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Use Literate.jl in generating examples (#816)
* Use in generating examples * Add GR ENV var * Remove unused images
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‎.github/workflows/docs.yml‎

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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
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DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} # If authenticating with SSH deploy key
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GKSwstype: "100" # https://discourse.julialang.org/t/generation-of-documentation-fails-qt-qpa-xcb-could-not-connect-to-display/60988
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run: julia --project=docs/ docs/make.jl

‎.gitignore‎

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examples/.ipynb_checkpoints/RectPDE Examples-checkpoint.ipynb
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examples/.ipynb_checkpoints/Wave and Klein-Gordon equation on a square-checkpoint.ipynb
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examples/.ipynb_checkpoints/Manipulate Helmholtz-checkpoint.ipynb
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examples/.ipynb_checkpoints/RectPDE Timings-checkpoint.ipynb
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examples/.ipynb_checkpoints/PDE Examples-checkpoint.ipynb
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examples/.ipynb_checkpoints/Periodic PDEs-checkpoint.ipynb
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examples/.ipynb_checkpoints/Rectangle PDEs-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Square-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Disk-checkpoint.ipynb
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examples/.ipynb_checkpoints/PDE Timings-checkpoint.ipynb
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examples/.ipynb_checkpoints/Disk PDEs-checkpoint.ipynb
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examples/.ipynb_checkpoints/Manipulate ODEs and PDEs-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Interval-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Square (Gadfly)-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Square (GLPlot)-checkpoint.ipynb
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examples/.ipynb_checkpoints/Untitled0-checkpoint.ipynb
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examples/.ipynb_checkpoints/Floquet-checkpoint.ipynb
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examples/.ipynb_checkpoints/Sampling-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Disk (GLPlot)-checkpoint.ipynb
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examples/.ipynb_checkpoints/Time Evolution PDEs on Square (PlotlyJS)-checkpoint.ipynb
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examples/gks.svg
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examples/.ipynb_checkpoints/Fractional Integral Equations-checkpoint.ipynb
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docs/build/index.md
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docs/build
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docs/src/generated
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src/.DS_Store
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.DS_Store

‎Project.toml‎

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DualNumbers = "0.6.2"
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FFTW = "1"
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FastTransforms = "0.13, 0.14"
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Plots = "1"
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RecipesBase = "1.0"
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Reexport = "1.0"
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SpecialFunctions = "1.1, 2"
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[extras]
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Aqua = "4c88cf16-eb10-579e-8560-4a9242c79595"
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
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Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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[targets]
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test = ["Aqua", "Documenter", "Test", "Random"]
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test = ["Aqua", "Documenter", "Plots", "Random", "Test"]

