#=
day5:
- Julia version: 1.7.0
- Author: Paul.Mealor
- Date: 2021-12-05
=#

using AoC, Test
using Base.Iterators: filter

parseLine(line) = split(line, r",| -> ") |> cc -> parse.(Int, cc) |> cc -> reshape(cc, 2,2)
@test parseLine("3,5 -> 6,8") == [[3, 5] [6,8]]

parseLines(lines) = cat((parseLine.(lines))...; dims=3)
@test parseLines(["3,5 -> 6,8", "4,6 -> 2,3"]) == [[3,5] [6,8] ;;; [4,6] [2,3]]

maxX(lines) = max(lines[1,:,:]...)
maxY(lines) = max(lines[2,:,:]...)
maxCoord(lines) = [maxX(lines), maxY(lines)]
@test maxCoord([[1,2] [3,4] ;;; [5,3] [1,2]]) == [5,4]

minX(lines) = min(lines[1,:,:]...)
minY(lines) = min(lines[2,:,:]...)
minCoord(lines) = [minX(lines), minY(lines)]
@test minCoord([[0,2] [3,4] ;;; [5,3] [1,2]]) == [0,2]

mapOffset(lines) = [1,1] - minCoord(lines)

mapSize(lines) = maxCoord(lines) + mapOffset(lines)

direction(line) = direction(line[:,1],line[:,2])
direction(start, finish) = sign.(finish - start)
@test direction([[1,1] [7,7]]) == [1,1]
@test direction([[1,7] [7,7]]) == [1,0]
@test direction([[7,1] [1,7]]) == [-1,1]

function createMap(lines)
    map = zeros(Int, mapSize(lines)...)
    offset = mapOffset(lines)

    for line in eachslice(lines; dims=3)
        point = line[:,1] + offset
        endPoint = line[:,2] + offset
        dir = direction(line)
        while true
            map[point...]+=1
            if point == endPoint
                break
            end
            point += dir
        end
    end

    return map
end

selectPart1(lines) = cat(filter(eachslice(lines, dims=3)) do line
    (start, finish) = (line[:,1], line[:,2])
    return start[1] == finish[1] || start[2] == finish[2]
end...; dims=3)
@test selectPart1([[2,2] [4,4] ;;; [2,2] [2,5]]) == [[2,2] [2,5] ;;;]

part1(lines) = parseLines(lines) |> selectPart1 |> createMap |> map -> count(x -> x>=2, map)
part2(lines) = parseLines(lines) |> createMap |> map -> count(x -> x>=2, map)

@test part1(exampleLines(5,1)) == 5
@test part2(exampleLines(5,1)) == 12

lines(5) |> ll -> @time part1(ll) |> show
lines(5) |> ll -> @time part2(ll) |> show


# Static Car Escape
T = Float64;

# model
model = Dynamics.DubinsCar()
n,m = size(model)
x0 = @SVector [2.5,2.5,0.]
xf = @SVector [7.5,2.5,0.]
N = 101
tf = 3.0

# cost
Q = (1e-3)*Diagonal(@SVector ones(n))
R = (1e-2)*Diagonal(@SVector ones(m))
Qf = 100.0*Diagonal(@SVector ones(n))
obj = LQRObjective(Q,R,Qf,xf,N)

# constraints
r = 0.5
s1 = 30; s2 = 50; s3 = 15

circles_escape = NTuple{3,Float64}[]

for i in range(0,stop=5,length=s1)
    push!(circles_escape,(0.,i,r))
end
for i in range(0,stop=5,length=s1)
    push!(circles_escape,(5.,i,r))
end
for i in range(0,stop=5,length=s1)
    push!(circles_escape,(10.,i,r))
end
for i in range(0,stop=10,length=s2)
    push!(circles_escape,(i,0.,r))
end
for i in range(0,stop=3,length=s3)
    push!(circles_escape,(i,5.,r))
end
for i in range(5,stop=8,length=s3)
    push!(circles_escape,(i,5.,r))
end

n_circles_escape = 3*s1 + s2 + 2*s3

circles_escape
x,y,r = collect(zip(circles_escape...))
x = SVector{n_circles_escape}(x)
y = SVector{n_circles_escape}(y)
r = SVector{n_circles_escape}(r)

obs = CircleConstraint(n,x,y,r)
con_obs = ConstraintVals(obs, 2:N-1)

bnd = BoundConstraint(n,m,u_min=-5.,u_max=5.)
con_bnd = ConstraintVals(bnd, 1:N-1)

goal = GoalConstraint(xf)
con_xf = ConstraintVals(goal, N:N)

conSet = ConstraintSet(n,m,[con_obs, con_bnd, con_xf], N)

# Build problem
U0 = [@SVector ones(m) for k = 1:N-1]

car_escape_static = Problem(model, obj, xf, tf;
    constraints=conSet, x0=x0)
initial_controls!(car_escape_static, U0);

X_guess = [2.5 2.5 0.;
           4. 5. .785;
           5. 6.25 0.;
           7.5 6.25 -.261;
           9 5. -1.57;
           7.5 2.5 0.]
X0_escape = interp_rows(N,tf,Array(X_guess'))
initial_states!(car_escape_static, X0_escape)


#多対一のケース
function deferred_acceptance(prop_prefs::Vector{Vector{Int}},
                             resp_prefs::Vector{Vector{Int}},
                             caps::Vector{Int})
    m = length(prop_prefs)
    n = length(resp_prefs)
    prop_matched = zeros(Int64,m)
    resp_matched = zeros(Int64,sum(caps))
    count = ones(Int64,m)
    accept = zeros(Int64,n)
    indptr = Array{Int}(n+1)
    indptr[1] = 1
    for i in 1:n
        indptr[i+1] = indptr[i] + caps[i]
    end

    for j in 1:n
        for i in 1:m
            if prop_matched[i] == 0 && count[i] <= length(prop_prefs[i])
                k = prop_prefs[i][count[i]]
                if findfirst(resp_prefs[k], i) != 0
                    if accept[k] < caps[k]
                        resp_matched[indptr[k+1]-1 - accept[k]] = i
                        prop_matched[i] = k
                        accept[k]+=1
                    else
                        list = resp_matched[indptr[k]:indptr[k+1]-1]
                        ranking = zeros(Int64,caps[k])
                        for l in 1:caps[k]
                            ranking[l] = findfirst(resp_prefs[k], list[l])
                        end
                        if 0 < findfirst(resp_prefs[k], i) < maximum(ranking)
                            resp_matched[indptr[k] + indmax(ranking) - 1] = i
                            prop_matched[list[indmax(ranking)]] = 0
                            prop_matched[i] = k
                        end
                    end
                end
                count[i]+=1
            end
        end
    end
    return prop_matched, resp_matched, indptr
end

#一対一のケース
function deferred_acceptance(prop_prefs::Vector{Vector{Int}},
                                resp_prefs::Vector{Vector{Int}})
    caps = ones(Int, length(resp_prefs))
    prop_matched, resp_matched, indptr =
        deferred_acceptance(prop_prefs, resp_prefs, caps)
    return prop_matched, resp_matched
end

function deferred_acceptance(m_prefs::Vector{Vector{Int}},f_prefs::Vector{Vector{Int}})
    m_size = length(m_prefs)
    f_size = length(f_prefs)
    m_matched = zeros(Int64,m_size)
    f_matched = zeros(Int64,f_size)
    m_prp = ones(Int64,m_size)
    for j in 1:f_size
        for i in 1:m_size
            if m_matched[i] == 0 && m_prp[i] <= length(m_prefs[i])
                if findfirst(f_prefs[m_prefs[i][m_prp[i]]], i) != 0
                    if f_matched[m_prefs[i][m_prp[i]]] == 0
                        f_matched[m_prefs[i][m_prp[i]]] = i
                        m_matched[i] = m_prefs[i][m_prp[i]]
                    else
                        if 0 < findfirst(f_prefs[m_prefs[i][m_prp[i]]], i) < findfirst(f_prefs[m_prefs[i][m_prp[i]]], f_matched[m_prefs[i][m_prp[i]]])
                            m_matched[f_matched[m_prefs[i][m_prp[i]]]] = 0
                            m_matched[i] = m_prefs[i][m_prp[i]]
                            f_matched[m_prefs[i][m_prp[i]]] = i
                        end
                    end
                end
                m_prp[i]+=1
            end
        end
    end
    return(m_matched,f_matched)
end


### A Pluto.jl notebook ###
# v0.15.1

using Markdown
using InteractiveUtils

# ╔═╡ d5d8aca7-1771-4bd9-9761-11b667f894fa
using Plots, Latexify, LaTeXStrings, DataFrames, PlutoUI

# ╔═╡ 7a45e73e-2bc0-489f-a7e1-4daa852589e1
md"# Máquinas de Fluxo - TR1
\
**Alunos:**
* Alessandro Cappellini Portugal
* Rodrigo Kenji Kuroda
* Jean Carlos Reisdorfer
* Laís Tussi
* Stella Colferai
"

# ╔═╡ c7180b99-37f6-4494-aafb-57bdec113524
html"""

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License."""

