diff --git a/.gitignore b/.gitignore
index a1e6a26..745deb1 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,4 +1,4 @@
-examples/.ipynb_checkpoints/Manipulate SIEs-checkpoint.ipynb
+examples/.ipynb_checkpoints/*.ipynb
src/CUDA/**/*.ptx
deps/**/*.o
deps/**/*.dll
diff --git a/examples/Scatteraux.jl b/examples/Scatteraux.jl
index 5291a05..a47fe34 100644
--- a/examples/Scatteraux.jl
+++ b/examples/Scatteraux.jl
@@ -17,6 +17,26 @@ if !isdefined(:scatteraux_loaded)
end
+ function logheatmap(A::AbstractMatrix,ϵ;kwds...)
+ ltϵ = log10(ϵ)
+ LGABSAt = [isinf(log10(abs(A[j,i]))) ? ltϵ : max(log10(abs(A[j,i])),ltϵ) for i=1:size(A,2),j=1:size(A,1)]
+ heatmap(flipdim(LGABSAt,1);kwds...)
+ end
+
+ function oscillate(u,T,glp)
+ push!(glp,u)
+ yield()
+ fps = 15
+ MLen = round(Int,T*fps)
+ t = inv(fps)
+ for k=1:MLen
+ u = u*exp(-im*2π*t)
+ push!(glp,u)
+ yield()
+ end
+ u
+ end
+
function makegif(x,y,u,L;plotfunction=Main.Plots.PyPlot.contourf,seconds=1,cmap="seismic",vert=1)
tm=string(time_ns())
dr = pwd()*"/"*tm*"mov"
diff --git a/examples/Time Evolution PDSIES.ipynb b/examples/Time Evolution PDSIES.ipynb
new file mode 100644
index 0000000..b7e2d4e
--- /dev/null
+++ b/examples/Time Evolution PDSIES.ipynb
@@ -0,0 +1,5932 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "\n",
+ " \n",
+ " \n",
+ "
"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ "WARNING: New definition \n",
+ " writemime(Base.IO, Base.Multimedia.MIME, Reactive.Signal) at /Users/Mikael/.julia/v0.4/Interact/src/IJulia/setup.jl:99\n",
+ "is ambiguous with: \n",
+ " writemime(Base.IO, Base.Multimedia.MIME{:text/plain}, Any) at replutil.jl:4.\n",
+ "To fix, define \n",
+ " writemime(Base.IO, Base.Multimedia.MIME{:text/plain}, Reactive.Signal)\n",
+ "before the new definition.\n"
+ ]
+ }
+ ],
+ "source": [
+ "using ApproxFun, SingularIntegralEquations, Plots,Interact,Reactive; gadfly();"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Benjamin-Ono equation with periodic boundary conditions $$u_t+uu_x+H\\{u_{xx}\\}=0$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[Plots.jl] Initializing backend: gadfly\n"
+ ]
+ },
+ {
+ "data": {
+ "image/svg+xml": [
+ "\n",
+ "\n"
+ ],
+ "text/html": [
+ "\n",
+ "\n"
+ ],
+ "text/plain": [
+ "Signal{Gadfly.Plot}(Gadfly.Plot([Gadfly.Layer(nothing,Dict{Symbol,Any}(:y=>[4.21324e-18,-1.48478e-12,-3.40575e-12,-6.18602e-12,-1.02387e-11,-1.5994e-11,-2.39481e-11,-3.47214e-11,-4.9113e-11,-6.81445e-11 … 9.30946e-11,6.81445e-11,4.9113e-11,3.47215e-11,2.39482e-11,1.5994e-11,1.02387e-11,6.18603e-12,3.40576e-12,1.48479e-12],:x=>[-5.0,-4.97647,-4.95294,-4.92941,-4.90588,-4.88235,-4.85882,-4.83529,-4.81176,-4.78824 … 