ApproxFun is a package for approximating functions. It is in a similar vein to the Matlab
package Chebfun
and the Mathematica package RHPackage
.
The ApproxFun Documentation
contains detailed information, or read on for a brief overview of the package.
Take your two favourite functions on an interval and create approximations to them as simply as:
using ApproxFun
x = Fun(identity,0..10)
f = sin(x^2)
g = cos(x)
Evaluating f(.1)
will return a high
accuracy approximation to sin(0.01)
. All the algebraic manipulations of functions
are supported and more. For example, we can add f
and g^2
together and compute
the roots and extrema:
h = f + g^2
r = roots(h)
rp = roots(h')
using Plots
plot(h)
scatter!(r,h(r))
scatter!(rp,h(rp))
Notice from above that to find the extrema, we used '
overridden for the differentiate
function. Several other Julia
base functions are overridden for the purposes of calculus. Because the exponential is its own
derivative, the norm
is small:
f = Fun(x->exp(x),-1..1)
norm(f-f')
Similarly, cumsum
defines an indefinite integration operator:
g = cumsum(f)
g = g + f(-1)
norm(f-g)
Algebraic and differential operations are also implemented where possible, and most of Julia's built-in functions are overridden to accept Fun
s:
x = Fun()
f = erf(x)
g = besselj(3,exp(f))
h = airyai(10asin(f)+2g)
Solve the Airy ODE u'' - x u = 0
as a BVP on [-1000,200]
:
x = Fun(identity,-1000..200)
d = domain(x)
D = Derivative(d)
B = dirichlet(d)
L = D^2 - x
u = [B;L] \ [airyai(d.a);airyai(d.b);0]
plot(u)
Solve a nonlinear boundary value problem satisfying the ODE 0.001u'' + 6*(1-x^2)*u' + u^2 = 1
with boundary conditions u(-1)==1
, u(1)==-0.5
on [-1,1]
:
x=Fun()
u0=0.0x
N=u->[u(-1)-1,u(1)+0.5,0.001u''+6*(1-x^2)*u'+u^2-1]
u=newton(N,u0)
plot(u)
There is also support for Fourier representations of functions on periodic intervals.
Specify the space Fourier
to ensure that the representation is periodic:
f = Fun(cos,Fourier(-π..π))
norm(f' + Fun(sin,Fourier(-π..π))
Due to the periodicity, Fourier representations allow for the asymptotic savings of 2/π
in the number of coefficients that need to be stored compared with a Chebyshev representation.
ODEs can also be solved when the solution is periodic:
s = Chebyshev(-π..π)
a = Fun(t-> 1+sin(cos(2t)),s)
L = Derivative() + a
f = Fun(t->exp(sin(10t)),s)
B = periodic(s,0)
uChebyshev = [B;L]\[0.;f]
s = Fourier(-π..π)
a = Fun(t-> 1+sin(cos(2t)),s)
L = Derivative() + a
f = Fun(t->exp(sin(10t)),s)
uFourier = L\f
ncoefficients(uFourier)/ncoefficients(uChebyshev),2/π
plot(uFourier)
Other operations including random number sampling using [Olver & Townsend 2013]. The
following code samples 10,000 from a PDF given as the absolute value of the sine function on [-5,5]
:
f = abs(Fun(sin,-5..5))
x = ApproxFun.sample(f,10000)
histogram(x;normed=true)
plot!(f/sum(f))
We can solve PDEs, the following solves Helmholtz Δu + 100u=0
with u(±1,y)=u(x,±1)=1
on a square. This function has weak singularities at the corner,
so we specify a lower tolerance to avoid resolving these singularities
completely.
d = Interval()^2 # Defines a rectangle
Δ = Laplacian(d) # Represent the Laplacian
f = ones(∂(d)) # one at the boundary
u = \([Dirichlet(d);Δ+100I],[f;0.]; # Solve the PDE
tolerance=1E-5)
surface(u) # Surface plot
Solving differential equations with high precision types is available. The following calculates e
to 300 digits by solving the ODE u' = u
:
setprecision(1000) do
d=BigFloat(0)..BigFloat(1)
D=Derivative(d)
u=[ldirichlet();D-I]\[1;0]
@test_approx_eq u(1) exp(BigFloat(1))
end
S. Olver & A. Townsend (2014), A practical framework for infinite-dimensional linear algebra, Proceedings of the 1st First Workshop for High Performance Technical Computing in Dynamic Languages, 57–62
A. Townsend & S. Olver (2014), The automatic solution of partial differential equations using a global spectral method, J. Comp. Phys., 299: 106–123
S. Olver & A. Townsend (2013), Fast inverse transform sampling in one and two dimensions, arXiv:1307.1223
S. Olver & A. Townsend (2013), A fast and well-conditioned spectral method, SIAM Review, 55:462–489