# ╔═╡ b42465ab-12ae-4e2e-a4dd-ccf375cd69f4
md"## NC 2733"

# ╔═╡ 67277e58-4e39-4ced-9135-ce059ed3fe51
begin
	e = [2.15, 3.11, 4.81, 7.06, 10, 13.1, 17.1, 22.1, 25.1, 28.4, 31.6, 34.7, 36.8, 41.7, 46.6, 51.7, 56]
	b₀ = [21280, 12260, 9312, 6937, 6219, 5105, 5363, 3960, 3063, 2967, 3113, 2449, 2237, 1979, 2266, 2266, 2266]
	b₁ = [1231, 973.3, 776.8, 697.9, 589.8, 572.8, 415.7, 362.1, 366, 333.3, 281.1, 302, 314, 325.3, 302, 302, 302]
end

# ╔═╡ 6de97699-ca2f-4d84-9c6c-0d1cf2fb9781
begin
	scatter(e,b₀,legend=true, xlabel="Thickness [m]", ylabel = "Unit pressure drop [Pa/m]", labels = "b₀", title = "NC 2733", xlim=(0,60),ylim=(0,25000))
	plot!(e, b₁, label = "b₁", seriestype = :scatter)
end

# ╔═╡ d9760d43-f85c-4574-b896-f098e52a5e9f
md"### Equação dos coeficientes e matriz A:

$b_{0}(e) = \frac{a_{0}}{e} + a_{1} = a_{0}\cdot e^{-1} + a_{1} = \begin{bmatrix} e^{-1} & 1 \end{bmatrix} \cdot \begin{bmatrix} a_{0}\\a_{1} \end{bmatrix}$

$A = \begin{bmatrix} e_{0} & 1 \\ e_{1} & 1 \\ \vdots & \vdots \\ e_{16} & 1 \end{bmatrix}$"

# ╔═╡ 0e89da5e-0028-4294-abca-62aa25435d4e
begin
	A = [e[i]^j for i in 1:17, j in -1:0]
	ê₀ = A \ b₀
	b̂₀(e) = ê₀' * [e^-1,1]
	ê₁ = A \ b₁
	b̂₁(e) = ê₁' * [e^-1,1]
	Print(ê₀, ê₁)
end

# ╔═╡ 183dfa02-3e9b-4d37-af9c-e14749d7bbed
begin
	plot!(b̂₀, label = "b₀(e)")
	plot!(b̂₁, label = "b₁(e)")
end

# ╔═╡ 7ca1454b-6b82-435f-8495-378634a92ddd
md"## NCX1116"

# ╔═╡ ce7b5b46-ce40-4fde-b919-4e7aa463621b
begin
	E = [5.12, 10, 13.2, 15.6, 20.4, 21.7, 25.3, 30.7, 37.5, 41.8, 44.9, 47.2, 50.8, 55, 59.9, 62.1]
	b₂ = [5665, 2689, 2144, 2236, 1592, 1872, 1151, 1196, 1334, 947, 1076, 1060, 1093, 888.7, 817.1, 949.2]
	b₃ = [256.2, 271.7, 235.9, 208.6, 171.3, 139.5, 155.8, 127.5, 93.9, 104.2, 91.3, 86.1, 79.9, 87.6, 86.7, 82.4]
end

# ╔═╡ 2a30f4cd-65cb-4fb6-bc45-7c3999e74ce0
begin
	scatter(E,b₂,legend=true, xlabel="Thickness [m]", ylabel = "Unit pressure drop [Pa/m]", label = "b₂", title = "NCX1116", xlim=(0,70), ylim=(0,6000))
	plot!(E, b₃, label = "b₃",seriestype = :scatter)
end

# ╔═╡ 63b5568d-8a7c-456d-912e-bf0f41821a27
md"### Equação dos coeficientes e matriz B:

$b_{2}(e) = \frac{a_{2}}{e} + a_{3} = a_{2}\cdot e^{-1} + a_{3} = \begin{bmatrix} e^{-1} & 1 \end{bmatrix} \cdot \begin{bmatrix} a_{2}\\a_{3} \end{bmatrix}$

$B = \begin{bmatrix} e_{0} & 1 \\ e_{1} & 1 \\ \vdots & \vdots \\ e_{15} & 1 \end{bmatrix}$"

# ╔═╡ 4c940207-3f1d-431b-9c01-52f05ab4e7d0
begin
	B = [E[i]^j for i in 1:16, j in -1:0]
	ê₂ = B \ b₂
	b̂₂(E) = ê₂' * [E^-1,1]
	ê₃ = B \ b₃
	b̂₃(E) = ê₃' * [E^-1,1]
	Print(ê₂, ê₃)
end

# ╔═╡ 80098407-d77f-4282-8776-93bb395e23ca
begin
	plot!(b̂₂, label = "b₂(e)")
	plot!(b̂₃, label = "b₃(e)")
end

# ╔═╡ fb25f078-e1fd-46e1-b231-5404f57edc15
begin
	compₐ1 = []
	erroₐ1 = []
	a_C1 = [1623.500, 39415.000, 284.200, 2177.600]
	a_AR1 = [ê₀[2], ê₀[1], ê₁[2], ê₁[1]]
	compₐ2 = []
	erroₐ2 = []
	a_C2 = [433.890, 25798.000, 83.501, 1230.200]
	a_AR2 = [ê₂[2], ê₂[1], ê₃[2], ê₃[1]]

	for i in 1:4
		append!(compₐ1, abs(a_AR1[i]-a_C1[i]))
		append!(erroₐ1, round(compₐ1[i]/a_C1[i]*100, digits=2))
		append!(compₐ2, abs(a_AR2[i]-a_C2[i]))
		append!(erroₐ2, round(compₐ2[i]/a_C2[i]*100, digits=2))
	end

	df1 = DataFrame(a=["a₀", "a₁", "a₂", "a₃"], artigo1=a_C1, experimento1=a_AR1, erro1=erroₐ1)

	rename!(df1, ["NC 2733", "Artigo", "Experimento", "Erro [%]"])

	df2 = DataFrame(a=["a₀", "a₁", "a₂", "a₃"], artigo2=a_C2, experimento2=a_AR2, erro2=erroₐ2)

	rename!(df2, ["NCX1116", "Artigo", "Experimento", "Erro [%]"])

	md"## Tabelas dos coeficientes e respectivas equações:"
end

# ╔═╡ c3eda44f-4bf3-4e6b-b9da-a730545118f7
df1

# ╔═╡ 18d0707c-e035-4c1e-8664-7a077648a9d3
L"$$
b_0(e) = 1624,43 + \frac{39375,80}{e}
$$
$$
b_1(e) = 284,31 + \frac{2174,91}{e}
$$"

# ╔═╡ fa429a6c-e740-4317-ba60-59a5b0d99b87
df2

# ╔═╡ cf4e0ac7-15d9-402d-9c40-a5eb600449f1
L"$$
b_2(e) = 434,34 + \frac{25760,50}{e}
$$
$$
b_3(e) = 83,50 + \frac{1228,77}{e}
$$"

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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Latexify = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"

[compat]
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "LibVPX_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "Pkg", "Zlib_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "3cc57ad0a213808473eafef4845a74766242e05f"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.3.1+4"

[[FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "35895cf184ceaab11fd778b4590144034a167a2f"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.1+14"

[[Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "cbd58c9deb1d304f5a245a0b7eb841a2560cfec6"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.1+5"

[[FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[Future]]
deps = ["Random"]
uuid = "9fa8497b-333b-5362-9e8d-4d0656e87820"

[[GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "dba1e8614e98949abfa60480b13653813d8f0157"
uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.5+0"

[[GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "b83e3125048a9c3158cbb7ca423790c7b1b57bea"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.57.5"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "e14907859a1d3aee73a019e7b3c98e9e7b8b5b3e"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.57.3+0"

[[GeometryBasics]]
deps = ["EarCut_jll", "IterTools", "LinearAlgebra", "StaticArrays", "StructArrays", "Tables"]
git-tree-sha1 = "15ff9a14b9e1218958d3530cc288cf31465d9ae2"
uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
version = "0.3.13"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "47ce50b742921377301e15005c96e979574e130b"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.68.1+0"

[[Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[HTTP]]
deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"]
git-tree-sha1 = "c6a1fff2fd4b1da29d3dccaffb1e1001244d844e"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "0.9.12"

[[IniFile]]
deps = ["Test"]
git-tree-sha1 = "098e4d2c533924c921f9f9847274f2ad89e018b8"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.0"

[[InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[InvertedIndices]]
deps = ["Test"]
git-tree-sha1 = "15732c475062348b0165684ffe28e85ea8396afc"
uuid = "41ab1584-1d38-5bbf-9106-f11c6c58b48f"
version = "1.0.0"

[[IterTools]]
git-tree-sha1 = "05110a2ab1fc5f932622ffea2a003221f4782c18"
uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
version = "1.3.0"

[[IteratorInterfaceExtensions]]
git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
uuid = "82899510-4779-5014-852e-03e436cf321d"
version = "1.0.0"

[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "642a199af8b68253517b80bd3bfd17eb4e84df6e"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.3.0"

[[JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
git-tree-sha1 = "81690084b6198a2e1da36fcfda16eeca9f9f24e4"
uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
version = "0.21.1"

[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
version = "2.1.0+0"

[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
version = "3.100.1+0"

[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
version = "2.10.1+0"

[[LaTeXStrings]]
git-tree-sha1 = "c7f1c695e06c01b95a67f0cd1d34994f3e7db104"
uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
version = "1.2.1"

[[Latexify]]
deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"]
git-tree-sha1 = "a4b12a1bd2ebade87891ab7e36fdbce582301a92"
uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.6"

[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[LibVPX_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "12ee7e23fa4d18361e7c2cde8f8337d4c3101bc7"
uuid = "dd192d2f-8180-539f-9fb4-cc70b1dcf69a"
version = "1.10.0+0"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+0"

[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29"
version = "1.3.0+3"

[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
version = "1.42.0+0"

[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
version = "1.16.1+1"

[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
version = "2.35.0+0"

[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "6a8a2a625ab0dea913aba95c11370589e0239ff0"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.6"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.0.3"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "4ea90bd5d3985ae1f9a908bd4500ae88921c5ce7"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.0"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+0"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.10+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "c8abc88faa3f7a3950832ac5d6e690881590d6dc"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "1.1.0"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "501c20a63a34ac1d015d5304da0e645f42d91c9f"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.11"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "f32cd6fcd2909c2d1cdd47ce55e1394b04a66fe2"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.18.2"