4.76471,4.78824,4.81176,4.83529,4.85882,4.88235,4.90588,4.92941,4.95294,4.97647]),Gadfly.StatisticElement[],Gadfly.Geom.LineGeometry(Gadfly.Stat.Identity(),true,2,symbol(\"\")),Gadfly.Theme(RGBA{Float64}(0.0,0.6056031366372086,0.9786801302529567,1.0),0.9mm,0.2645502645502646mm,Any[],nothing,nothing,1.0,nothing,5.0mm,RGB{U8}(0.816,0.816,0.878),[0.5mm,0.5mm],RGB{U8}(0.627,0.627,0.627),0.2mm,\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.424,0.376,0.42),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",3.880555555555555mm,RGB{U8}(0.337,0.29,0.333),\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",3.880555555555555mm,RGB{U8}(0.212,0.165,0.208),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),40,-0.05mm,1.0mm,3.0mm,Gadfly.default_stroke_color,0.3mm,Gadfly.default_discrete_highlight_color,Gadfly.default_continuous_highlight_color,(anonymous function),1.0,Gadfly.default_middle_color,0.6mm,:left,:square,:right,RGB{Float64}(0.0,0.6056031366372086,0.9786801302529567),[Compose.circle,Gadfly.square,Gadfly.diamond,Gadfly.cross,Gadfly.xcross,Gadfly.utriangle,Gadfly.dtriangle,Gadfly.star1,Gadfly.star2,Gadfly.hexagon,Gadfly.octagon],2.0mm,1000,10.0,0.5,0.2,4),0)],0x0 DataFrames.DataFrame\n",
+ ",Data(\n",
+ " titles=Dict{Symbol,AbstractString}()\n",
+ ")\n",
+ ",Gadfly.ScaleElement[Gadfly.Scale.ContinuousScale([:x,:xmin,:xmax,:xintercept,:xviewmin,:xviewmax],Gadfly.Scale.ContinuousScaleTransform(identity,identity,Gadfly.Scale.identity_formatter),-5,5,2,10,nothing,nothing,true),Gadfly.Scale.ContinuousScale([:y,:ymin,:ymax,:yintercept,:middle,:upper_fence,:lower_fence,:upper_hinge,:lower_hinge,:yviewmin,:yviewmax],Gadfly.Scale.ContinuousScaleTransform(identity,identity,Gadfly.Scale.identity_formatter),-1.0,1.0,2,10,nothing,nothing,true)],Gadfly.StatisticElement[],Gadfly.Coord.Cartesian([:x,:xmin,:xmax,:xintercept],[:y,:ymin,:ymax,:yintercept,:middle,:lower_hinge,:upper_hinge,:lower_fence,:upper_fence,:outliers],-5,5,-1.0,1.0,false,false,false,nothing,false),Gadfly.GuideElement[Gadfly.Guide.ManualColorKey{ColorTypes.Color{T,N}}(\"\",AbstractString[\"y1\"],ColorTypes.Color[RGB{Float64}(0.0,0.6056031366372086,0.9786801302529567)]),Gadfly.Guide.XLabel(\"\",:auto),Gadfly.Guide.YLabel(\"\",:auto),Gadfly.Guide.Title(\"\")],Gadfly.Theme(LCHab{Float32}(70.0f0,60.0f0,240.0f0),0.9mm,0.3mm,nothing,nothing,nothing,1.0,RGBA{U8}(1.0,1.0,1.0,1.0),1.0mm,RGB{U8}(0.816,0.816,0.878),[0.5mm,0.5mm],RGB{U8}(0.627,0.627,0.627),0.2mm,\"Helvetica\",2.822222222222222mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"Helvetica\",3.880555555555555mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),\"Helvetica\",3.880555555555555mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"Helvetica\",2.822222222222222mm,RGBA{U8}(0.0,0.0,0.0,1.0),40,-0.05mm,1.0mm,3.0mm,Gadfly.