From acb7be120ecf0316b39d289b1ec7ce8c01f2d741 Mon Sep 17 00:00:00 2001 From: gdannay Date: Fri, 12 Apr 2019 18:45:00 +0200 Subject: [PATCH 1/8] Question 1 --- Project.toml | 8 + q1.png | Bin 0 -> 21793 bytes src/HWfuncapp.jl | 28 ++- src/Manifest.toml | 522 ++++++++++++++++++++++++++++++++++++++++++++++ src/Project.toml | 4 + 5 files changed, 560 insertions(+), 2 deletions(-) create mode 100644 q1.png create mode 100644 src/Manifest.toml create mode 100644 src/Project.toml diff --git a/Project.toml b/Project.toml index 516f264..90b67fc 100644 --- a/Project.toml +++ b/Project.toml @@ -3,6 +3,14 @@ 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" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PyPlot = 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a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -13,9 +13,33 @@ using Plots ChebyT(x,deg) = cos(acos(x)*deg) unitmap(x,lb,ub) = 2 .* (x .- lb) ./ (ub .- lb) .- 1 #[a,b] -> [-1,1] -function q1(n) +export q1, q2, ChebyT + +function q1(n=15) f(x) = x .+ 2x.^2 - exp.(-x) + nd(x) = cos.((2 .* x .- 1) .* pi ./ (2 .* n)) + + p = n:-1:1 + x = unitmap(range(-3, stop = 3, length = n), -3, 3) + cheb = zeros(n, n) + y = f(x) + node = nd(p) + + for i in 1:n + cheb[:, i] = ChebyT.(node, i - 1) + end + c = (cheb' * cheb)^-1 * cheb' * y + y2 = zeros(n) + for i in 1:n + y2 += c[i] * ChebyT.(x, i - 1) + end + err = sum(y - y2) + + PyPlot.plot(x, y) + PyPlot.plot(x, y2) + xlabel("x") + ylabel("y") # without using PyPlot, just erase the `PyPlot.` part PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q1.png")) return Dict(:error=>maximum(abs,err)) @@ -24,7 +48,7 @@ end function q2(b::Number) @assert b > 0 # use ApproxFun.jl to do the same: - + Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q2.png")) end diff --git a/src/Manifest.toml b/src/Manifest.toml new file mode 100644 index 0000000..7ceea6f --- /dev/null +++ b/src/Manifest.toml @@ -0,0 +1,522 @@ +# 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" + +[[ApproxFun]] +deps = ["AbstractFFTs", "BandedMatrices", "BlockArrays", "BlockBandedMatrices", "Calculus", "DSP", "DomainSets", "DualNumbers", "FFTW", "FastGaussQuadrature", "FastTransforms", "FillArrays", "InfiniteArrays", "IntervalSets", "LazyArrays", "LinearAlgebra", "LowRankApprox", "Random", "RecipesBase", "SparseArrays", "SpecialFunctions", "StaticArrays", "Statistics", "Test", "ToeplitzMatrices"] +git-tree-sha1 = "2fbd592ab81eec53b7f256e771507a10ae38a3e6" +uuid = "28f2ccd6-bb30-5033-b560-165f7b14dc2f" +version = "0.10.4" + +[[ArnoldiMethod]] +deps = ["DelimitedFiles", 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b/src/HWfuncapp.jl @@ -17,27 +17,29 @@ export q1, q2, ChebyT function q1(n=15) f(x) = x .+ 2x.^2 - exp.(-x) - nd(x) = cos.((2 .* x .- 1) .* pi ./ (2 .* n)) + nd(x) = 3 * cos.((2 .* x .- 1) .* pi ./ (2 .* n)) p = n:-1:1 - x = unitmap(range(-3, stop = 3, length = n), -3, 3) + x = range(-3, stop = 3, length = n) cheb = zeros(n, n) y = f(x) node = nd(p) for i in 1:n - cheb[:, i] = ChebyT.(node, i - 1) + cheb[:, i] = ChebyT.(unitmap(node, -3, 3), i - 1) end c = (cheb' * cheb)^-1 * cheb' * y y2 = zeros(n) for i in 1:n - y2 += c[i] * ChebyT.(x, i - 1) + y2 += c[i] * ChebyT.