[[PlutoUI]]
deps = ["Base64", "Dates", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "Suppressor"]
git-tree-sha1 = "44e225d5837e2a2345e69a1d1e01ac2443ff9fcb"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.9"

[[PooledArrays]]
deps = ["DataAPI", "Future"]
git-tree-sha1 = "cde4ce9d6f33219465b55162811d8de8139c0414"
uuid = "2dfb63ee-cc39-5dd5-95bd-886bf059d720"
version = "1.2.1"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[PrettyTables]]
deps = ["Crayons", "Formatting", "Markdown", "Reexport", "Tables"]
git-tree-sha1 = "0d1245a357cc61c8cd61934c07447aa569ff22e6"
uuid = "08abe8d2-0d0c-5749-adfa-8a2ac140af0d"
version = "1.1.0"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RecipesBase]]
git-tree-sha1 = "b3fb709f3c97bfc6e948be68beeecb55a0b340ae"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.1"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "2a7a2469ed5d94a98dea0e85c46fa653d76be0cd"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.3.4"

[[Reexport]]
git-tree-sha1 = "5f6c21241f0f655da3952fd60aa18477cf96c220"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.1.0"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "a43a7b58a6e7dc933b2fa2e0ca653ccf8bb8fd0e"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.6"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "2f6792d523d7448bbe2fec99eca9218f06cc746d"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.8"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "000e168f5cc9aded17b6999a560b7c11dda69095"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.0"

[[Suppressor]]
git-tree-sha1 = "a819d77f31f83e5792a76081eee1ea6342ab8787"
uuid = "fd094767-a336-5f1f-9728-57cf17d0bbfb"
version = "0.2.0"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "8ed4a3ea724dac32670b062be3ef1c1de6773ae8"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.4.4"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "acc685bcf777b2202a904cdcb49ad34c2fa1880c"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.14.0+4"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7a5780a0d9c6864184b3a2eeeb833a0c871f00ab"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "0.1.6+4"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d713c1ce4deac133e3334ee12f4adff07f81778f"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2020.7.14+2"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "487da2f8f2f0c8ee0e83f39d13037d6bbf0a45ab"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.0.0+3"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─7a45e73e-2bc0-489f-a7e1-4daa852589e1
# ╟─c7180b99-37f6-4494-aafb-57bdec113524
# ╟─b42465ab-12ae-4e2e-a4dd-ccf375cd69f4
# ╠═d5d8aca7-1771-4bd9-9761-11b667f894fa
# ╠═67277e58-4e39-4ced-9135-ce059ed3fe51
# ╠═6de97699-ca2f-4d84-9c6c-0d1cf2fb9781
# ╟─d9760d43-f85c-4574-b896-f098e52a5e9f
# ╠═0e89da5e-0028-4294-abca-62aa25435d4e
# ╠═183dfa02-3e9b-4d37-af9c-e14749d7bbed
# ╟─7ca1454b-6b82-435f-8495-378634a92ddd
# ╠═ce7b5b46-ce40-4fde-b919-4e7aa463621b
# ╠═2a30f4cd-65cb-4fb6-bc45-7c3999e74ce0
# ╟─63b5568d-8a7c-456d-912e-bf0f41821a27
# ╠═4c940207-3f1d-431b-9c01-52f05ab4e7d0
# ╠═80098407-d77f-4282-8776-93bb395e23ca
# ╟─fb25f078-e1fd-46e1-b231-5404f57edc15
# ╟─c3eda44f-4bf3-4e6b-b9da-a730545118f7
# ╟─18d0707c-e035-4c1e-8664-7a077648a9d3
# ╟─fa429a6c-e740-4317-ba60-59a5b0d99b87
# ╟─cf4e0ac7-15d9-402d-9c40-a5eb600449f1
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002


include("CrossEntropyLog.jl")
include("jacobian.jl")
using Base.Test

X = rand(Float32, (5, 3))
t = rand(Float32, (5, 3))
l = CrossEntropyLog{Float32}()

# Test grad. wrt. input X
target!(l, t)
Dfwd = jacobian_fwd(l, X)
Dbwd = jacobian_bwd(l, X)

@test_approx_eq Dfwd Dbwd


require("Ito")

using Calendar
using Ito.Time
using Ito.TermStructure

@test_approx_eq 1.7 compound_factor(0.7, :Compounded, Ito.TermStructure.Annual, Actual360(), today(), today()+days(360))
@test_approx_eq 1.35^2 compound_factor(0.7, :Compounded, Ito.TermStructure.Semiannual, Actual360(), today(), today()+days(360))

@test_approx_eq 1/1.7 discount_factor(0.7, :Compounded, Ito.TermStructure.Annual, Actual360(), today(), today()+days(360))


#=== Simplex ===#
struct Simplex{N, P} <: AbstractSimplex
    vs::NTuple{N, P}
    hash::UInt64
    Simplex{N,P}(vals::NTuple{N, P}) where {N, P} = new(vals, hash(Set(vals)))
    Simplex{0,P}() where {P} = new((), zero(UInt))
end
Simplex(splx::P...) where {P} = Simplex{length(splx),P}(splx)
Simplex(splx::Vector{P}) where {P} = Simplex(splx...)
Simplex(::Type{P}) where {P} = Simplex{0,P}()

# Private methods

show(io::IO, splx::Simplex) = show(io, "σ$(splx.vs)")
eltype(splx::Simplex{N,P}) where {N,P} = P
eltype(::Type{Simplex{N,P}}) where {N,P} = P

# Public methods

dim(splx::Simplex{N,P}) where {N,P} = N-1
hash(splx::Simplex) = splx.hash
==(a::Simplex, b::Simplex) = a.hash == b.hash

function vertices(splx::Simplex)
    vtxs = typeof(hash(splx))[]
    for v in splx.vs
        push!(vtxs, hash(Simplex(v)))
    end
    return vtxs
end

union(u::Simplex, v::Simplex) = Simplex(collect(u.vs ∪ v.vs))

# TODO make generated
function faces(σ::Simplex)
    d = dim(σ)
    d == 0 && return Simplex[]
    (Simplex([ifelse(j < i, σ.vs[j], σ.vs[j+1]) for j in 1:d]) for i in 1:(d+1))
end

function boundary(::Type{R}, σ::Simplex) where {R}
    d = dim(σ)
    ch = Chain(d-1, UInt, R)
    d == 0 && return ch
    o = one(R)
    for (i,face) in enumerate(faces(σ))
        push!(ch, hash(face)=>( isodd(i) ? o : -o))
    end
    return ch
end

# Public methods for AbstractSimplex

indices(splx::Simplex) = collect(splx.vs)

# Misc. methods

function volume(S::AbstractMatrix)
    d, vc = size(S)
    @assert d == vc-1 "Number of vertices in simplex should be dim+1"
    v0 = S[:,1]
    return abs(det(S[:,2:end] .- v0))/prod(1:d)
end
volume(s::Simplex{N,Int}, X::AbstractMatrix) where {N}= volume(X[:, indices(s)])

parse(::Type{Simplex{N,P}}, str::AbstractString) where {N,P} =
    Simplex(map(e->parse(P, e), split(str, ' ')))


using Distributions, Plots, Turing, DataFrames, MCMCChains, StatsPlots

mutable struct QLearningAgent
    q::Array{Real, 1}
    α::Real
    β::Real
    function QLearningAgent(α::Real, β::Real, K::Int64)
        q = Array{Real, 1}(zeros(K))
        new(q, α, β)
    end
end

function update(agent::QLearningAgent, action::Real, reward::Real)
    agent.q[action] += agent.α * (reward - agent.q[action])
end

function policy(agent::QLearningAgent)
    qmax = maximum(agent.q)
    qexp = exp.(agent.β .* (agent.q .- qmax))
    return qexp ./ sum(qexp)
end

struct Bandit
    p::Array{Real, 1}
end

function step(env::Bandit, action::Int64)
    k = length(env.p)
    rewards = env.p .>= rand(k)
    return rewards[action]
end

function step(agent::QLearningAgent, env::Bandit)
    p = policy(agent)
    action = rand(Categorical(p))
    reward = step(env, action)
    update(agent, action, reward)
    d = DataFrame(action = action, reward = reward)
    return hcat(d, DataFrame(agent.q', :auto))
end

function run(agent::QLearningAgent, env::Bandit, trial::Int64)
    return reduce(vcat, map(_ -> step(agent, env), 1:trial))
end

bandit = Bandit([0.2, 0.8])
k = length(bandit.p)
agent = QLearningAgent(0.05, 2., k)
result = run(agent, bandit, 500)

plot(result.x1, label="Q1")
plot!(result.x2, label="Q2")
scatter!(result.action .- 1,
         markersize=1,
         markershape=:vline,
         label="Response")

@model QLearningModel(actions::Array{Real, 1}, rewards::Array{Real, 1}, k::Int64) = begin
    T = length(actions)
    α ~ Beta(1, 1)
    β ~ Gamma(1, 100)
    agent = QLearningAgent(α, β, k)
    for t in 1:T
        action = actions[t]
        reward = rewards[t]
        p = policy(agent)
        actions[t] ~ Categorical(p)
        update(agent, action, reward)
    end
end

actions = Array{Real, 1}(result.action)
rewards = Array{Real, 1}(result.reward)
chains = sample(QLearningModel(actions, rewards, 2), NUTS(), 1000)
plot(chains)

chains = sample(QLearningModel(actions, rewards, 2), NUTS(), MCMCThreads(), 1000, 4)
plot(chains)


function compute_mutual_information(boundary_ps, applied_p, prior, sigma, interior_q)
  critical_fs = get_critical_values(boundary_ps, applied_p, prior, interior_q)
  compute_mutual_information(critical_fs, prior, sigma)
end

function compute_mutual_information(critical_fs, prior, sigma)
  nu = sigma^2

  belief = prior[:]
  field = critical_fs[:]

  normals = normal_matrix(field, nu)

  compute_mutual_information(belief, normals)
end

function compute_mutual_information(belief::Array{Float64,1},
                                    normals::Array{Float64,2})
  i1 = integral_cross(belief, normals)
  i2 = integral_joint(belief, normals)
  i3 = integral_marginals(belief, normals)

  out::Float64 = -2*log(e, i1) + log(e, i2) + log(e, i3)

  out
end

function normal_matrix(field::Array{Float64,1}, nu::Float64)
  two_nu::Float64 = 2 * nu

  n = length(field)
  normals = zeros(n, n)

  nhalf_over_nu = - 0.5 / two_nu
  over_sqrt2pinu = 1/sqrt(2*pi*two_nu)

