From e6097ae40ea051ac6909db2424f839c6a9fc00ba Mon Sep 17 00:00:00 2001 From: Thomas Pellet Date: Tue, 9 Apr 2019 12:41:53 +0200 Subject: [PATCH 1/6] initial commit TP JS --- Project.toml | 10 +++++ q1.png | Bin 0 -> 42699 bytes q2.png | Bin 0 -> 25915 bytes src/HWfuncapp.jl | 95 ++++++++++++++++++++++++++++++++++++++++++----- test/runtests.jl | 6 ++- 5 files changed, 100 insertions(+), 11 deletions(-) create mode 100644 q1.png create mode 100644 q2.png diff --git a/Project.toml b/Project.toml index 516f264..6b73a7f 100644 --- a/Project.toml +++ b/Project.toml @@ -3,6 +3,16 @@ uuid = "ce1ef771-b552-5ca6-80e4-7358b8c578e2" authors = ["Florian Oswald "] version = "0.1.0" +[deps] +ApproXD = "f0e97480-066f-5960-9ba5-e027352bc8af" +ApproxFun = "28f2ccd6-bb30-5033-b560-165f7b14dc2f" +BasisMatrices = "08854c51-b66b-5062-a90d-8e7ae4547a49" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838" +Plots = 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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 BasisMatrices using ApproxFun using Plots +export 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] +abmap(x,lb,ub) = 0.5 .* (ub .+ lb) .+ 0.5 .* (ub .- lb) .* x # [-1,1] -> [a,b] +function chebpol(x, n) + Φ = zeros(size(x,1),n) # chebyshev polynomials basis + for i in 1:size(x,1) + for j in 1:n + Φ[i,j] = ChebyT(x[i],j-1) + # Φ[i,j] = abmap(Φ[i,j], lb, ub) + end + end + return Φ +end -function q1(n) - f(x) = x .+ 2x.^2 - exp.(-x) +function q1(n=15) + lb = -3 + ub = 3 + x = range(lb, stop = ub ,length = n) + # generate Chebyshev nodes + S, y = gausschebyshev(n) + # Scale Chebyshev nodes on -3 to 3 domain + ϕ = abmap(S, lb, ub) # Chebyshev nodes on -3, 3 + # define function + f(x) = x .+ 2x.^2 - exp.(-x) + # Evaluate function at chebyshev nodes + Y = f(ϕ) + + # Build Function grid + Φ = chebpol(S, n) + + # Estimate parameters of approximation + c = Φ\Y + + # Evaluate approximation + Yhat = Φ*c + + # Test accuracy on larger sample + n_new = range(lb, stop = ub ,length = 100) + # Evaluate function + Y_new = f(n_new) + # Build Polynomial grid + Φ_new = chebpol(unitmap(n_new,lb,ub), n) + + # Evaluate using fitted coefficients + Yhat_new = Φ_new*c + + # Plot + p = Any[] + push!(p,Plots.plot(n_new, [Y_new, Yhat_new], label=["True value" "Approximation"], marker=([:none :diamond]))) + err = Y_new .- Yhat_new + push!(p, Plots.plot(n_new, err, label="Approximation error")) + Plots.plot(p...) # without using PyPlot, just erase the `PyPlot.` part - PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q1.png")) - return Dict(:error=>maximum(abs,err)) + Plots.savefig(Plots.plot(p...), joinpath(dirname(@__FILE__),"..","q1.png")) + return Dict("error"=>maximum(abs,err)) end -function q2(b::Number) + +function q2(b=4) @assert b > 0 + n = 15 # use ApproxFun.jl to do the same: + deg = n - 1 + ub = b + lb = -b + S = Chebyshev(lb..ub) + p = range(lb,stop=ub,length=n) # a non-default grid + # define function + f(x) = x .+ 2x.^2 - exp.(-x) + # evaluate function + v = f(p) # values at the non-default grid + V = Array{Float64}(undef,n,deg + 1) # Create a Vandermonde matrix by evaluating the basis at the grid - Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q2.png")) + for k = 1:deg+1 + V[:,k] = Fun(S,[zeros(k-1);1]).(p) + end + V + g = Fun(S,V\v); + @show g(1.1) + @show f(1.1) + + n_new = range(lb, stop = ub, length = 100) + p = Any[] + push!(p,Plots.plot(n_new, [f(n_new), g.(n_new)], label=["True value" "Approximation"], marker=([:none :diamond]))) + err = f(n_new) .- g.(n_new) + push!