(unitmap(x, -3, 3), i - 1) end err = sum(y - y2) - + subplot(122) PyPlot.plot(x, y) PyPlot.plot(x, y2) + subplot(121) + PyPlot.plot(x, y - y2) xlabel("x") ylabel("y") # without using PyPlot, just erase the `PyPlot.` part From 0d8652b4e24b91c5b016d4c846c1cca849a4f3cc Mon Sep 17 00:00:00 2001 From: gdannay Date: Sat, 13 Apr 2019 01:48:14 +0200 Subject: [PATCH 3/8] HW --- Project.toml | 2 ++ q1.png | Bin 26460 -> 24208 bytes q2.png | Bin 0 -> 24051 bytes q3.png | Bin 0 -> 48760 bytes q4.png | Bin 0 -> 122127 bytes src/HWfuncapp.jl | 71 +++++++++++++++++++++++++++++++++++------------ 6 files changed, 55 insertions(+), 18 deletions(-) create mode 100644 q2.png create mode 100644 q3.png create mode 100644 q4.png diff --git a/Project.toml b/Project.toml index 90b67fc..d492c5c 100644 --- a/Project.toml +++ b/Project.toml @@ -6,10 +6,12 @@ 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 = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee" +SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b" [extras] Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" diff --git a/q1.png b/q1.png index 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+6,7 @@ using FastGaussQuadrature # to get chebyshevnodes # you will see me often qualify code with `PyPlot.plot` or so. using PyPlot import ApproXD: getBasis, BSpline +using LinearAlgebra using Distributions using ApproxFun using Plots @@ -13,56 +14,90 @@ using Plots ChebyT(x,deg) = cos(acos(x)*deg) unitmap(x,lb,ub) = 2 .* (x .- lb) ./ (ub .- lb) .- 1 #[a,b] -> [-1,1] -export q1, q2, ChebyT +export q1, q2, q3, q4, ChebyT function q1(n=15) f(x) = x .+ 2x.^2 - exp.(-x) - nd(x) = 3 * cos.((2 .* x .- 1) .* pi ./ (2 .* n)) - p = n:-1:1 - x = range(-3, stop = 3, length = n) - cheb = zeros(n, n) - y = f(x) - node = nd(p) + deg = n:-1:1 + precision = 10000 + points = range(-3, 3, length=precision) + + node = 3 * cos.((2 .* deg .- 1) .* pi ./ (2 .* n)) + y = f(node) + cheb = zeros(n, n) for i in 1:n cheb[:, i] = ChebyT.(unitmap(node, -3, 3), i - 1) end + c = cheb^-1 * y - c = (cheb' * cheb)^-1 * cheb' * y - y2 = zeros(n) + estim = zeros(precision) for i in 1:n - y2 += c[i] * ChebyT.(unitmap(x, -3, 3), i - 1) + estim += c[i] * ChebyT.(unitmap(points, -3, 3), i - 1) end - err = sum(y - y2) + subplot(122) - PyPlot.plot(x, y) - PyPlot.plot(x, y2) - subplot(121) - PyPlot.plot(x, y - y2) + PyPlot.plot(points, f.(points)) + PyPlot.scatter(points, estim, color="red", s=2) xlabel("x") ylabel("y") + subplot(121) + PyPlot.plot(points, f.(points) - estim) # without using PyPlot, just erase the `PyPlot.` part PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q1.png")) + + err = sum(abs.(f.(points) - estim)) return Dict(:error=>maximum(abs,err)) end function q2(b::Number) @assert b > 0 + f(x) = x .+ 2x.^2 - exp.(-x) + + precision = 10000 + points = range(-3, 3, length=precision) # use ApproxFun.jl to do the same: - - Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q2.png")) + x = Fun(f, Chebyshev(-b..b)) + estim = x.(points) + + subplot(122) + PyPlot.plot(points, f.(points)) + PyPlot.scatter(points, estim, color="red", s=2) + xlabel("x") + ylabel("y") + subplot(121) + PyPlot.plot(points, f.