  @inbounds for ii = 1:n
    @fastmath @simd for jj = ii+1:n
      normals[jj,ii] = normal(field[ii] - field[jj], nhalf_over_nu, over_sqrt2pinu)
      normals[ii,jj] = normals[jj,ii]
    end
  end

  @inbounds @fastmath @simd for ii = 1:n
    normals[ii,ii] = normal(0.0, nhalf_over_nu, over_sqrt2pinu)
  end

  normals
end

function integral_cross(belief, normals)
  out::Float64 = 0.0

  @inbounds for ii = 1:length(belief)
    @fastmath @simd for jj = 1:length(belief)
      out += belief[ii]^2 * belief[jj] * normals[jj,ii]
    end
  end
  out /= length(belief)^2

  out
end

function integral_joint(belief, normals)
  out::Float64 = 0.0

  @fastmath @inbounds @simd for ii = 1:length(belief)
    out += belief[ii]^2 * normals[ii,ii]
  end
  out /= length(belief)

  out
end

function integral_marginals(belief, normals)
  val1::Float64 = sum(belief.^2) / length(belief)

  val2::Float64 = 0.0
  @inbounds for ii = 1:length(belief)
    @fastmath @simd for jj = 1:length(belief)
      val2 += belief[ii] * belief[jj] * normals[jj,ii]
    end
  end
  val2 /= length(belief)^2

  out::Float64 = val1 * val2

  out
end


function octree(charts::Vector{S} where {S <: CompScienceMeshes.Simplex})

    ncells = length(charts)

    T = coordtype(charts[1])
    P = eltype(charts[1].vertices)

    points = zeros(P, ncells)
    radii = zeros(T, ncells)

    for (i,ch) in enumerate(charts)
        points[i] = cartesian(center(ch))
        radii[i] = maximum(norm(v-points[i]) for v in ch.vertices)
    end

    return Octree(points, radii)
end

"""
    Hi
"""
function grideval(points, coeffs, basis; type=nothing)

    # charts: active charts
    # ad: assembly data (active_cell_idx, local_shape_idx) -> [dof1, dfo2, ...]
    # ag: active_cell_idx -> global_cell_idx
    charts, ad, ag = assemblydata(basis)
    refs = refspace(basis)

    V = valuetype(refs, eltype(charts))
    T = promote_type(eltype(coeffs), eltype(V))
    P = similar_type(V, T)

    type != nothing && (P = type)

    values = zeros(P, size(points))

    chart_tree = BEAST.octree(charts)
    for (j,point) in enumerate(points)
        i = CompScienceMeshes.findchart(charts, chart_tree, point)
        if i != nothing
            # @show i
            chart = charts[i]
            u = carttobary(chart, point)
            vals = refs(neighborhood(chart,u))
            for r in 1 : numfunctions(refs)
                for (m,w) in ad[i, r]
                    values[j] += w * coeffs[m] * vals[r][1]
                end
            end
        end
    end
    return values
end


using Flux, LinearAlgebra, Polynomials, Plots

const n = 100
# target polynomial
c = ChebyshevT([-1,0,-2,0,1,0,1,2,3])
target = convert(Polynomial, c)
plot(target, (-1.,1.)...,label="target")

length(target.coeffs)

# mapping polynomial to circulant matrix
S = circshift(Matrix{Float64}(I, n, n),(1,0))
param = zeros(n)
param[1:9] = target.coeffs
Circulant = param
for k in 1:n-1
    Circulant = hcat(Circulant, S^k*param)
end

# creating dataset
bs = 3000
x = randn(Float32,n,1,bs)
y = convert(Array{Float32},reshape(transpose(Circulant)*dropdims(x;dims=2) ,(n,1,bs)))
data = [(x,y)]

# padding function to work modulo n
function pad_cycl(x;l=1,r=1)
    last = size(x,1)
    xl = selectdim(x,1,last-l+1:last)
    xr = selectdim(x,1,1:r)
    cat(xl, x, xr, dims=1)
end

# neural network with 7 convolution layers
model = Chain(
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros()),
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros()),
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros()),
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros()),
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros()),
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros()),
    x -> pad_cycl(x,l=0,r=2),
    CrossCor((3,),1=>1,bias=Flux.Zeros())
)

loss(x, y) = Flux.Losses.mse(model(x), y)
ps = Flux.params(model)
loss_vector = Vector{Float32}()
logging_loss() = push!(loss_vector, loss(x, y))
opt = ADAM(0.2)
n_epochs = 1700
for epochs in 1:n_epochs
    Flux.train!(loss, ps, data, opt, cb=logging_loss)
    if epochs % 50 == 0
        println("Epoch: ", epochs, " | Loss: ", loss(x,y))
    end
end

plot(loss_vector)

pred = Polynomial([1])
for p in ps
    if typeof(p) <: Array
        pred *= Polynomial([p...])
    end
end
plot(target, (-1.,1.)...,label="target")
ylims!((-10,10))
plot!(pred, (-1.,1.)...,label="pred")


## Methods for canonical, varfunc and devresid2 for distributions in the exponential family

@doc """
Returns the canonical Link type for an instance of a distribution in the exponential family
""" ->
canonical(::Bernoulli) = LogitLink()
canonical(::Binomial) = LogitLink()
canonical(::Gamma) = InverseLink()
canonical(::Normal) = IdentityLink()
canonical(::Poisson) = LogLink()

@doc """
Evaluate `y*log(y/μ)` with the correct limit as `y` approaches zero from above
"""->
ylogydμ{T<:AbstractFloat}(y::T,μ::T) = y > zero(T) ? y*log(y/μ) : zero(T)

varfunc(::Bernoulli,μ) = μ*(one(μ)-μ)
varfunc(::Binomial,μ) = μ*(one(μ)-μ)
varfunc(::Gamma,μ) = abs2(μ)
varfunc(::Normal,μ) = one(μ)
varfunc(::Poisson,μ) = μ

two(y) = one(y) + one(y)                # equivalent to convert(typeof(y),2)

@doc """
Evaluate the squared deviance residual for a distribution instance and values of `y` and `μ`
"""->
devresid2(::Bernoulli,y,μ) = two(y)*(ylogydμ(y,μ) + ylogydμ(one(y)-y,one(μ)-μ))
devresid2(::Binomial,y,μ) = devresid2(Bernoulli(),y,μ)
devresid2(::Gamma,y,μ) =  two(y)*((y-μ)/μ - (y == zero(y) ? y : log(y/μ)))
devresid2(::Normal,y,μ) = abs2(y-μ)
devresid2(::Poisson,y,μ) = two(y)*(ylogydμ(y,μ)-(y-μ))

@doc """
Initial μ value from the y and wt
""" ->
mustart{T<:AbstractFloat}(::Bernoulli,y::T,wt::T) = (wt*y + convert(T,0.5))/(wt + one(T))
mustart{T<:AbstractFloat}(::Binomial,y::T,wt::T) = (wt*y + convert(T,0.5))/(wt + one(T))
mustart(::Gamma,y,wt) = y
mustart(::Normal,y,wt) = y
mustart{T<:AbstractFloat}(::Poisson,y::T,wt::T) = convert(T,1.1)*y


function part1and2()
	# Input file is small enough that you can just enter it manually instead of parsing the file
	xmin = 257
	xmax = 286
	ymin = -101
	ymax = -57

	highestheight = 0
	validvelocities = 0

	for xvel in 1:xmax
		for yvel in ymin:100
			x, y = 0, 0
			xvelcur = xvel
			yvelcur = yvel
			highesty = 0
			success = true
			while !(xmin <= x <= xmax && ymin <= y <= ymax)
				x += xvelcur
				y += yvelcur
				if xvelcur > 0
					xvelcur -= 1
				elseif xvelcur < 0
					xvelcur += 1
				end
				yvelcur -= 1

				if y > highesty
					highesty = y
				end

				if x > xmax || y < ymin
					success = false
					break
				end
			end

			if success
				if highesty > highestheight
					highestheight = highesty
				end
				validvelocities += 1
			end
		end
	end

	return (highestheight, validvelocities)
end

println(part1and2())


##
using AeroMDAO

## Test function
function doublet_source_case(foil, uniform)
    @time system = solve_case(foil,
                              uniform;
                              num_panels = 80)

    cl = lift_coefficient(system)
    cls, cms, _ = surface_coefficients(system)

    println("Cl: $cl")
    println("Σᵢ Clᵢ: $(sum(cls))")
    println("Σᵢ Cmᵢ: $(sum(cms))")
end

uniform = Uniform2D(angle = 5)

## Foil processing
naca_foil = naca4(2,4,1,2) # NACA 4-digit

doublet_source_case(naca_foil, uniform)

## Cosine spacing
cos_foil = cosine_interpolation(naca_foil, 81)

doublet_source_case(cos_foil, uniform)

## Camber-thickness transformations
xcamthick = camber_thickness(cos_foil)
foiler = Foil(camber_thickness_to_coordinates(xcamthick[:,1], xcamthick[:,2], xcamthick[:,3]))

doublet_source_case(foiler, uniform)

## Fitting to CST
num_dv = 8

# Coordinates fitting
up, low  = split_surface(naca_foil)
alpha_u  = coordinates_to_CST(up, num_dv)
alpha_l  = coordinates_to_CST(low, num_dv)
cst_foil = kulfan_CST(alpha_u, alpha_l, (0., 0.), (0., 0.))