(p, Plots.plot(n_new, err, label="Approximation error")) + fig = Plots.plot(p...) + + Plots.savefig(fig,joinpath(dirname(@__FILE__),"..","q2.png")) end function q3(b::Number) @@ -43,7 +120,7 @@ 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 +156,7 @@ 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..9d13d54 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -3,8 +3,10 @@ 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 + @testset "q1" begin + out = q1() + @test out["error"] < 1e-9 end From d43e3a2b4eec54c71e1e2b41d16b0689165e172f Mon Sep 17 00:00:00 2001 From: tpellet Date: Tue, 9 Apr 2019 14:35:32 +0200 Subject: [PATCH 2/6] TP JS 2 --- Project.toml | 2 ++ src/HWfuncapp.jl | 20 ++++++++++++++++++-- 2 files changed, 20 insertions(+), 2 deletions(-) diff --git a/Project.toml b/Project.toml index 6b73a7f..5fba901 100644 --- a/Project.toml +++ b/Project.toml @@ -9,9 +9,11 @@ ApproxFun = "28f2ccd6-bb30-5033-b560-165f7b14dc2f" BasisMatrices = "08854c51-b66b-5062-a90d-8e7ae4547a49" Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" PyCall = "438e738f-606a-5dbb-bf0a-cddfbfd45ab0" PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee" +SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b" [extras] Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" diff --git a/src/HWfuncapp.jl b/src/HWfuncapp.jl index 3d5ff93..f9cc187 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -8,6 +8,7 @@ using PyPlot import ApproXD: getBasis, BSpline using Distributions using BasisMatrices +using LinearAlgebra, SpecialFunctions using ApproxFun using Plots @@ -88,7 +89,8 @@ function q2(b=4) V = Array{Float64}(undef,n,deg + 1) # Create a Vandermonde matrix by evaluating the basis at the grid for k = 1:deg+1 - V[:,k] = Fun(S,[zeros(k-1);1]).(p) + V[:,k] = Fun(S,[zeros(k-1);1]).(p) # Evaluate Chebyshev basis at p + # [zeros(k-1);1] is the identity matrix, all observations have same k degree Chebyshev basis function, evaluated at a different point end V g = Fun(S,V\v); @@ -105,8 +107,22 @@ function q2(b=4) Plots.savefig(fig,joinpath(dirname(@__FILE__),"..","q2.png")) end -function q3(b::Number) +function q3(b=10) + + x = Fun(identity,-b..b) + f = sin(x^2) + g = cos(x) + h = f - g + r = roots(h) + + Plots.plot(h, label="h(x)") + p = Plots.scatter!(r,h.(r), label="Roots") + + # xbis = Fun(identity,-b..0) + g = cumsum(h) # indefinite integral + g = g + h(-b) # definite integral with constant of integration + integral = norm(g(0) - g(-b)) # definite integral from -b to 0 # p is your plot return 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+187,42 @@ function q6() end -function q7() +function q7(n=13) + f(x) = abs.(x).^0.5 + + # Regular grid + bs = BSpline(13,3,-1,1) #13 knots, degree 3 in [0,1], no multiplicity + # 3 knots grid + multiknots = vcat(range(-1,stop = -0.1,length = 5),0,0,0, range(0.1,stop = 1,length =5)) + bs2 = BSpline(multiknots, 3) + + # Evaluate function of interest + x = range(-1,stop =1.0, length = 65) + Y = f(x) + + # Evaluate basis on grid + B = Array(getBasis(collect(x),bs)) + B2 = Array(getBasis(collect(x),bs2)) + # Solve for parameters + c = B\Y + c2 = B2\Y - PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q7.png")) + # Compute fitted values + Yhat = B*c + Yhatbis = B2*c2 + + # Compute fitting errors + err = Y .- Yhat + errbis = Y .- Yhatbis + + # Plot + p = Any[] + push!(p,Plots.plot(x, Y, title = "True function")) + push!(p,Plots.plot(x, [Yhat, Yhatbis], label = ["uniform" "multiplicity"], title="Approximations")) + push!