(points) - estim) + + PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q2.png")) end function q3(b::Number) + x = Fun(identity, -b..b) + f = sin(x^2) + g = cos(x) + h = f - g + + r = roots(h) + p = Plots.plot(h, labels="h") + Plots.scatter!(r, h.(r), labels="roots.h") + Plots.savefig(joinpath(dirname(@__FILE__),"..","q3.png")) # p is your plot + g = cumsum(h) + integral = g(b) - g(-b) return (p,integral) end # optinal function q4() - + precision = 10000 + points = range(-1, stop = 1, length=precision) + fig = PyPlot.plot(points, ChebyT.(points, 0)) + for i in 1:8 + fig = PyPlot.plot(points, ChebyT.(points, i)) + end + PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q4.png")) return fig end From adc6de6055217f038a5221b2ff574c0796b023c7 Mon Sep 17 00:00:00 2001 From: gdannay Date: Sat, 13 Apr 2019 12:16:51 +0200 Subject: [PATCH 4/8] End --- Project.toml | 1 + q1.png | Bin 24208 -> 32389 bytes q2.png | Bin 24051 -> 32674 bytes q4.png | Bin 122127 -> 127717 bytes q5.png | Bin 0 -> 60518 bytes q6.png | Bin 0 -> 72979 bytes q7.png | Bin 0 -> 53778 bytes src/HWfuncapp.jl | 86 +++++++++++++++++++++++++++++++++++++++-------- test/runtests.jl | 11 ++++-- 9 files changed, 82 insertions(+), 16 deletions(-) create mode 100644 q5.png create mode 100644 q6.png create mode 100644 q7.png diff --git a/Project.toml b/Project.toml index d492c5c..2d0aa20 100644 --- a/Project.toml +++ b/Project.toml @@ -9,6 +9,7 @@ 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" PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee" SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b" diff --git a/q1.png b/q1.png index 280cce49790f816d9503c59f87197c6305c2201a..90abceffcf47873db588378149fd317452af99bc 100644 GIT binary patch literal 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index dc2415e..fcb20f9 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -14,13 +14,14 @@ using Plots ChebyT(x,deg) = cos(acos(x)*deg) unitmap(x,lb,ub) = 2 .* (x .- lb) ./ (ub .- lb) .- 1 #[a,b] -> [-1,1] -export q1, q2, q3, q4, q5, q7, runall +export q1, q2, q3, q4, q5, q6, q7, runall function q1(n=15) f(x) = x .+ 2x.^2 - exp.(-x) + n_new = 100 deg = n:-1:1 - points = range(-3, 3, length=n) + points = range(-3, 3, length=n_new) node = 3 * cos.((2 .* deg .- 1) .* pi ./ (2 .* n)) y = f(node) @@ -31,18 +32,21 @@ function q1(n=15) end c = cheb^-1 * y - estim = zeros(n) + estim = zeros(n_new) for i in 1:n estim += c[i] * ChebyT.(unitmap(points, -3, 3), i - 1) end subplot(122) - PyPlot.plot(points, f.(points)) - PyPlot.scatter(points, estim, color="red", s=2) + PyPlot.plot(points, f.(points), label="True function") + PyPlot.scatter(points, estim, color="red", s=2, label="approximation") xlabel("x") ylabel("y") + title("Chebyshev approximation") + legend() subplot(121) PyPlot.plot(points, f.(points) - estim) + title("Error") # without using PyPlot, just erase the `PyPlot.` part PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q1.png")) @@ -61,12 +65,15 @@ function q2(b::Number) estim = x.(points) subplot(122) - PyPlot.plot(points, f.(points)) - PyPlot.scatter(points, estim, color="red", s=2) + PyPlot.plot(points, f.(points), label="True function") + PyPlot.scatter(points, estim, color="red", s=2, label="approximation") xlabel("x") ylabel("y") + title("Chebyshev approximation") + legend() subplot(121) PyPlot.plot(points, f.