doublet_source_case(cst_foil, uniform)

## Camber-thickness fitting
alpha_c, alpha_t = camber_thickness_to_CST(cos_foil, num_dv)
cam_foil = camber_CST(alpha_c, alpha_t, 0., 0)

doublet_source_case(cam_foil, uniform)

## Control surface deflection
con_foil = control_surface(cos_foil; angle = 8., hinge = 0.75)

doublet_source_case(con_foil, uniform)

## Plotting library
using Plots
gr(dpi = 300);

## Cosine and camber-thickness
plot(aspect_ratio = 1)
plot!(naca_foil.x, naca_foil.y, label = "Original")
plot!(cos_foil.x, cos_foil.y, label = "Cosine")
plot!(xcamthick[:,1], xcamthick[:,2], ls = :dot, label = "Camber")
plot!(xcamthick[:,1], xcamthick[:,3], ls = :dot, label = "Thickness")
plot!(foiler.x, foiler.y, label = "Inverse Camber-Thickness")
plot!(cst_foil.x, cst_foil.y, label = "CST Coordinates Fit")
plot!(cam_foil.x, cam_foil.y, label = "CST Camber-Thickness Fit")
plot!(con_foil.x, con_foil.y, ls = :dash, label = "Control Surface Deflection")


using Test, GenericLinearAlgebra, LinearAlgebra, Quaternions

@testset "Singular value decomposition" begin
    @testset "Problem dimension ($m,$n)" for
        (m,n) in ((6,5)     , (6,6)     , (5,6),
                  (60, 50)  , (60, 60)  , (50, 60),
                  (200, 150), (200, 200), (150, 200))

        vals = reverse(collect(1:min(m,n)))
        U = qr(Quaternion{Float64}[Quaternion(randn(4)...) for i = 1:m, j = 1:min(m,n)]).Q
        V = qr(Quaternion{Float64}[Quaternion(randn(4)...) for i = 1:n, j = 1:min(m,n)]).Q

        # FixMe! Using Array here shouldn't be necessary. Can be removed once
        # the bug in LinearAlgebra is fixed
        A = U*Array(Diagonal(vals))*V'

        @test size(A) == (m, n)
        @test vals ≈ svdvals(A)

        F = svd(A)
        @test vals ≈ F.S
        @show norm(F.U'*A*F.V - Diagonal(F.S), Inf)
        @test F.U'*A*F.V ≈ Diagonal(F.S)
    end

    @testset "The Ivan Slapničar Challenge" begin
        # This matrix used to hang (for n = 70). Thanks to Ivan Slapničar for reporting.
        n = 70
        J = Bidiagonal(0.5 * ones(n), ones(n-1), :U)
        @test GenericLinearAlgebra._svdvals!(copy(J)) ≈ svdvals(J)
        @test GenericLinearAlgebra._svdvals!(copy(J))[end] / svdvals(J)[end] - 1 < n*eps()
    end

    @testset "Compare to Base methods. Problem dimension ($m,$n)" for
        (m, n) in ((10,  9), # tall
                   (10, 10), # square
                   (9 , 10)) # wide

        A    = randn(m,n)
        Abig = big.(A)
        @test svdvals(A) ≈ Vector{Float64}(svdvals(Abig))
        @test cond(A)    ≈ Float64(cond(Abig))

        F    = svd(A)
        Fbig = svd(Abig)
        @test abs.(F.U'Float64.(Fbig.U)) ≈ I
        @test abs.(F.V'Float64.(Fbig.V)) ≈ I

        F    = svd(A, full=true)
        Fbig = svd(Abig, full=true)
        @test abs.(F.U'Float64.(Fbig.U)) ≈ I
        @test abs.(F.V'Float64.(Fbig.V)) ≈ I
    end

    @testset "Issue 54" begin
        U0, _, V0 = svd(big.(reshape(0:15, 4, 4)))
        A = U0[:, 1:3] * V0[:, 1:3]'

        U, S, V = svd(A)
        @test A ≈ U*Diagonal(S)*V'
    end

    @testset "Very small matrices. Issue 79" begin
        A = randn(1, 2)
        FA = svd(A)
        FAb = svd(big.(A))
        FAtb = svd(big.(A'))
        @test FA.S ≈ Float64.(FAb.S) ≈ Float64.(FAtb.S)
        @test abs.(FA.U'*Float64.(FAb.U))  ≈ I
        @test abs.(FA.U'*Float64.(FAtb.V)) ≈ I
        @test abs.(FA.V'*Float64.(FAb.V))  ≈ I
        @test abs.(FA.V'*Float64.(FAtb.U)) ≈ I
    end

    @testset "Issue 81" begin
        m = [1 0 0 0; 0 2 1 0; 0 1 2 0; 0 0 0 -1]
        @test Float64.(svdvals(big.(m))) ≈ svdvals(m)
    end
end


# Matern 3/2 ARD covariance function

"""
    Mat32Ard <: MaternARD

ARD Matern 3/2 kernel (covariance)
```math
k(x,x') = σ²(1 + √3|x-x'|/L)\\exp(- √3|x-x'|/L)
```
with length scale ``ℓ = (ℓ₁, ℓ₂, …)`` and signal standard deviation ``σ`` where
``L = diag(ℓ₁, ℓ₂, …)``.
"""
mutable struct Mat32Ard <: MaternARD
    "Inverse squared length scale"
    iℓ2::Vector{Float64}
    "Signal variance"
    σ2::Float64
    "Priors for kernel parameters"
    priors::Array

    """
        Mat32Ard(ll::Vector{Float64}, lσ::Float64)

    Create `Mat32Ard` with length scale `exp.(ll)` and signal standard deviation `exp(lσ)`.
    """
    Mat32Ard(ll::Vector{Float64}, lσ::Float64) = new(exp.(-2 .* ll), exp(2 * lσ), [])
end

function set_params!(mat::Mat32Ard, hyp::VecF64)
    length(hyp) == num_params(mat) || throw(ArgumentError("Mat32 kernel only has $(num_params(mat)) parameters"))
    @views @. mat.iℓ2 = exp(-2 * hyp[1:(end-1)])
    mat.σ2 = exp(2 * hyp[end])
end

get_params(mat::Mat32Ard) = [-log.(mat.iℓ2) / 2; log(mat.σ2) / 2]
get_param_names(mat::Mat32Ard) = [get_param_names(mat.iℓ2, :ll); :lσ]
num_params(mat::Mat32Ard) = length(mat.iℓ2) + 1

Statistics.cov(mat::Mat32Ard, r::Float64) =
    (s = √3 * r; mat.σ2 * (1 + s) * exp(-s))

dk_dll(mat::Mat32Ard, r::Float64, wdiffp::Float64) = 3 * mat.σ2 * wdiffp * exp(-√3 * r)


# Seed
Random.seed!(42) # in case solvers use stochastic algorithms

# Setup
StoppingProblem = @with_kw (μ_bar = -0.01,
                          σ_bar = 0.01,
                          S_bar = 10.0,
                          γ = 0.5, # u(x) = x^γ
                          ρ = 0.05, # discount rate
                          x_min = 0.01,
                          x_max = 5.0,
                          M = 15) # num of grid points

function LCP_objects(sp)
# setup
  @unpack μ_bar, σ_bar, S_bar, γ, ρ, x_min, x_max, M = sp
  grid = range(x_min, x_max, length = M)
  grid_extended = [grid[1] - diff(grid)[1]; grid; grid[end] + diff(grid)[end]]
  μ = μ_bar
  S(x) = S_bar
  u(x) = x^γ
  σ = σ_bar
# construct operator
  L = μ * L₁₋bc(grid_extended, (Reflecting(), Reflecting())) + σ^2/2 * L₂bc(grid_extended, (Reflecting(), Reflecting()))
# construct/return LCP objects
  B = ρ*I - L
  S_vec = S.(grid)
  q_vec = -u.(grid) + B*S_vec
  return (L = L, B = B, S = S_vec, q = q_vec)
end

function LCPsolve(sp)
  @unpack L, B, S, q = LCP_objects(sp)
  f = z -> B*z + q
  n = sp.M
  lb = zeros(n)
  ub = 300*ones(n) # a reasonable guess?
  options(convergence_tolerance = 1e-12, output = :no, time_limit = 600) # 10 minute budget
  exit_code, sol_z, sol_f = @suppress solveLCP(f, lb, ub);
end

@testset "Backward Case, Nonzero S" begin
  code, sol, f = LCPsolve(StoppingProblem())
  @test code == :Solved
  @test sol[1] ≈ 0.0 atol = 1e-5
  @test sol[5] ≈ 12.066957816758809 atol = 1e-5
  @test f[1] ≈ 0.39946441597137766 atol = 1e-5
end

@testset "Backward Case, Zero S" begin
  code, sol, f = LCPsolve(StoppingProblem(S_bar = 0.))
  @test code == :Solved
  @test sol[1] ≈ 2.050665004133949 atol = 1e-5
  @test sol[8] ≈ 30.258918534086924 atol = 1e-5
  @test f[1] ≈ 0.0 atol = 1e-5
end

function LCP_split(S)
  # setup
    u(x) = x^2
    ρ = 0.75
    μ(x) = -x
    σ = 0.1
    M = 300
    x = range(-5., 5., length = M)
    x̄ = [x[1] - diff(x)[1]; x; x[end] + diff(x)[end]]
    bc = (Reflecting(), Reflecting())
    L₁ = Diagonal(min.(μ.(x), 0.0)) * L₁₋bc(x̄, bc) + Diagonal(max.(μ.(x), 0.0)) * L₁₊bc(x̄, bc)
  # operator construction
    L = L₁+ σ^2/2 * L₂bc(x̄, bc)
  # LCP stuff
    B = ρ*I - L
    S_vec = S * ones(M)
    q_vec = -u.(x) + B*S_vec
    return (L = L, B = B, S = S_vec, q = q_vec)
end