default_stroke_color,0.3mm,Gadfly.default_discrete_highlight_color,Gadfly.default_continuous_highlight_color,Gadfly.default_lowlight_color,0.6,Gadfly.default_middle_color,0.6mm,:left,:square,:right,nothing,[Compose.circle,Gadfly.square,Gadfly.diamond,Gadfly.cross,Gadfly.xcross,Gadfly.utriangle,Gadfly.dtriangle,Gadfly.star1,Gadfly.star2,Gadfly.hexagon,Gadfly.octagon],2.0mm,1000,10.0,0.5,0.2,4),Dict{Any,Any}()), nactions=1)"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {
+ "comm_id": "ecd8b24e-d687-4a10-b802-ab17b9884b7b",
+ "reactive": true
+ },
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "a,b = -5,5\n",
+ "u0,t=Fun(x->sinpi(x)*exp(-x^2),Laurent(PeriodicInterval(a,b))),0.0\n",
+ "sp=space(u0);D=Derivative(sp);H=Hilbert(sp)\n",
+ "L = -H*D^2\n",
+ "N(u,t) = -u*u'\n",
+ "x=Signal(u0);map(y->plot(real(y);xlims=(a,b),ylims=(-1.,1.)).o,x)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "u0,t=ETDRK4(L,N,u0,t,0.01,200,x,1e-8);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Here is a 1-periodic wave solution"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ "\n",
+ "\n"
+ ],
+ "text/html": [
+ "\n",
+ "\n"
+ ],
+ "text/plain": [
+ "Signal{Gadfly.Plot}(Gadfly.Plot([Gadfly.Layer(nothing,Dict{Symbol,Any}(:y=>[0.666667,0.666837,0.667347,0.668199,0.669393,0.670932,0.672817,0.675052,0.67764,0.680585 … 0.683892,0.680585,0.67764,0.675052,0.672817,0.670932,0.669393,0.668199,0.667347,0.666837],:x=>[-31.4159,-31.3377,-31.2594,-31.1812,-31.1029,-31.0247,-30.9464,-30.8682,-30.79,-30.7117 … 30.6335,30.7117,30.79,30.8682,30.9464,31.0247,31.1029,31.1812,31.2594,31.3377]),Gadfly.StatisticElement[],Gadfly.Geom.LineGeometry(Gadfly.Stat.Identity(),true,2,symbol(\"\")),Gadfly.Theme(RGBA{Float64}(0.0,0.6056031366372086,0.9786801302529567,1.0),0.9mm,0.2645502645502646mm,Any[],nothing,nothing,1.0,nothing,5.0mm,RGB{U8}(0.816,0.816,0.878),[0.5mm,0.5mm],RGB{U8}(0.627,0.627,0.627),0.2mm,\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.424,0.376,0.42),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",3.880555555555555mm,RGB{U8}(0.337,0.29,0.333),\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",3.880555555555555mm,RGB{U8}(0.212,0.165,0.208),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),40,-0.05mm,1.0mm,3.0mm,Gadfly.default_stroke_color,0.3mm,Gadfly.default_discrete_highlight_color,Gadfly.default_continuous_highlight_color,(anonymous function),1.0,Gadfly.default_middle_color,0.6mm,:left,:square,:right,RGB{Float64}(0.0,0.6056031366372086,0.9786801302529567),[Compose.circle,Gadfly.square,Gadfly.diamond,Gadfly.cross,Gadfly.xcross,Gadfly.utriangle,Gadfly.dtriangle,Gadfly.star1,Gadfly.star2,Gadfly.hexagon,Gadfly.octagon],2.0mm,1000,10.0,0.5,0.2,4),0)],0x0 DataFrames.DataFrame\n",
