# ApproxFun.jl

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.

## Introduction

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))
```

## Differentiation and integration

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)
```

## Solving ordinary differential equations

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)
```

## Nonlinear Boundary Value problems

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)
```

## Periodic functions

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)
```

## Sampling

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))
```

## Solving partial differential equations

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
```

## High precision

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 u(1) ≈ exp(BigFloat(1))
end
```

## References

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