‎README.md‎

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[![Join the chat at https://gitter.im/JuliaApproximation/ApproxFun.jl](https://badges.gitter.im/JuliaApproximation/ApproxFun.jl.svg)](https://gitter.im/JuliaApproximation/ApproxFun.jl?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)
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ApproxFun is a package for approximating functions. It is in a similar vein to the Matlab
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package [`Chebfun`](http://www.chebfun.org) and the Mathematica package [`RHPackage`](https://github.com/dlfivefifty/RHPackage).
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The [`ApproxFun Documentation`](https://JuliaApproximation.github.io/ApproxFun.jl/latest) contains detailed information, or read on for a brief overview of the package.
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The [`ApproxFun Documentation`](https://JuliaApproximation.github.io/ApproxFun.jl/latest) contains detailed information, or read on for a brief overview of the package. The documentation contains examples of usage, such as solving ordinary and partial differential equations.
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The [`ApproxFun Examples`](https://github.com/JuliaApproximation/ApproxFunExamples) contains many examples of
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using this package, in Jupyter notebooks and Julia scripts.
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The [`ApproxFun Examples`](https://github.com/JuliaApproximation/ApproxFunExamples) repo contains many examples of
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using this package, in Jupyter notebooks and Julia scripts. Note that this is independently maintained, so it might not always be in sync with the latest version of `ApproxFun`. We recommend checking the examples in the documentation first, as these will always be compatible with the latest version of the package.
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## Introduction
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### Approximating Functions
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Take your two favourite functions on an interval and create approximations to them as simply as:
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<img src=https://github.com/JuliaApproximation/ApproxFun.jl/raw/master/images/extrema.png width=500 height=400>
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## Differentiation and integration
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### Differentiation and integration
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Notice from above that to find the extrema, we used `'` overridden for the `differentiate` function. Several other `Julia`
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base functions are overridden for the purposes of calculus. Because the exponential is its own
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derivative, the `norm` is small:
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base functions are overridden for the purposes of calculus. We may check that the exponential is its own derivative, by evaluating the norm of the difference and checking that it is small:
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```julia
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f = Fun(x->exp(x), -1..1)
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f = Fun(exp, -1..1)
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norm(f-f') # 4.4391656415701095e-14
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```
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norm(f-g) # 3.4989733283850415e-15d
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```
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Algebraic and differential operations are also implemented where possible, and most of Julia's built-in functions are overridden to accept `Fun`s:
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Algebraic and differential operations are also implemented where possible, and most of Julia's built-in functions (and special functions from [`SpecialFunctions.jl`](https://github.com/JuliaMath/SpecialFunctions.jl)) are overridden to accept `Fun`s:
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```julia
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x = Fun()
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h = airyai(10asin(f)+2g)
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```
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## Examples of Usage
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## Solving ordinary differential equations
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We can also solve differential equations. Consider the Airy ODE `u'' - x u = 0` as a boundary value problem on `[-1000,200]` with conditions `u(-1000) = 1` and `u(200) = 2`. The unique solution is a linear combination of Airy functions. We can calculate it as follows:
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```julia
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x = Fun(identity, -1000..200) # the function x on the interval -1000..200
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D = Derivative() # The derivative operator
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B = Dirichlet() # Dirichlet conditions
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L = D^2 - x # the Airy operator
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u = [B;L] \ [[1,2],0] # Calculate u such that B*u == [1,2] and L*u == 0
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plot(u; label="u")
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```
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<img src=https://github.com/JuliaApproximation/ApproxFun.jl/raw/master/images/airy.png width=500 height=400>
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## Nonlinear Boundary Value problems
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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]`:
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```julia
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x = Fun()
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u₀ = 0.0x # initial guess
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N = u -> [u(-1)-1, u(1)+0.5, 0.001u'' + 6*(1-x^2)*u' + u^2 - 1]
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u = newton(N, u₀) # perform Newton iteration in function space
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plot(u)
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```
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<img src=https://github.com/JuliaApproximation/ApproxFun.jl/raw/master/images/nbvp.png width=500 height=400>
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One can also solve a system nonlinear ODEs with potentially nonlinear boundary conditions:
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```julia
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x = Fun(identity, 0..1)
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N = (u1,u2) -> [u1'(0) - 0.5*u1(0)*u2(0);
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u2'(0) + 1;
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u1(1) - 1;
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u2(1) - 1;
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u1'' + u1*u2;
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u2'' - u1*u2]
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u10 = one(x)
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u20 = one(x)
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u1,u2 = newton(N, [u10,u20])
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plot(u1, label="u1")
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plot!(u2, label="u2")
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```
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<img src=images/example_nonlin.png width=500 height=400>
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## Periodic functions
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There is also support for Fourier representations of functions on periodic intervals.
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Specify the space `Fourier` to ensure that the representation is periodic:
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```julia
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f = Fun(cos, Fourier(-π..π))
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norm(f' + Fun(sin, Fourier(-π..π))) # 5.923502902288505e-17
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```
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Due to the periodicity, Fourier representations allow for the asymptotic savings of `2/π`
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in the number of coefficients that need to be stored compared with a Chebyshev representation.
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ODEs can also be solved when the solution is periodic:
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```julia
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s = Chebyshev(-π..π)
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a = Fun(t-> 1+sin(cos(2t)), s)
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L = Derivative() + a
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f = Fun(t->exp(sin(10t)), s)
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B = periodic(s,0)
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uChebyshev = [B;L] \ [0.;f]
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s = Fourier(-π..π)
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a = Fun(t-> 1+sin(cos(2t)), s)
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L = Derivative() + a
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f = Fun(t->exp(sin(10t)), s)
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uFourier = L\f
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ncoefficients(uFourier)/ncoefficients(uChebyshev),2/π
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plot(uFourier)
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```
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<img src=https://github.com/JuliaApproximation/ApproxFun.jl/raw/master/images/periodic.png width=500 height=400>
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## Sampling
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Other operations including random number sampling using [Olver & Townsend 2013]. The
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following code samples 10,000 from a PDF given as the absolute value of the sine function on `[-5,5]`:
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```julia
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f = abs(Fun(sin, -5..5))
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x = ApproxFun.sample(f,10000)
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histogram(x;normed=true)
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plot!(f/sum(f))
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```
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<img src=https://github.com/JuliaApproximation/ApproxFun.jl/raw/master/images/sample.png width=500 height=400>
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## Solving partial differential equations
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We can solve PDEs, the following solves Helmholtz `Δu + 100u=0` with `u(±1,y)=u(x,±1)=1`
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on a square. This function has weak singularities at the corner,
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so we specify a lower tolerance to avoid resolving these singularities
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completely.
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```julia
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d = ChebyshevInterval()^2 # Defines a rectangle
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Δ = Laplacian(d) # Represent the Laplacian
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f = ones(∂(d)) # one at the boundary
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u = \([Dirichlet(d); Δ+100I], [f;0.]; # Solve the PDE
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tolerance=1E-5)
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surface(u) # Surface plot
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```
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<img src=https://github.com/JuliaApproximation/ApproxFun.jl/raw/master/images/helmholtz.png width=500 height=400>
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<!-- We can also evolve PDEs. The following solves advection—diffusion
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`u_t = 0.01Δu - 4u_x -3u_y` on a rectangle
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```julia
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d = ChebyshevInterval()^2
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u0 = Fun((x,y)->exp(-40(x-.1)^2-40(y+.2)^2),d)
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B = dirichlet(d)
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D = Derivative(ChebyshevInterval())
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L = (0.01D^2-4D)⊗I + I⊗(0.01D^2-3D)
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h = 0.002
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timeevolution(B,L,u0,h) # Requires GLPlot
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``` -->
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## High precision
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Solving differential equations with high precision types is available. The following calculates `e` to 300 digits by solving the ODE `u' = u`:
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```julia
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setprecision(1000) do
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d = BigFloat(0)..BigFloat(1)
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D = Derivative(d)
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u = [ldirichlet(); D-I] \ [1; 0]
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@test u(1) ≈ exp(BigFloat(1))
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end
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```
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## Self-adjoint eigenvalue problems
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This solves the confined anharmonic oscillator, `[-𝒟² + V(x)] u = λu`, where `u(±10) = 0`, `V(x) = ω*x² + x⁴`, and `ω = 25`.
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```julia
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n = 3000
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ω = 25.0
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d = Segment(-10..10)
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S = Ultraspherical(0.5, d)
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NS = NormalizedPolynomialSpace(S)
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V = Fun(x->ω*x^2+x^4, S)
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L = -Derivative(S, 2) + V
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C = Conversion(domainspace(L), rangespace(L))
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B = Dirichlet(S)
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QS = QuotientSpace(B)
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Q = Conversion(QS, S)
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D1 = Conversion(S, NS)
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D2 = Conversion(NS, S)
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R = D1*Q
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P = cache(PartialInverseOperator(C, (0, ApproxFun.bandwidth(L, 1) + ApproxFun.bandwidth(R, 1) + ApproxFun.bandwidth(C, 2))))
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A = R'D1*P*L*D2*R
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B = R'R
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SA = Symmetric(A[1:n,1:n], :L)
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SB = Symmetric(B[1:n,1:n], :L)
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λ = eigvals(SA, SB)[1:round(Int, 3n/5)]
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```
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Check the [documentation](https://JuliaApproximation.github.io/ApproxFun.jl/latest) for examples of usage.
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## References
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‎docs/Project.toml‎

