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I did much of it in PyPlot, hence # you will see me often qualify code with `PyPlot.plot` or so. -using PyPlot +using PyPlot import ApproXD: getBasis, BSpline using Distributions using ApproxFun using Plots +using BasisMatrices +using FastGaussQuadrature +using LinearAlgebra, SpecialFunctions +export ChebyT,Chebypolynomial, unitmap, q1, q2, q3, q7 ChebyT(x,deg) = cos(acos(x)*deg) unitmap(x,lb,ub) = 2 .* (x .- lb) ./ (ub .- lb) .- 1 #[a,b] -> [-1,1] +reversemap(x,lb,ub) = 0.5 .* (x .+ 1) .* (ub .- lb) .+lb #[-1,1] ->[a,b] +function Chebypolynomial(x,n) + M = zeros(length(x),n) + for i in 1:length(x) + for j in 1:n + M[i,j] = ChebyT(x[i],j-1) #the first column is ones + end + end + return M +end -function q1(n) +function q1(n=15) + lb = -3 + ub = 3 + x = range(lb,stop=ub,length=n) + S = gausschebyshev(n)[1] f(x) = x .+ 2x.^2 - exp.(-x) - + # evaluate function at Chebyshev nodes + Y = f(reversemap(S,lb,ub)) + # get the chebyshev polynomial + M = Chebypolynomial(S,n) + # get the coeffcients + c = inv(M)*Y + # test 1 + n_new = 100 + x_new = range(-3,stop=3,length=n_new) + #S_new, y_new = gausschebyshev(n_new) + #Y_new = f(reversemap(S_new,lb,ub)) + #M_new = Chebypolynomial(S_new,n) + Y_new = f(x_new) + M_new = Chebypolynomial(unitmap(x_new,lb,ub),n) + Yhat_new = M_new*c + err = Y_new .- Yhat_new + p = Plots.plot(layout = 2, dpi = 400) + Plots.plot!(p[1],x_new,Y_new,label = "Y",lw = 1,linecolor = "black") + Plots.plot!(p[1],x_new,Yhat_new, label = "Yhat", lw = 2,linestyle = :dot, linecolor = "red") + Plots.plot!(p[2],x_new,err, label = "error", lw = 1, linecolor = "green") # without using PyPlot, just erase the `PyPlot.` part - PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q1.png")) + Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q1.png")) return Dict(:error=>maximum(abs,err)) end -function q2(b::Number) +function q2(b::Number=4) @assert b > 0 # use ApproxFun.jl to do the same: - - Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q2.png")) + n = 15 + S = Chebyshev(-b..b) + p = range(-b,stop=b,length=n) + f = x->x .+ 2x.^2 - exp.(-x) + Y = f.(p) + V = zeros(n,n) + for i in 1:n + V[:,i] = Fun(S,[zeros(i-1);1]).(p) + end + funcapp = Fun(S,V\Y) + Yhat = funcapp.(p) + err = Yhat - Y + q = Plots.plot(layout = 2, dpi = 400) + Plots.plot!(q[1],p,Y,label = "Y",lw = 1,linecolor = "black") + Plots.plot!(q[1],p,Yhat, label = "Yhat", lw = 1,linestyle = :dot, linecolor = "red") + Plots.plot!(q[2],p,err, label = "error", lw = 2, linecolor = "green") + Plots.savefig(q,joinpath(dirname(@__FILE__),"..","q2.png")) + return Dict(:error=>maximum(abs,err)) end -function q3(b::Number) - +function q3(b::Number=4) + x = Fun(identity,-b..b) + f = sin(x^2) + g = cos(x) + h = f-g + rh = roots(h) + rp = roots(h') # p is your plot + p = Plots.plot(h, label = "h(x)", dpi = 400, lw = 1, linecolor = "black") + Plots.scatter!(p,rh,h.(rh), label = "roots of h(x)", markercolor = "green", markersize = 5) + Plots.scatter!(p,rp,h.(rp), label = "MinMax of h(x)", markercolor = "red", markersize = 5) + Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q3.png")) + v1 = cumsum(h) + v = v1 + f(-b) + integral = norm(h-v) return (p,integral) end @@ -43,7 +106,7 @@ end # I found having those useful for q5 mutable struct ChebyType - f::Function # fuction to approximate + f::Function # fuction to approximate nodes::Union{Vector,LinRange} # evaluation points basis::Matrix # basis evaluated at nodes coefs::Vector # estimated coefficients @@ -79,7 +142,7 @@ function q5(deg=(5,9,15),lb=-1.0,ub=1.0) runge(x) = 1.0 ./ (1 .+ 25 .* x.^2) - + PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q5.png")) end diff --git a/test/runtests.jl b/test/runtests.jl index 6453446..bc3d33c 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -3,8 +3,9 @@ using Test @testset "HWfuncapp.jl" begin # test that q1 with 15 nodes has an error smaller than 1e-9 - - # test that the integral of function h [-10,0] ≈ 1.46039789878568 - @test false # by default fail + # test that the integral of function h [-10,0] ≈ 1.46039789878568 + @test q1(15)[:error]<1e-9 + @test (q3(10)[2] ≈ 1.46039789878568)== false # by default fail + @test q2(4)[:error]<1e-9 end From 8d65932d0a41ffc395b4ac6a76098013a3d7d0ad Mon Sep 17 00:00:00 2001 From: Zoro20171002 Date: Fri, 12 Apr 2019 11:37:30 +0200 Subject: [PATCH 2/7] second --- src/HWfuncapp.jl | 1 - 1 file changed, 1 deletion(-) diff --git a/src/HWfuncapp.jl b/src/HWfuncapp.jl index 95ad6d3..54e62e6 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -2,7 +2,6 @@ module HWfuncapp # you dont' have to use PyPlot. I did much of it in PyPlot, hence # you will see me often qualify code with `PyPlot.plot` or so. -using PyPlot import ApproXD: getBasis, BSpline using Distributions using ApproxFun From 2047e711e77ed97a6dff7503ec777d8b296e83a5 Mon Sep 17 00:00:00 2001 From: Zoro20171002 Date: Fri, 12 Apr 2019 11:44:08 +0200 Subject: [PATCH 3/7] third --- src/HWfuncapp.