@testset "Split Case, Nonzero S" begin
  # setup
  @unpack L, B, S, q = LCP_split(0.125)
  f = z -> B*z + q
  n = 300
  lb = zeros(n)
  ub = 300*ones(n) # a reasonable guess?
  options(convergence_tolerance = 1e-12, output = :no, time_limit = 600) # 10 minute budget
  exit_code, sol_z, sol_f = @suppress solveLCP(f, lb, ub);

  # tests
  @test exit_code == :StationaryPointFound
  @test sol_z[3] ≈ 8.774937148286833 atol = 1e-5
  @test sol_z[120] ≈ 0.3021168981772892 atol = 1e-5
  @test sol_z[270] ≈ 5.729541001488482 atol = 1e-5
  @test sol_f[end-1] ≈ 3.197442310920451e-14 atol = 1e-5
end

@testset "Split Case, Zero S" begin
  # setup
  @unpack L, B, S, q = LCP_split(0.)
  f = z -> B*z + q
  n = 300
  lb = zeros(n)
  ub = 300*ones(n) # a reasonable guess?
  options(convergence_tolerance = 1e-12, output = :no, time_limit = 600) # 10 minute budget
  exit_code, sol_z, sol_f = @suppress solveLCP(f, lb, ub);

  # tests
  @test exit_code == :StationaryPointFound
  @test sol_z[3] ≈ 8.888461348456772 atol = 1e-5
  @test sol_z[76] ≈ 2.279767804635279 atol = 1e-5
  @test sol_z[150] ≈ 0.005770703117189083 atol = 1e-5
  @test sol_z[269] ≈ 5.744079221450249 atol = 1e-5
  @test sol_f[123] ≈ 1.7763568394002505e-15 atol = 1e-5
end


x = range(0, stop=π/2, length=100)
d1 = plot(x, sin.(x), lw=0.05)
d2 = plot!(x, cos.(x), lw=0.05)
fill_between!(x, sin.(x), cos.(x), alpha=0.2)
x = x[1:5:end]
d3 = scatter!(x, sin.(x) .* cos.(x), msize=0.3, marker="fdiamond")
legend([d1, d2, d3], ["sine", "cosine", "scatter"])
xlim(0,π/2)


@testset "679.24-game.jl" begin
    @test judgePoint24([1, 2, 1, 2]) == false

    # julia> 8 / (2 - (5 / 3))
    # 24.000000000000007
    @test judgePoint24([8, 2, 5, 3]) == true
end


@testset "transition rate matrix checks" begin
    @test_throws ArgumentError TransitionRateMatrix(ones(3, 4))  # non-square
    @test_throws DomainError TransitionRateMatrix(-ones(3, 3))   # negative
end

@testset "transition matrix creation (dense)" begin
    for i in 1:100
        n = rand(5:10)
        A = abs.(randn(n, n))
        # copying constructor
        Q = TransitionRateMatrix(A)
        @test size(Q) ≡ (n, n)
        @test Base.IndexStyle(typeof(Q)) ≡ Base.IndexStyle(typeof(A))
        for i in axes(Q, 1)
            @test sum(Q[i, :]) ≈ 0 atol = n*10*eps()
        end

        # non-copying constructor
        Q2 = TransitionRateMatrix!(A)
        @test size(Q2) ≡ (n, n)
        @test Base.IndexStyle(typeof(Q2)) ≡ Base.IndexStyle(typeof(A))
        for i in axes(Q2, 1)
            @test sum(Q2[i, :]) ≈ 0 atol = n*10*eps()
        end

        π = stationary_distribution(Q)
        π2 = stationary_distribution(Q2)
        @test π == π2           # exact equivalence
        @test sum(π) ≈ 1
        @test all(π .≥ 0)
        @test Q'*π ≈ zero(π) atol = 10*eps()
    end
end

@testset "transition matrix creation (sparse)" begin
    for i in 1:100
        n = rand(5:10)
        A = abs.(sprandn(n, n, 1/n))
        # copying constructor
        Q = TransitionRateMatrix(A)
        @test size(Q) ≡ (n, n)
        @test Base.IndexStyle(typeof(Q)) ≡ Base.IndexStyle(typeof(A))
        for i in axes(Q, 1)
            @test sum(Q[i, :]) ≈ 0 atol = n*10*eps()
        end

        # non-copying constructor
        Q2 = TransitionRateMatrix!(A)
        @test size(Q2) ≡ (n, n)
        @test Base.IndexStyle(typeof(Q2)) ≡ Base.IndexStyle(typeof(A))
        for i in axes(Q2, 1)
            @test sum(Q2[i, :]) ≈ 0 atol = n*10*eps()
        end

        # FIXME tests commented out, cf
        # https://github.com/tpapp/ContinuousTimeMarkov.jl/issues/2

        # π = stationary_distribution(Q)
        # π2 = stationary_distribution(Q2)
        # @test π == π2           # exact equivalence
        # @test sum(π) ≈ 1
        # @test all(π .≥ 0)
        # @test Q'*π ≈ zero(π) atol = 10*eps()
    end
end


@testset "Test Linear Basis" begin

    bas1 = BasisMatrices.Basis(BasisMatrices.Lin(), [0.0,4.0], 5)

    @testset "test type" begin
        bas2 = BasisMatrices.Basis(BasisMatrices.Lin(), [0.0,2.0,4.0], 5)
        @test  bas1.params[1].breaks  ==  [0.0,1.0,2.0,3.0,4.0]
        @test  bas2.params[1].evennum  ==  3
        @test_throws ErrorException BasisMatrices.Basis(BasisMatrices.Lin(), [0.0], 5)
        @test_throws ErrorException BasisMatrices.Basis(BasisMatrices.Lin(), [0.0,1.5,4.0], 5)
    end

    @testset "test constructor" begin

       valbas_lin = @inferred BasisMatrices.evalbase(bas1.params[1])
       valbas_spl = @inferred BasisMatrices.evalbase(BasisMatrices.SplineParams([0.0,1.0,2.0,3.0,4.0],0,1))

       @test  valbas_lin  ==  valbas_spl

    end

    @testset "test derivative" begin

    	#test derivative of basis indirectly, by fitting exponential function
		bas2 = Basis(LinParams([0,1.0],1000000))
		coeffs = funfitf(bas2,exp)
		points = rand(10000)

		d1 = funeval(coeffs,bas2,points,1)

        @test_approx_eq_eps d1 exp(points) 1e-7

    end

end


import Base.convert

function convert(::Type{Array{ComplexF64,2}}, p::PauliString)

end

#
#	many-body pauli-matrices for operator decomposition
#
function get_Sz(N::Int, site::Int)
	# Important convention we start chain with site : 1 ..... and end with site N
	site = site - 1 # so site 1 corresponds to a bitshift of 0

	vec_nidx = [0:1:(2^N-1)...] # vector with all nidx labels 0,1,2,3 for 0..0, 10..0, 010..0, 110..0,...

	vec = vec_nidx.>>site  # shift bit by site to the left so if there was a 1 at site:
			 	# site in orginal it would be at leading bit now.

	vec = Vector{Int}(-1 .*(2 .*(vec.%2).-1)) # check if there is a 1 at bit at site: site it is now at the leading bit.
	 				    # we can check if vec entries are now odd or not .. if odd the bit at site :site was
					    # previously 1 otherwise it was 0 ---> for 1 assigne +1 for 0 assigne -1.

	#vec = Vector{Int}(1 .*(2 .*(vec.%2).-1)) # check if there is a 1 at bit at site: site it is now at the leading bit
	 				    # we can check if vec entries are now odd or not .. if odd the bit at site :site was
					    # previously 1 otherwise it was 0 ---> for 1 assigne +1 for 0 assigne -1

	return sparse(vec_nidx.+1, vec_nidx.+1, vec) # .+1 since julia starts array count with 1
end

function get_Sy(N::Int, site::Int)
	# Important convention we start chain with site : 1 ..... and end with site N
	site = site - 1 # so site 1 corresponds to a bitshift of 0

	vec_nidx = [0:1:(2^N-1)...] # vector with all nidx labels 0,1,2,3 for 0..0, 10..0, 010..0, 110..0,...

	# well Sy ≈ S+ - S-
	# so we can incorporate this with an xor operation:

	vec_applied = vec_nidx .⊻ Int(2^site)
	# if there is a bit 1 at site in vec_nidx it will remove that bit 1 but if there isnt it will creat one!

	# according to if their was a spin or not we have -1 or +1
	vec_data = vec_nidx.>>site # see comment at the beginning
	vec_data = 2 .*(vec_data.%2).-1
	vec_data = 1im.*vec_data
	#vec_data = -1im.*vec_data

	 return sparse(vec_nidx.+1, vec_applied.+1, vec_data)
end

function get_Sx(N::Int, site::Int)
	# Important convention we start chain with site : 1 ..... and end with site N
	site = site - 1 # so site 1 corresponds to a bitshift of 0

	vec_nidx = [0:1:(2^N-1)...] # vector with all nidx labels 0,1,2,3 for 0..0, 10..0, 010..0, 110..0,...