+ ",Data(\n",
+ " titles=Dict{Symbol,AbstractString}()\n",
+ ")\n",
+ ",Gadfly.ScaleElement[Gadfly.Scale.ContinuousScale([:x,:xmin,:xmax,:xintercept,:xviewmin,:xviewmax],Gadfly.Scale.ContinuousScaleTransform(identity,identity,Gadfly.Scale.identity_formatter),-31.41592653589793,31.41592653589793,2,10,nothing,nothing,true),Gadfly.Scale.ContinuousScale([:y,:ymin,:ymax,:yintercept,:middle,:upper_fence,:lower_fence,:upper_hinge,:lower_hinge,:yviewmin,:yviewmax],Gadfly.Scale.ContinuousScaleTransform(identity,identity,Gadfly.Scale.identity_formatter),-0.0,2.0,2,10,nothing,nothing,true)],Gadfly.StatisticElement[],Gadfly.Coord.Cartesian([:x,:xmin,:xmax,:xintercept],[:y,:ymin,:ymax,:yintercept,:middle,:lower_hinge,:upper_hinge,:lower_fence,:upper_fence,:outliers],-31.41592653589793,31.41592653589793,-0.0,2.0,false,false,false,nothing,false),Gadfly.GuideElement[Gadfly.Guide.ManualColorKey{ColorTypes.Color{T,N}}(\"\",AbstractString[\"y1\"],ColorTypes.Color[RGB{Float64}(0.0,0.6056031366372086,0.9786801302529567)]),Gadfly.Guide.XLabel(\"\",:auto),Gadfly.Guide.YLabel(\"\",:auto),Gadfly.Guide.Title(\"\")],Gadfly.Theme(LCHab{Float32}(70.0f0,60.0f0,240.0f0),0.9mm,0.3mm,nothing,nothing,nothing,1.0,RGBA{U8}(1.0,1.0,1.0,1.0),1.0mm,RGB{U8}(0.816,0.816,0.878),[0.5mm,0.5mm],RGB{U8}(0.627,0.627,0.627),0.2mm,\"Helvetica\",2.822222222222222mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"Helvetica\",3.880555555555555mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),\"Helvetica\",3.880555555555555mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"Helvetica\",2.822222222222222mm,RGBA{U8}(0.0,0.0,0.0,1.0),40,-0.05mm,1.0mm,3.0mm,Gadfly.default_stroke_color,0.3mm,Gadfly.default_discrete_highlight_color,Gadfly.default_continuous_highlight_color,Gadfly.default_lowlight_color,0.6,Gadfly.default_middle_color,0.6mm,:left,:square,:right,nothing,[Compose.circle,Gadfly.square,Gadfly.diamond,Gadfly.cross,Gadfly.xcross,Gadfly.utriangle,Gadfly.dtriangle,Gadfly.star1,Gadfly.star2,Gadfly.hexagon,Gadfly.octagon],2.0mm,1000,10.0,0.5,0.2,4),Dict{Any,Any}()), nactions=1)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {
+ "comm_id": "bce6e728-f61d-4040-acf3-323e8e994cd8",
+ "reactive": true
+ },
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "a,b = -10π,10π\n",
+ "u0,t=Fun(x->1/(1-0.5cos(x/2)),Laurent(PeriodicInterval(a,b))),0.0\n",
+ "sp=space(u0);D=Derivative(sp);H=Hilbert(sp)\n",
+ "L = -H*D^2\n",
+ "N(u,t) = -u*u'\n",
+ "x=Signal(u0);map(y->plot(real(y);xlims=(a,b),ylims=(-0.,2.)).o,x)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "# It moves very slowly to the right.\n",