(p,Plots.plot(x, [err, errbis], label = ["uniform" "multiplicity"], title="Errors")) + return Plots.plot(p...) + Plots.savefig(joinpath(dirname(@__FILE__),"..","q7.png")) end From 1a7169230d535efe1dad68244279ec053bcd09a1 Mon Sep 17 00:00:00 2001 From: tpellet Date: Sun, 14 Apr 2019 16:45:12 +0200 Subject: [PATCH 4/6] Travis --- .gitignore | 1 - 1 file changed, 1 deletion(-) diff --git a/.gitignore b/.gitignore index adb2aef..089c1bf 100644 --- a/.gitignore +++ b/.gitignore @@ -2,4 +2,3 @@ *.jl.cov *.jl.mem .DS_Store -/Manifest.toml From 793306adb54c63bd58ae9bf2f72a2d8018f8d1dc Mon Sep 17 00:00:00 2001 From: tpellet Date: Mon, 15 Apr 2019 13:32:10 +0200 Subject: [PATCH 5/6] Solving Travis problem --- Manifest.toml | 614 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 614 insertions(+) create mode 100644 Manifest.toml diff --git a/Manifest.toml b/Manifest.toml new file mode 100644 index 0000000..639496b --- /dev/null +++ b/Manifest.toml @@ -0,0 +1,614 @@ +# This file is machine-generated - editing it directly is not advised + 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"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" From 55320351d502f63ddadad210fb5a400c7d674a92 Mon Sep 17 00:00:00 2001 From: tpellet Date: Mon, 15 Apr 2019 16:37:29 +0200 Subject: [PATCH 6/6] No Pyplot --- Manifest.toml | 20 ++++++++++---------- src/HWfuncapp.jl | 2 +- 2 files changed, 11 insertions(+), 11 deletions(-) diff --git a/Manifest.toml b/Manifest.toml index 639496b..4f01be6 100644 --- a/Manifest.toml +++ b/Manifest.toml @@ -1,10 +1,10 @@ # This file is machine-generated - editing it directly is not advised [[AbstractFFTs]] -deps = ["Compat", "LinearAlgebra"] -git-tree-sha1 = "8d59c3b1463b5e0ad05a3698167f85fac90e184d" +deps = ["LinearAlgebra", "Test"] +git-tree-sha1 = "dfaf23dba016254bb0eed6de326510caea2889bf" uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" -version = "0.3.2" +version = "0.4.0" [[ApproXD]] deps = ["LinearAlgebra", "SparseArrays", "StatsBase"] @@ -280,10 +280,10 @@ uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" version = "1.0.3" [[LazyArrays]] -deps = ["FillArrays", "LinearAlgebra", "StaticArrays", "Test"] -git-tree-sha1 = "737472254720be95800f859a7c2aaf62488fd53f" +deps = ["FillArrays", "LinearAlgebra", "MacroTools", "StaticArrays", "Test"] +git-tree-sha1 = "4439e840fe68cbcde806fcc625d05166227a56a5" uuid = "5078a376-72f3-5289-bfd5-ec5146d43c02" -version = "0.7.1" +version = "0.8.0" [[LibGit2]] uuid = "76f85450-5226-5b5a-8eaa-529ad045b433" @@ -385,9 +385,9 @@ version = "0.18.1" [[OrderedCollections]] deps = ["Random", "Serialization", "Test"] -git-tree-sha1 = "85619a3f3e17bb4761fe1b1fd47f0e979f964d5b" +git-tree-sha1 = "c4c13474d23c60d20a67b217f1d7f22a40edf8f1" uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d" -version = "1.0.2" +version = "1.1.0" [[PDMats]] deps = ["Arpack", "LinearAlgebra", "SparseArrays", "SuiteSparse", "Test"] @@ -413,9 +413,9 @@ version = "0.3.0" [[PlotUtils]] deps = ["Colors", "Dates", "Printf", "Random", "Reexport", "Test"] -git-tree-sha1 = "5da9a79a138db46558e6e08f649865b4c1f1ce93" +git-tree-sha1 = "8e87bbb778c26f575fbe47fd7a49c7b5ca37c0c6" uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043" -version = "0.5.7" +version = "0.5.8" [[Plots]] deps = ["Base64", "Contour", "Dates", "FixedPointNumbers", "GR", "JSON", "LinearAlgebra", "Measures", "NaNMath", "Pkg", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "Reexport", "Requires", "Showoff", "SparseArrays", "StaticArrays", "Statistics", "StatsBase", "Test", "UUIDs"] diff --git a/src/HWfuncapp.jl b/src/HWfuncapp.jl index 1e3ad11..b60db13 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -4,7 +4,7 @@ using FastGaussQuadrature # to get chebyshevnodes # 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 +# using PyPlot import ApproXD: getBasis, BSpline using Distributions using BasisMatrices