	# well Sx ≈ S+ + S-
	# so we can incorporate this with an xor operation:
	vec_applied = vec_nidx .⊻ Int(2^site)

	vec_data = ones(2^N)

	return sparse(vec_nidx.+1, vec_applied.+1, vec_data)
end

#
#	convert to PauliString
#


#=
this is a canonical implementation
the implemented version benchmarked slightly better under a variety of inputs (julia v1.0.1)
@inline function two_sum(a::T, b::T) where {T<:AbstractFloat}
    hi = a + b
    v  = hi - a
    lo = (a - (hi - v)) + (b - v)
    return hi, lo
end
=#

const FloatWithFMA = Union{Float64, Float32, Float16}

"""
    two_sum(a, b)
Computes `hi = fl(a+b)` and `lo = err(a+b)`.
"""
@inline function two_sum(a::T, b::T) where {T<:FloatWithFMA}
    hi = a + b
    a1 = hi - b
    b1 = hi - a1
    lo = (a - a1) + (b - b1)
    return hi, lo
end

"""
    three_sum(a, b, c)

Computes `hi = fl(a+b+c)` and `md = err(a+b+c), lo = err(md)`.
"""
function three_sum(a::T, b::T, c::T) where {T<:FloatWithFMA}
    s, t   = two_sum(b, c)
    hi, u  = two_sum(a, s)
    md, lo = two_sum(u, t)
    hi, md = two_hilo_sum(hi, md)
    return hi, md, lo
end

"""
    four_sum(a, b, c, d)

Computes `hi = fl(a+b+c+d)` and `hm = err(a+b+c+d), ml = err(hm), lo = err(ml)`.
"""
function four_sum(a::T,b::T,c::T,d::T) where {T<:FloatWithFMA}
    t0, t1 = two_sum(a ,  b)
    t0, t2 = two_sum(t0,  c)
    hi, t3 = two_sum(t0,  d)
    t0, t1 = two_sum(t1, t2)
    hm, t2 = two_sum(t0, t3) # here, t0 >= t3
    ml, lo = two_sum(t1, t2)
    return hi, hm, ml, lo
end

"""
    two_sumof3(a, b, c)

Computes `hi = fl(a+b+c)` and `lo = err(a+b+c)`.
"""
function two_sumof3(a::T, b::T, c::T) where {T<:FloatWithFMA}
    s, t = two_sum(b, c)
    x, u = two_sum(a, s)
    y    = u + t
    x, y = two_sum(x, y)
    return x, y
end

@inline two_sum(a::T, b::T, c::T) where {T<:FloatWithFMA} = two_sumof3(a, b, c)

"""
    two_sumof4(a, b, c, d)

Computes `hi = fl(a+b+c+d)` and `lo = err(a+b+c+d)`.
"""
function two_sumof4(a::T, b::T, c::T, d::T) where {T<:FloatWithFMA}
    t0, t1 = two_sum(a ,  b)
    t0, t2 = two_sum(t0,  c)
    a,  t3 = two_sum(t0,  d)
    t0  = t1 + t2
    b   = t0 + t3
    return a, b
end

@inline two_sum(a::T, b::T, c::T, d::T) where {T<:FloatWithFMA} = two_sumof4(a, b, c, d)

function three_sumof4(a::T,b::T,c::T,d::T) where {T<:FloatWithFMA}
    t0, t1 = two_sum(a ,  b)
    t0, t2 = two_sum(t0,  c)
    hi, t3 = two_sum(t0,  d)
    t0, t1 = two_sum(t1, t2)
    hm, t2 = two_sum(t0, t3) # here, t0 >= t3
    ml     = t1 + t2
    return hi, hm, ml
end

@inline three_sum(a::T, b::T, c::T, d::T) where {T<:FloatWithFMA} = thre_sumof4(a, b, c, d)

"""
    two_diff(a, b)

Computes `hi = fl(a-b)` and `lo = err(a-b)`.
"""
@inline function two_diff(a::T, b::T) where {T<:FloatWithFMA}
    hi = a - b
    a1 = hi + b
    b1 = hi - a1
    lo = (a - a1) - (b + b1)
    return hi, lo
end

"""
    two_diffof3(a, b, c)

Computes `hi = fl(a-b-c)` and `lo = err(a-b-c)`.
"""
function two_diffof3(a::T, b::T, c::T) where {T<:FloatWithFMA}
    s, t = two_diff(-b, c)
    x, u = two_sum(a, s)
       y = u + t
    x, y = two_sum(x, y)
    return x, y
end

@inline two_diff(a::T, b::T, c::T) where {T<:FloatWithFMA} = two_diffof3(a, b, c)

"""
    two_hilo_sum(a, b)

*unchecked* requirement `|a| ≥ |b|`
Computes `s = fl(a+b)` and `e = err(a+b)`.
"""
@inline function two_hilo_sum(a::T, b::T) where {T<:FloatWithFMA}
    s = a + b
    e = b - (s - a)
    return s, e
end

"""
    three_hilo_sum(a, b, c)

*unchecked* requirement `|a| ≥ |b| ≥ |c|`
Computes `s = fl(a+b+c)` and `e1 = err(a+b+c), e2 = err(e1)`.
"""
function three_hilo_sum(a::T,b::T,c::T) where {T<:FloatWithFMA}
    s, t = two_hilo_sum(b, c)
    x, u = two_hilo_sum(a, s)
    y, z = two_hilo_sum(u, t)
    x, y = two_hilo_sum(x, y)
    return x, y, z
end

"""
    two_hilo_sumof3(a, b, c)

*unchecked* requirement `|a| ≥ |b| ≥ |c|`
Computes `s = fl(a+b+c)` and `e1 = err(a+b+c)`.
"""
function two_hilo_sumof3(a::T,b::T,c::T) where {T<:FloatWithFMA}
    s, t = two_hilo_sum(b, c)
    x, u = two_hilo_sum(a, s)
    y    = u + t
    x, y = two_hilo_sum(x, y)
    return x, y
end

"""
    four_hilo_sum(a, b, c, d)

*unchecked* requirement `|a| ≥ |b| ≥ |c| ≥ |d|`
Computes `s = fl(a+b+c+d)` and `e1 = err(a+b+c+d), e2 = err(e1), e3 = err(e2)`.
"""
function four_hilo_sum(a::T,b::T,c::T,d::T) where {T<:FloatWithFMA}
    t0, t1 = two_hilo_sum(a ,  b)
    t0, t2 = two_hilo_sum(t0,  c)
    hi, t3 = two_hilo_sum(t0,  d)
    t0, t1 = two_hilo_sum(t1, t2)
    hm, t2 = two_hilo_sum(t0, t3) # here, t0 >= t3
    ml, lo = two_hilo_sum(t1, t2)
    return hi, hm, ml, lo
end

"""
    three_hilo_sumof4(a, b, c, d)

*unchecked* requirement `|a| ≥ |b| ≥ |c| ≥ |d|`
Computes `s = fl(a+b+c+d)` and `e1 = err(a+b+c+d), e2 = err(e1)`.
"""
function three_hilo_sumof4(a::T,b::T,c::T,d::T) where {T<:FloatWithFMA}
    t0, t1 = two_hilo_sum(a ,  b)
    t0, t2 = two_hilo_sum(t0,  c)
    hi, t3 = two_hilo_sum(t0,  d)
    t0, t1 = two_hilo_sum(t1, t2)
    md, t2 = two_hilo_sum(t0, t3) # here, t0 >= t3
    lo     = t1 + t2
    return hi, md, lo
end

"""
    two_hilo_sumof4(a, b, c, d)

*unchecked* requirement `|a| ≥ |b| ≥ |c| ≥ |d|`
Computes `s = fl(a+b+c+d)` and `e1 = err(a+b+c+d)`.
"""
function two_hilo_sumof4(a::T,b::T,c::T,d::T) where {T<:FloatWithFMA}
    t0, t1 = two_hilo_sum(a ,  b)
    t0, t2 = two_hilo_sum(t0,  c)
    hi, t3 = two_hilo_sum(t0,  d)
    t0     = t1 + t2
    lo     = t0 + t3
    return hi, lo
end

@inline function two_prod(a::T, b::T) where {T<:FloatWithFMA}
    s = a * b
    t = fma(a, b, -s)
    return s, t
end


# # [Multirate Infinitesimal Step (MIS) Timestepping](@id MIS-Timestepping)

# In this tutorial, we shall explore the use of Multirate Infinitesimal Step
# (MIS) methods for the solution of nonautonomous (or non time-invariant) equations.
# For our model problem, we shall reuse the rising thermal bubble
# tutorial. See its [tutorial page](@ref Rising-Thermal-Bubble-Configuration)
# for details on the model and parameters. For the purposes of this tutorial,
# we will only run the experiment for a total of 500 simulation seconds.

using ClimateMachine
const clima_dir = dirname(dirname(pathof(ClimateMachine)));
include(joinpath(
    clima_dir,
    "tutorials",
    "Numerics",
    "TimeStepping",
    "tutorial_risingbubble_config.jl",
))

FT = Float64;

# Referencing the formulation introduced in the previous
# [Multirate RK methods tutorial](@ref Multirate-RK-Timestepping), we can
# describe Multirate Infinitesimal Step (MIS) methods by

# ```math
# \begin{align}
# v_i (0)
#   &= q^n + \sum_{j=1}^{i-1} \alpha_{ij} (Q^{(j)} - q^n) \\
# \frac{dv_i}{d\tau}
#   &= \sum_{j=1}^{i-1} \frac{\gamma_{ij}}{d_i \Delta t} (Q^{(j)} - q^n)
#     + \sum_{j=1}^i \frac{\beta_{ij}}{d_i} \mathcal{T}_S (Q^{(j)}, t + \Delta t c_i)
#     + \mathcal{T}_F(v_i, t^n +  \Delta t \tilde c_i + \frac{c_i - \tilde c_i}{d_i} \tau),
# \quad \tau \in [0, \Delta t d_i] \\
# Q^{(i)} &= v_i(\Delta t d_i),
# \end{align}
# ```
#
# where we have used the the stage values ``Q^{(i)} = v_i(\tau_i)`` as the
# solution to the _inner_ ODE problem, ``{\mathcal{T}_{s}}``
# for the slow component, and ``{\mathcal{T}_{f}}` for the fast
# one, as in the [Multirate RK methods tutorial](@ref Multirate-RK-Timestepping).