+ "u0,t=ETDRK4(L,N,u0,t,0.01,200,x,1e-8);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Benjamin-Ono equation integrable on $\\mathbb{R}$, $$u_t+uu_x+H\\{u_{xx}\\}=0$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": false,
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ "\n",
+ "\n"
+ ],
+ "text/html": [
+ "\n",
+ "\n"
+ ],
+ "text/plain": [
+ "Signal{Gadfly.Plot}(Gadfly.Plot([Gadfly.Layer(nothing,Dict{Symbol,Any}(:y=>[0.15944,0.317074,0.471115,0.619818,0.761499,0.894551,1.01747,1.12886,1.22745,1.31214 … -1.31214,-1.22745,-1.12886,-1.01747,-0.894551,-0.761499,-0.619818,-0.471115,-0.317074,-0.15944],:x=>[-18.7625,-9.35462,-6.20676,-4.62386,-3.66689,-3.02282,-2.55749,-2.20379,-1.92446,-1.69709 … 1.69709,1.92446,2.20379,2.55749,3.02282,3.66689,4.62386,6.20676,9.35462,18.7625]),Gadfly.StatisticElement[],Gadfly.Geom.LineGeometry(Gadfly.Stat.Identity(),true,2,symbol(\"\")),Gadfly.Theme(RGBA{Float64}(0.0,0.6056031366372086,0.9786801302529567,1.0),0.9mm,0.2645502645502646mm,Any[],nothing,nothing,1.0,nothing,5.0mm,RGB{U8}(0.816,0.816,0.878),[0.5mm,0.5mm],RGB{U8}(0.627,0.627,0.627),0.2mm,\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.424,0.376,0.42),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",3.880555555555555mm,RGB{U8}(0.337,0.29,0.333),\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",3.880555555555555mm,RGB{U8}(0.212,0.165,0.208),\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),40,-0.05mm,1.0mm,3.0mm,Gadfly.default_stroke_color,0.3mm,Gadfly.default_discrete_highlight_color,Gadfly.default_continuous_highlight_color,(anonymous function),1.0,Gadfly.default_middle_color,0.6mm,:left,:square,:right,RGB{Float64}(0.0,0.6056031366372086,0.9786801302529567),[Compose.circle,Gadfly.square,Gadfly.diamond,Gadfly.cross,Gadfly.xcross,Gadfly.utriangle,Gadfly.dtriangle,Gadfly.star1,Gadfly.star2,Gadfly.hexagon,Gadfly.octagon],2.0mm,1000,10.0,0.5,0.2,4),0)],0x0 DataFrames.DataFrame\n",
+ ",Data(\n",
+ " titles=Dict{Symbol,AbstractString}()\n",
+ ")\n",
+ ",Gadfly.ScaleElement[Gadfly.Scale.ContinuousScale([:x,:xmin,:xmax,:xintercept,:xviewmin,:xviewmax],Gadfly.Scale.ContinuousScaleTransform(identity,identity,Gadfly.Scale.identity_formatter),-10,10,2,10,nothing,nothing,true),Gadfly.Scale.ContinuousScale([:y,:ymin,:ymax,:yintercept,:middle,:upper_fence,:lower_fence,:upper_hinge,:lower_hinge,:yviewmin,:yviewmax],Gadfly.Scale.ContinuousScaleTransform(identity,identity,Gadfly.Scale.identity_formatter),-2.0,4.0,2,10,nothing,nothing,true)],Gadfly.StatisticElement[],Gadfly.Coord.Cartesian([:x,:xmin,:xmax,:xintercept],[:y,:ymin,:ymax,:yintercept,:middle,:lower_hinge,:upper_hinge,:lower_fence,:upper_fence,:outliers],-10,10,-2.0,4.0,false,false,false,nothing,false),Gadfly.GuideElement[Gadfly.Guide.ManualColorKey{ColorTypes.Color{T,N}}(\"\",AbstractString[\"y1\"],ColorTypes.Color[RGB{Float64}(0.0,0.6056031366372086,0.9786801302529567)]),Gadfly.Guide.XLabel(\"\",:auto),Gadfly.Guide.YLabel(\"\",:auto),Gadfly.Guide.Title(\"\")],Gadfly.Theme(LCHab{Float32}(70.0f0,60.0f0,240.0f0),0.9mm,0.3mm,nothing,nothing,nothing,1.0,RGBA{U8}(1.0,1.0,1.0,1.0),1.0mm,RGB{U8}(0.816,0.816,0.878),[0.5mm,0.5mm],RGB{U8}(0.627,0.627,0.627),0.2mm,\"Helvetica\",2.822222222222222mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"Helvetica\",3.880555555555555mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\",2.822222222222222mm,RGB{U8}(0.298,0.251,0.294),\"Helvetica\",3.880555555555555mm,RGBA{U8}(0.0,0.0,0.0,1.0),\"Helvetica\",2.822222222222222mm,RGBA{U8}(0.0,0.0,0.0,1.0),40,-0.05mm,1.0mm,3.0mm,Gadfly.default_stroke_color,0.3mm,Gadfly.default_discrete_highlight_color,Gadfly.default_continuous_highlight_color,Gadfly.default_lowlight_color,0.6,Gadfly.default_middle_color,0.6mm,:left,:square,:right,nothing,[Compose.circle,Gadfly.square,Gadfly.diamond,Gadfly.cross,Gadfly.xcross,Gadfly.utriangle,Gadfly.dtriangle,Gadfly.star1,Gadfly.star2,Gadfly.hexagon,Gadfly.octagon],2.0mm,1000,10.0,0.5,0.2,4),Dict{Any,Any}()), nactions=1)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {
+ "comm_id": "688fde5e-f6d7-49f7-88a2-8f4d846310b2",
+ "reactive": true
+ },
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "a,b = -10,10\n",
+ "sp,spd = Laurent(PeriodicLine()),LaurentDirichlet(PeriodicLine())\n",
+ "u0=Fun(x->-3x/(1+x^2),spd)\n",
+ "H=Hilbert(spd);D=Derivative(spd);\n",
+ "L = -D^2*H\n",
+ "N(u) = -u*u'\n",
+ "x=Signal(u0);map(y->plot(real(y);xlims=(a,b),ylims=(-2.,4.)).o,x)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "u0=BDF2(L,N,u0,0.01,100,x,1e-8);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 0.4.1-pre",
+ "language": "julia",
+ "name": "julia-0.4"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "0.4.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
diff --git a/src/GreensFun/CauchyWeight.jl b/src/GreensFun/CauchyWeight.jl
index 29d00eb..a47a654 100644
--- a/src/GreensFun/CauchyWeight.jl
+++ b/src/GreensFun/CauchyWeight.jl
@@ -7,6 +7,7 @@ immutable CauchyWeight{O,SV,T,DD} <: AbstractProductSpace{SV,T,DD,2}
end
CauchyWeight{SV,T,DD}(space::TensorSpace{SV,T,DD,2},O) = CauchyWeight{O,SV,T,DD}(space)
+CauchyWeight(domain::ProductDomain,O) = CauchyWeight(Space(domain),O)
order{O}(C::CauchyWeight{O}) = O
domain(C::CauchyWeight) = domain(C.space)
diff --git a/src/Operators/Hilbert.jl b/src/Operators/Hilbert.jl
index cf2d89a..00b010e 100644
--- a/src/Operators/Hilbert.jl
+++ b/src/Operators/Hilbert.jl
@@ -41,7 +41,6 @@ for Op in (:PseudoHilbert,:Hilbert,:SingularIntegral)
end
bandinds{s,DD}(::$ConcOp{Hardy{s,DD}})=0,0
- domainspace{s,DD}(H::$ConcOp{Hardy{s,DD}})=H.space