(points) - estim) + title("Error") PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q2.png")) end @@ -181,9 +188,34 @@ end function q6() - + f(x) = abs(x)^0.5 + precision = 63 + x = range(-1, 1, length=precision) + y = f.(x) # compare 2 knot vectors with runge's function + myknots = vcat(range(-1,stop = -0.1,length = 5), 0, 0, 0, range(0.1,stop = 1,length =5)) + bs = BSpline(13, 3, -5, 5) + bs2 = BSpline(myknots, 3) + B = Array(getBasis(collect(x), bs)) + B2 = Array(getBasis(collect(x), bs2)) + c = B \ y + c2 = B2 \ y + subplot(311) + PyPlot.plot(x, y, label="True function") + PyPlot.scatter(x, B * c, color="red", s=2, label="Approximation") + title("Uniform knot vector") + legend() + subplot(312) + PyPlot.plot(x, y, label="True function") + PyPlot.scatter(x, B2 * c2, color="red", s=2, label="Approximation") + title("Knot multiplicity x=0") + legend() + subplot(313) + PyPlot.plot(x, y - B * c, color="blue", label="Uniform") + PyPlot.plot(x, y - B2 * c2, color="red", label="Knot multiplicity") + title("Error plots") + legend() PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q6.png")) end From 646b2d0871aedc29b2d4d8d445459609b524319e Mon Sep 17 00:00:00 2001 From: gdannay Date: Sat, 13 Apr 2019 15:25:41 +0200 Subject: [PATCH 7/8] Correc --- q2.png | Bin 54498 -> 39900 bytes q4.png | Bin 69893 -> 127717 bytes q5.png | Bin 79163 -> 60518 bytes q7.png | Bin 66544 -> 53778 bytes 4 files changed, 0 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b/src/HWfuncapp.jl index 3ddff9d..dc2415e 100644 --- a/src/HWfuncapp.jl +++ b/src/HWfuncapp.jl @@ -14,14 +14,13 @@ using Plots ChebyT(x,deg) = cos(acos(x)*deg) unitmap(x,lb,ub) = 2 .* (x .- lb) ./ (ub .- lb) .- 1 #[a,b] -> [-1,1] -export q1, q2, q3, q4, ChebyT +export q1, q2, q3, q4, q5, q7, runall function q1(n=15) f(x) = x .+ 2x.^2 - exp.(-x) deg = n:-1:1 - precision = 10000 - points = range(-3, 3, length=precision) + points = range(-3, 3, length=n) node = 3 * cos.((2 .* deg .- 1) .* pi ./ (2 .* n)) y = f(node) @@ -32,7 +31,7 @@ function q1(n=15) end c = cheb^-1 * y - estim = zeros(precision) + estim = zeros(n) for i in 1:n estim += c[i] * ChebyT.(unitmap(points, -3, 3), i - 1) end @@ -55,7 +54,7 @@ function q2(b::Number) @assert b > 0 f(x) = x .+ 2x.^2 - exp.(-x) - precision = 10000 + precision = 100 points = range(-3, 3, length=precision) # use ApproxFun.jl to do the same: x = Fun(f, Chebyshev(-b..b)) @@ -85,7 +84,7 @@ function q3(b::Number) Plots.savefig(joinpath(dirname(@__FILE__),"..","q3.png")) # p is your plot g = cumsum(h) - integral = g(b) - g(-b) + integral = g(0) - g(-b) return (p,integral) end @@ -93,10 +92,12 @@ end function q4() precision = 10000 points = range(-1, stop = 1, length=precision) - fig = PyPlot.plot(points, ChebyT.(points, 0)) + fig = PyPlot.plot(points, ChebyT.(points, 0), label=0) for i in 1:8 - fig = PyPlot.plot(points, ChebyT.(points, i)) + fig = PyPlot.plot(points, ChebyT.(points, i), label=i) end + title("Chebyshev basis") + legend() PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q4.png")) return fig end @@ -117,7 +118,7 @@ mutable struct ChebyType function ChebyType(_nodes::Union{Vector,LinRange},_deg,_lb,_ub,_f::Function) n = length(_nodes) y = _f(_nodes) - _basis = Float64[ChebyT(unitmap(_nodes[i],_lb,_ub),j) for i=1:n,j=0:_deg] + _basis = Float64[ChebyT(unitmap(_nodes[i],_lb,_ub),j) for i=1:n,j=0:_deg - 1] _coefs = _basis \ y # type `?