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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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DomainSets = "5b8099bc-c8ec-5219-889f-1d9e522a28bf"
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306"
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Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
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SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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[compat]
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Documenter = "0.27"
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Literate = "2"
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Plots = "1"
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SpecialFunctions = "2"

‎docs/make.jl‎

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using Documenter, ApproxFun, BandedMatrices, DomainSets, LinearAlgebra
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using Literate
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# Generate examples using Literate
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# See https://github.com/fredrikekre/Literate.jl/blob/master/docs/make.jl
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example_dir = joinpath(@__DIR__, "..", "examples")
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output_dir = joinpath(@__DIR__, "src/generated")
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for example in ["ODE.jl", "PDE.jl", "Sampling.jl", "Periodic.jl",
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"Eigenvalue.jl", "NonlinearBVP.jl"]
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filename = joinpath(example_dir, example)
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Literate.markdown(filename, output_dir, documenter=true)
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end
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makedocs(
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doctest = false,
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"Operators" => "usage/operators.md",
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"Linear Equations" => "usage/equations.md"
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],
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"Examples" => [
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"generated/ODE.md",
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"generated/PDE.md",
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"generated/Sampling.md",
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"generated/Periodic.md",
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"generated/Eigenvalue.md",
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"generated/NonlinearBVP.md",
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],
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"Frequently Asked Questions" => "faq.md",
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"Library" => "library.md"
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]

‎examples/Eigenvalue.jl‎

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# # Self-adjoint Eigenvalue Problem
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# Ref:
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# [J. L. Aurentz & R. M. Slevinsky (2019), arXiv:1903.08538](https://arxiv.org/abs/1903.08538)
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# We solve the confined anharmonic oscillator
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# ```math
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# \left[-\frac{d^2}{dx^2} + V(x)\right] u = λu,
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# ```
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# where ``u(\pm 10) = 0``, ``V(x) = ωx^2 + x^4``, and ``ω = 25``.
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using ApproxFun
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using LinearAlgebra
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# Define parameters
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ω = 25.0
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d = -10..10;
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S = Legendre(d)
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NS = NormalizedPolynomialSpace(S) # NormalizedLegendre
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D1 = Conversion(S, NS)
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D2 = Conversion(NS, S);
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# Construct the differential operator
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V = Fun(x -> ω * x^2 + x^4, S)
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L = -Derivative(S, 2) + V;
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# Basis recombination to transform to one that satisfies Dirichlet boundary conditions
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B = Dirichlet(S)
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QS = QuotientSpace(B)
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Q = Conversion(QS, S)
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R = D1*Q;
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# This inversion is computed approximately, such that
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# $\mathrm{C}^-1 * \mathrm{C} ≈ \mathrm{I}$ up to a certain bandwidth
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C = Conversion(domainspace(L), rangespace(L))
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P = cache(PartialInverseOperator(C, (0, ApproxFun.bandwidth(L, 1) + ApproxFun.bandwidth(R, 1) + ApproxFun.bandwidth(C, 2))));
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A = R'D1*P*L*D2*R
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B = R'R;
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# We impose a cutoff to obtain approximate finite matrix representations
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n = 3000
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SA = Symmetric(A[1:n,1:n], :L)
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SB = Symmetric(B[1:n,1:n], :L)
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λ = eigvals(SA, SB)[1:round(Int, 3n/5)];
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# We plot the eigenvalues
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import Plots
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Plots.plot(λ; title = "Eigenvalues", legend=false)

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