spherefun(@(x,y,z) sin(50*x.*y.*z)); u = Poisson(f,0,150,150);
When are spherical harmonic expansions needed?
Alex Townsend @ MIT
22/26
Disk computations
Active biological fluids on the disk NOT the sphere
High-order variant of Navier–Stokes:
Bv
` pv ¨ ∇q v “ ´∇p ` Γ0 ∇2 v ´ Γ2 ∇4 v ` Γ4 ∇6 v,
Bt
on the unit disk
∇¨v “0
Bacillus subtilis
Lab vorticity Numerical vorticity
Jonasz Slomka Jörn Dunkel
For periodic bcs, a collective “swarming” motion.
What is the impact of various geometries on the vortices?
Alex Townsend @ MIT
[Slomka & Dunkel, 2015]
23/26
Disk computations
The same symmetry holds for the disk
1
1
g
0
´π
h
π
flipph q
flippg q
´1
´π
Unit disk
r´π, πs ˆ r0, 1s
π
r´π, πs ˆ r´1, 1s
We use Chebyshev technology in r and Fourier in θ.
Thanks to OOP, diskfun and spherefun use the same code in many places.
f = diskfun(@(x,y) 2.ˆ(-cos(4*x)) + sin(7*pi*x.*sin(y-.1)), ’cart’);
Alex Townsend @ MIT
24/26
Disk computations
The same symmetry holds for the disk
1
1
g
0
´π
h
π
flipph q
flippg q
´1
´π
Unit disk
r´π, πs ˆ r0, 1s
π
r´π, πs ˆ r´1, 1s
We use Chebyshev technology in r and Fourier in θ.
Thanks to OOP, diskfun and spherefun use the same code in many places.
f = diskfun(@(x,y) 2.ˆ(-cos(4*x)) + sin(7*pi*x.*sin(y-.1)), ’cart’);
Alex Townsend @ MIT
24/26
Disk computations
The same symmetry holds for the disk
1
1
g
0
´π
h
π
flipph q
flippg q
´1
´π
Unit disk
r´π, πs ˆ r0, 1s
π
r´π, πs ˆ r´1, 1s
We use Chebyshev technology in r and Fourier in θ.
Thanks to OOP, diskfun and spherefun use the same code in many places.
f = diskfun(@(x,y) 2.ˆ(-cos(4*x)) + sin(7*pi*x.*sin(y-.1)), ’cart’);
Alex Townsend @ MIT
24/26
Disk computations
The same symmetry holds for the disk
1
1
g
0
´π
h
π
flipph q
flippg q
´1
´π
Unit disk
r´π, πs ˆ r0, 1s
π
r´π, πs ˆ r´1, 1s
We use Chebyshev technology in r and Fourier in θ.
Thanks to OOP, diskfun and spherefun use the same code in many places.
f = diskfun(@(x,y) 2.ˆ(-cos(4*x)) + sin(7*pi*x.*sin(y-.1)), ’cart’);
Alex Townsend @ MIT
24/26
Demo
Demo
Alex Townsend @ MIT
25/26
Thank you
Sphere and disk computations
without worrying about the
underlying discretization
More information in: T., H. Wilber, and G. Wright, “Computing with functions in
spherical and polar geometries I. The sphere”, submitted to SISC.
Code soon to be part of Chebfun: http://www.chebfun.org/
Alex Townsend @ MIT
26/26