\` to find out more about the backslash operator. depending the args given, it performs a different operation # create a ChebyComparer with those values new(_f,_nodes,_basis,_coefs,_deg,_lb,_ub) @@ -128,8 +129,8 @@ end function predict(Ch::ChebyType,x_new) true_new = Ch.f(x_new) - basis_new = Float64[ChebyT(unitmap(x_new[i],Ch.lb,Ch.ub),j) for i=1:length(x_new),j=0:Ch.deg] - basis_nodes = Float64[ChebyT(unitmap(Ch.nodes[i],Ch.lb,Ch.ub),j) for i=1:length(Ch.nodes),j=0:Ch.deg] + basis_new = Float64[ChebyT(unitmap(x_new[i],Ch.lb,Ch.ub),j) for i=1:length(x_new),j=0:Ch.deg - 1] + basis_nodes = Float64[ChebyT(unitmap(Ch.nodes[i],Ch.lb,Ch.ub),j) for i=1:length(Ch.nodes),j=0:Ch.deg - 1] preds = basis_new * Ch.coefs preds_nodes = basis_nodes * Ch.coefs @@ -139,8 +140,40 @@ end function q5(deg=(5,9,15),lb=-1.0,ub=1.0) runge(x) = 1.0 ./ (1 .+ 25 .* x.^2) + f(x) = x .+ 2x.^2 - exp.(-x) + precision = 10000 + points = range(lb, ub, length=precision) + cheb_m = zeros(precision, length(deg)) + uni_m = zeros(precision, length(deg)) + + for (i, d) in enumerate(deg) + degse = d:-1:1 + node = 1/2 .* (ub .+ lb) .+ 1/2 .* (ub - lb) .* cos.((2 .* degse .- 1) .* pi ./ (2 .* d)) + uni = LinRange(lb, ub, d) + cheb = ChebyType(node, d, lb, ub, runge) + unic = ChebyType(uni, d, lb, ub, runge) + + pred_cheb = predict(cheb, points) + pred_uni = predict(unic, points) + + cheb_m[:, i] = pred_cheb["preds"] + uni_m[:, i] = pred_uni["preds"] + end - + subplot(121) + PyPlot.plot(points, runge.(points), label="True") + for (i, d) in enumerate(deg) + PyPlot.plot(points, cheb_m[:, i,], label=d) + end + title("Chebyshev nodes") + legend() + subplot(122) + PyPlot.plot(points, runge.(points), label="True") + for (i, d) in enumerate(deg) + PyPlot.plot(points, uni_m[:, i,], label=d) + end + title("Uniform nodes") + legend() PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q5.png")) end @@ -156,8 +189,33 @@ function q6() end function q7() - - + f(x) = abs(x)^0.5 + precision = 63 + x = range(-1, 1, length=precision) + myknots = vcat(range(-1,stop = -0.1,length = 5), 0, 0, 0, range(0.1,stop = 1,length =5)) + + y = f.(x) + bs = BSpline(13, 3, -1, 1) + bs2 = BSpline(myknots, 3) + B = Array(getBasis(collect(x), bs)) + B2 = Array(getBasis(collect(x), bs2)) + c = B \ y + c2 = B2 \ y + subplot(311) + PyPlot.plot(x, y, label="True function") + PyPlot.scatter(x, B * c, color="red", s=2, label="Approximation") + title("Uniform knot vector") + legend() + subplot(312) + PyPlot.plot(x, y, label="True function") + PyPlot.scatter(x, B2 * c2, color="red", s=2, label="Approximation") + title("Knot multiplicity x=0") + legend() + subplot(313) + PyPlot.plot(x, y - B * c, color="blue", label="Uniform") + PyPlot.plot(x, y - B2 * c2, color="red", label="Knot multiplicity") + title("Error plots") + legend() PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q7.png")) end diff --git a/test/runtests.jl b/test/runtests.jl index 6453446..d7694f2 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -3,8 +3,15 @@ using Test @testset "HWfuncapp.jl" begin # 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