rangespace{s,DD}(H::$ConcOp{Hardy{s,DD}})=H.space
bandinds{DD}(::$ConcOp{Laurent{DD}})=0,0
@@ -49,9 +48,14 @@ for Op in (:PseudoHilbert,:Hilbert,:SingularIntegral)
rangespace{DD}(H::$ConcOp{Laurent{DD}})=H.space
bandinds{DD}(H::$ConcOp{Fourier{DD}})=-H.order,H.order
- domainspace{DD}(H::$ConcOp{Fourier{DD}})=H.space
rangespace{DD}(H::$ConcOp{Fourier{DD}})=H.space
+ bandinds{DD<:PeriodicInterval}(H::$ConcOp{CosSpace{DD}})=(0,1)
+ bandinds{DD<:PeriodicInterval}(H::$ConcOp{SinSpace{DD}})=(-1,0)
+ rangespace{DD<:PeriodicInterval}(H::$ConcOp{CosSpace{DD}})=SinSpace(domain(H))
+ rangespace{DD<:PeriodicInterval}(H::$ConcOp{SinSpace{DD}})=CosSpace(domain(H))
+
+
function rangespace{DD}(H::$ConcOp{JacobiWeight{Chebyshev{DD},DD}})
@assert domainspace(H).α==domainspace(H).β==-0.5
H.order==0 ? Chebyshev(domain(H)) : Ultraspherical(H.order,domain(H))
@@ -290,6 +294,71 @@ function getindex{DD<:Circle,OT,T}(H::ConcreteSingularIntegral{Fourier{DD},OT,T}
end
end
+## PeriodicInterval
+
+function getindex{DD<:PeriodicInterval,OT,T}(H::ConcreteHilbert{Taylor{DD},OT,T},k::Integer,j::Integer)
+ d = domain(H)
+ a,b = d.a,d.b
+ if H.order == 1
+ if k>1
+ if k == j
+ T(im)
+ else
+ zero(T)
+ end
+ else
+ zero(T)
+ end
+ else
+ error("Hilbert order $(H.order) not implemented for Taylor")
+ end
+end
+
+function getindex{DD<:PeriodicInterval,OT,T}(H::ConcreteHilbert{Hardy{false,DD},OT,T},k::Integer,j::Integer)
+ d = domain(H)
+ a,b = d.a,d.b
+ if H.order == 1
+ if k>1
+ if k == j
+ T(-im)
+ else
+ zero(T)
+ end
+ else
+ zero(T)
+ end
+ else
+ error("Hilbert order $(H.order) not implemented for Hardy{false}")
+ end
+end
+
+function getindex{DD<:PeriodicInterval,OT,T}(H::ConcreteHilbert{CosSpace{DD},OT,T},k::Integer,j::Integer)
+ d = domain(H)
+ a,b = d.a,d.b
+ if H.order == 1
+ if k == j-1
+ -one(T)
+ else
+ zero(T)
+ end
+ else
+ error("Hilbert order $(H.order) not implemented for CosSpace")
+ end
+end
+
+function getindex{DD<:PeriodicInterval,OT,T}(H::ConcreteHilbert{SinSpace{DD},OT,T},k::Integer,j::Integer)
+ d = domain(H)
+ a,b = d.a,d.b
+ if H.order == 1
+ if k == j+1
+ one(T)
+ else
+ zero(T)
+ end
+ else
+ error("Hilbert order $(H.order) not implemented for CosSpace")
+ end
+end
## JacobiWeight
diff --git a/src/clustertree.jl b/src/clustertree.jl
index afafc44..cc0ad38 100644
--- a/src/clustertree.jl
+++ b/src/clustertree.jl
@@ -3,7 +3,7 @@ export clustertree
#
# clustertree partitions a collection x by maximizing distances normalized by the
# minimal diameter along coordinate axes in 2D. It uses complex numbers to represent
-# points in the plane R². The Admissibility condition for low-rank approximants to
+# points in the plane ℝ². The Admissibility condition for low-rank approximants to