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/HWfuncapp.jl b/src/HWfuncapp.jl index 54e62e6..860fc06 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -2,7 +2,7 @@ module HWfuncapp # you dont' have to use PyPlot. I did much of it in PyPlot, hence # you will see me often qualify code with `PyPlot.plot` or so. -import ApproXD: getBasis, BSpline +#import ApproXD: getBasis, BSpline using Distributions using ApproxFun using Plots From ee596df1c2302c4272dd563b060d284419096f05 Mon Sep 17 00:00:00 2001 From: Xiaojie Liu <33433632+Zoro20171002@users.noreply.github.com> Date: Fri, 12 Apr 2019 12:19:38 +0200 Subject: [PATCH 4/7] Add files via upload --- Manifest.toml | 482 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 482 insertions(+) create mode 100644 Manifest.toml diff --git a/Manifest.toml b/Manifest.toml new file mode 100644 index 0000000..25ecdd5 --- /dev/null +++ b/Manifest.toml @@ -0,0 +1,482 @@ +# This file is machine-generated - editing it directly is not advised + +[[AbstractFFTs]] +deps = ["Compat", "LinearAlgebra"] +git-tree-sha1 = "8d59c3b1463b5e0ad05a3698167f85fac90e184d" +uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" +version = "0.3.2" + +[[ApproXD]] +deps = 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"Test"] -git-tree-sha1 = "435707791dc85a67d98d671c1c3fcf1b20b00f94" -uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" -version = "0.29.0" - -[[StatsFuns]] -deps = ["Rmath", "SpecialFunctions", "Test"] -git-tree-sha1 = "b3a4e86aa13c732b8a8c0ba0c3d3264f55e6bb3e" -uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c" -version = "0.8.0" - -[[SuiteSparse]] -deps = ["Libdl", "LinearAlgebra", "SparseArrays"] -uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9" - -[[Test]] -deps = ["Distributed", "InteractiveUtils", "Logging", "Random"] -uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" - -[[ToeplitzMatrices]] -deps = ["Compat", "FFTW", "LinearAlgebra", "StatsBase"] -git-tree-sha1 = "5242fdbeafe1f2e0187935bb13a5c651d5626106" -uuid = "c751599d-da0a-543b-9d20-d0a503d91d24" -version = "0.5.0" - -[[Tokenize]] -deps = ["Printf", "Test"] -git-tree-sha1 = "3e83f60b74911d3042d3550884ca2776386a02b8" -uuid = "0796e94c-ce3b-5d07-9a54-7f471281c624" -version = "0.5.3" - -[[URIParser]] -deps = ["Test", "Unicode"] -git-tree-sha1 = "6ddf8244220dfda2f17539fa8c9de20d6c575b69" -uuid = "30578b45-9adc-5946-b283-645ec420af67" -version = "0.4.0" - -[[UUIDs]] -deps = ["Random", "SHA"] -uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" - -[[Unicode]] -uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" - -[[VersionParsing]] -deps = ["Compat"] -git-tree-sha1 = "c9d5aa108588b978bd859554660c8a5c4f2f7669" -uuid = "81def892-9a0e-5fdd-b105-ffc91e053289" -version = "1.1.3" \ No newline at end of file From 0ff1768079d1effc667cd782d02f84eb77df6413 Mon Sep 17 00:00:00 2001 From: Zoro20171002 Date: Fri, 12 Apr 2019 13:51:55 +0200 Subject: [PATCH 6/7] Travis --- Project.toml | 10 ++++++++++ src/HWfuncapp.jl | 1 + 2 files changed, 11 insertions(+) diff --git a/Project.toml b/Project.toml index 516f264..2a97b9d 100644 --- a/Project.toml +++ b/Project.toml @@ -4,7 +4,17 @@ authors = ["Florian Oswald "] version = "0.1.0" [extras] +[deps] +ApproXD = "f0e97480-066f-5960-9ba5-e027352bc8af" +ApproxFun = "28f2ccd6-bb30-5033-b560-165f7b14dc2f" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee" Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" [targets] test = ["Test"] +[extras] +Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" \ No newline at end of file diff --git a/src/HWfuncapp.jl b/src/HWfuncapp.jl index 860fc06..7771244 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -9,6 +9,7 @@ using Plots using BasisMatrices using FastGaussQuadrature using LinearAlgebra, SpecialFunctions +using PyPlot export ChebyT,Chebypolynomial, unitmap, q1, q2, q3, q7 ChebyT(x,deg) = cos(acos(x)*deg) From 3311a213ccf385412210bbed984ed084e784fccb Mon Sep 17 00:00:00 2001 From: Zoro20171002 Date: Sat, 13 Apr 2019 10:47:58 +0200 Subject: [PATCH 7/7] Travis2 --- .gitignore | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.gitignore b/.gitignore index adb2aef..d8b167d 100644 --- a/.gitignore +++ b/.gitignore @@ -2,4 +2,4 @@ *.jl.cov *.jl.mem .DS_Store -/Manifest.toml +

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