# asymptotically smooth kernels is:
#
# Adm_η(σ×τ) = min{diam(Γ_σ),diam(Γ_τ)} ≤ 2η dist(Γ_σ,Γ_τ),
diff --git a/src/periodicline.jl b/src/periodicline.jl
index 469b4fe..bf33c55 100644
--- a/src/periodicline.jl
+++ b/src/periodicline.jl
@@ -11,7 +11,7 @@ end
function hilbert{L<:PeriodicLine,SS}(S::Space{SS,L},f,z::Number)
S2=setdomain(S,Circle())
- hilbert(S2,f,mappoint(domain(f),Circle(),z))-hilbert(S2,f,-1)
+ hilbert(S2,f,mappoint(domain(S),Circle(),z))-hilbert(S2,f,-1)
end
diff --git a/src/stieltjesmoment.jl b/src/stieltjesmoment.jl
index a357214..b7e77bb 100644
--- a/src/stieltjesmoment.jl
+++ b/src/stieltjesmoment.jl
@@ -28,7 +28,7 @@ stieltjesjacobimoment(α::Real,β::Real,z) = stieltjesjacobimoment(α,β,0,z)
function logjacobimoment(α::Real,β::Real,n::Int,z)
x = 2./(1-z)
if n == 0
- 2normalization(0,α,β)*(log(z-1)-dualpart(_₂F₁(dual(zero(α)+eps(α+β),one(β)),α+1,α+β+2,x)))
+ 2normalization(0,α,β)*(log.(z-1)-dualpart.(_₂F₁(dual(zero(α)+eps(α+β),one(β)),α+1,α+β+2,x)))
# For testing purposes only, should be equivalent to above within radius of convergence
#2normalization(0,α,β)*(log(z-1)-(α+1)/(α+β+2)*x.*_₃F₂(α+2,α+β+3,x))
else
diff --git a/test/logkerneltest.jl b/test/logkerneltest.jl
index 5bc5c26..50d4ac2 100644
--- a/test/logkerneltest.jl
+++ b/test/logkerneltest.jl
@@ -1,6 +1,6 @@
using Base.Test, ApproxFun, SingularIntegralEquations
import ApproxFun: ∞, testbandedoperator, testfunctional, testblockbandedoperator, testraggedbelowoperator, JacobiZ
- import SingularIntegralEquations: testsies, testsieeval, stieltjesmoment, Directed, _₂F₁, ⁺, ⁻, value
+ import SingularIntegralEquations: testsies, testsieeval, stieltjesmoment, Directed, _₂F₁, ⁺, ⁻, value, joukowskyinverse
testsieeval(Jacobi(0,0))
@@ -33,6 +33,11 @@ H=OffSingularIntegral(S,Chebyshev(3..4),0)
H=OffSingularIntegral(S,Chebyshev(3..4.0+im),0)
@test (H*f)(3.5+0.5im) ≈ logkernel(f,3.5+0.5im)
+x = Fun()
+f = chebyshevt(10)/sqrt(1-x^2)
+logkernel(f,0.123+.456im) ≈ -real(joukowskyinverse(Val{true},0.123+.456im)^10)/10
+f = chebyshevt(11)/sqrt(1-x^2)
+logkernel(f,0.123+.456im) ≈ -real(joukowskyinverse(Val{true},0.123+.456im)^11)/11
## Circle
diff --git a/test/runtests.jl b/test/runtests.jl
index fc0d5f6..87104b7 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -31,6 +31,17 @@ for s in (true,false)
end
+#
+# Hilbert transform on the real line for periodic functions on [a,b).
+# It is involutory, so H^2*f = - f, except it zeros the constant.
+#
+
+H = Hilbert()
+f = Fun(ones(20),Laurent(PeriodicInterval()))
+@test norm(H^2*f+f-1) == 0.0
+f = Fun(ones(20),Fourier(PeriodicInterval()))
+@test norm(H^2*f+f-1) == 0.0
+
x=Fun()
@test sum(logabs(x-2.0)) ≈ logabslegendremoment(2.0)
@test sum(logabs(x-(2.0+im))) ≈ logabslegendremoment(2.0+im)