# Referencing the canonical form introduced in [Time integration](@ref
# Time-integration), both ``{\mathcal{T}_{f}}`` and ``{\mathcal{T}_{s}}``
# could be discretized either explicitly or implicitly, hence, they could
# belong to either ``\mathcal{F}(t, \boldsymbol{q})`` or ``\mathcal{G}(t, \boldsymbol{q})``
# term.
#
# The method is defined in terms of the lower-triangular matrices ``\alpha``,
# ``\beta`` and ``\gamma``, with ``d_i = \sum_j \beta_{ij}``,
# ``c_i = (I - \alpha - \gamma)^{-1} d`` and ``\tilde c = \alpha c``.
# More details can be found in [WenschKnothGalant2009](@cite) and
# [KnothWensch2014](@cite).

ode_solver = ClimateMachine.MISSolverType(;
    mis_method = MIS2,
    fast_method = LSRK144NiegemannDiehlBusch,
    nsubsteps = (40,),
)

timeend = FT(500)
CFL = FT(20)
run_simulation(ode_solver, CFL, timeend);

# The reader can compare the Courant number (denoted by `CFL` in the code snippet)
# used in this example, with the adopted in the
# [single-rate explicit timestepping tutorial page](@ref Single-rate-Explicit-Timestepping)
# in which we use the same scheme as the fast method employed in this case,
# and notice that with this MIS method we are able to take a much larger
# Courant number.

# ## References
# - [WenschKnothGalant2009](@cite)
# - [KnothWensch2014](@cite)


# This file is a part of RunStatistics.jl, licensed under the MIT License (MIT).

export squares_pvalue, squares_cdf

const log_factorial = Vector{Float64}(undef, 0)

function cachechi2(T_obs::Real, N::Int)

    @argcheck 0 < N

    res = zeros(N + 1)
    res[1] = NaN

    for i = firstindex(res) + 1 :lastindex(res)

        res[i] = logcdf(Chisq(i - 1), T_obs)
    end

    return res
end

"""
    squares_cdf(T_obs::Real, N::Integer)

Compute P(T < T_obs | N), the value of the cumulative distribution of the Squares statistic `T` at the
value `T_obs` observed in `N` independent trials with gaussian probability.

`T_obs` is the value of the test statistic for the observed data set; i.e., the largest `χ^2` of
any run of consecutive observed values above the expectation in a sequence of `N` independent trials
with Gaussian uncertainty.

`N` is the total number of data points.

The calculation implements equations (16) and (17) from

Frederik Beaujean and Allen Caldwell. *A Test Statistic for
Weighted Runs.* Journal of Statistical Planning and Inference 141,
no. 11 (November 2011): 3437–46. doi:10.1016/j.jspi.2011.04.022

https://arxiv.org/abs/1005.3233.

"""
function squares_cdf(T_obs::Real, N::Integer)

    T = promote_type(typeof(T_obs), typeof(N))

    if N >= length(log_factorial)

        sizehint!(log_factorial, N)

        if isempty(log_factorial)
            push!(log_factorial, 0.0)
        end

        for i = length(log_factorial):N
            push!(log_factorial, last(log_factorial) + log(i))
        end

    end

    log_cumulative = cachechi2(T_obs, N)

    logpow2N1 = (N <= (8 * sizeof(Int) - 1)) ? log((1 << N) - 1) : N * log(2)

    p = zero(T)

    for r = 1:N

        Mmax = min(r, N - r + 1)
        poch = 0.0

        for M = 1:Mmax

            poch += log(N - r + 2 - M)
            scale = poch - logpow2N1
            ppi = 0.0

            g = init_partition(r, M)
            n = g.c
            y = g.y

            done = false

            while !done

                h = g.h
                ppartition = 0.0

                for l = 2:(h + 1)
                    ppartition += n[l] * log_cumulative[y[l]+1] - log_factorial[n[l]+1]
                end

                ppi += exp(ppartition)

                done = next_partition!(g)
            end

            p += exp(scale + log(ppi))
        end
    end

    @assert p < 1
    return p
end

"""
    squares_pvalue(T_obs::Real, N::Integer)

Compute P(T >= `T_obs` | `N`), the p value for the Squares test statistic `T` being larger or equal to `T_obs`,
the value of the Squares statistic observed in `N` datapoints.

The Squares statistic `T` denotes the largest `χ^2` of any run of consecutive successes (above expectation) in a
sequence of `N` independent trials with Gaussian uncertainty.

Via `squares_cdf()` this function implements equations (16) and (17) from

Frederik Beaujean and Allen Caldwell. *A Test Statistic for
Weighted Runs.* Journal of Statistical Planning and Inference 141,
no. 11 (November 2011): 3437–46. doi:10.1016/j.jspi.2011.04.022

https://arxiv.org/abs/1005.3233.

"""
function squares_pvalue(T_obs::Real, N::Integer)

    return 1 - squares_cdf(T_obs, N)

end


"""
A type representing a conjunctive normal form.
"""
struct CNF
    N::Int
    M::Int
    clauses::Vector{Vector{Int}}
end

function CNF(clauses::Vector{Vector{Int}})
    M = length(clauses)
    N = maximum(maximum(abs.(c)) for c in clauses)
    return CNF(N, M, clauses)
end

"""
    randomcnf(; N=100, k=3, α=0.1, seed=-1, planted = Vector{Vector{Int}}())

Generates a random instance of the k-SAT problem, with `N` variables
and `αN` clauses.

Any configuration in `planted` is guaranteed to be a solution of the problem.
"""
function randomcnf(; N::Int = 100, k::Int = 3, α::Float64 = 0.1, seed::Int=-1,
                    planted = Vector{Vector{Int}}())
    seed > 0 && Random.seed!(seed)
    M = round(Int, N*α)
    clauses = Vector{Vector{Int}}()
    for p in planted
        @assert length(p) == N   "Wrong size for planted configurations ($N != $(lenght(p)) )"
    end
    for a=1:M
        while true
            c = rand(1:N, k)
            length(union(c)) != k && continue
            c = c .* rand([-1,1], k)

            # reject if not satisfies the planted solutions
            sat = Bool[any(i -> i>0, sol[abs.(c)] .* c) for sol in planted]
            !all(sat) && continue

            push!(clauses, c)
            break
        end
    end
    return CNF(N, M, clauses)
end

"""
    readcnf(fname::String)

Reads a CNF from file `fname`.
"""
function readcnf(fname::String)
    f = open(fname, "r")
    head = split(readline(f))
    N, M = parse(Int, head[3]), parse(Int, head[4])
    clauses = Vector{Vector{Int}}()
    for i=1:M
        line = readline(f)
        c = [parse(Int64, e) for e in split(line)[1:end-1]]
        push!(clauses, c)
    end
    return CNF(N, M, clauses)
end

"""
    writecnf(fname::String, cnf::CNF)

Writes `cnf` to file `fname`.
"""
function writecnf(fname::String, cnf::CNF)
    f = open(fname, "w")
    println(f, "p cnf $(cnf.N) $(cnf.M)")
    for c in cnf.clauses
        for i in c
            print(f, i, " ")
        end
        print(f, "0\n")
    end
    close(f)
end

"""
    energy(cnf, σ)

Counts the number of violated clauses.
"""
function energy(cnf::CNF, σ::Vector{Int})
    E = 0
    for c in cnf.clauses
        issatisfied = false
        for i in c
            if sign(i) == σ[abs(i)]
                issatisfied = true
                break
            end
        end
        E += issatisfied ? 0 : 1
    end
    E
end

"""
    adjlist(cnf::CNF)

Returns a vector containing for each variable a vector of the adjacent clause indexes.
"""
function adjlist(cnf::CNF)
    adj = Vector{Vector{Int}}(cnf.N)
    for (μ, c) in enumerate(cnf.clauses)
        for i in c
            push!(adj[abs(i)], μ)
        end
    end
end


# ------------------------------------------------------------------
# Licensed under the MIT License. See LICENSE in the project root.
# ------------------------------------------------------------------

"""
    Rectangle(start, finish)

A N-dimensional rectangle with lower left corner at `start`
and upper right corner at `finish`.
"""
struct Rectangle{T,N} <: AbstractGeometry{T,N}
  start::SVector{N,T}
  finish::SVector{N,T}
end

Rectangle(start::NTuple{N,T}, finish::NTuple{N,T}) where {N,T} =
  Rectangle{T,N}(start, finish)

Rectangle(start::Vec{N,T}, finish::Vec{N,T}) where {N,T} =
  Rectangle{T,N}(start, finish)

in(x, r::Rectangle) = all(r.start .≤ x .≤ r.finish)

"""
    extrema(rectangle)

Return the corners of the `rectangle`.
"""
Base.extrema(r::Rectangle) = r.start, r.finish

"""
    sides(rectangle)

Return all the sides of the `rectangle` as a tuple.
"""
sides(r::Rectangle) = r.finish - r.start

"""
    center(rectangle)

Return the center of the `rectangle`.
"""
center(r::Rectangle) = (r.start + r.finish) / 2

"""
    diagonal(rectangle)

Return the diagonal of the `rectangle`.
"""
diagonal(r::Rectangle) = norm(r.finish - r.start)

"""
    volume(rectangle)

Return the volume of the `rectangle`.
"""
volume(r::Rectangle) = prod(r.finish - r.start)


using GenericLinearAlgebra

@testset "Test sign of eigenvalues" begin
    n = 20
    T = SymTridiagonal(randn(n), randn(n-1))
    @test numnegevals(T) == count(x->x<0, eigvals(T))
end


using LinearAlgebra
@testset "Linear fit" begin
    function graph_test_graph()
        (graph, cref) = graph_monomial(zeros(2))
        add_ldiv!(graph, :invA, :A, :I)
        add_lincomb!(graph, :Gout, 1.0, :P2, 0.0, :invA)
        graph.outputs[1] = :Gout
        push!(cref, (:Gout, 2))
        return (graph, cref)
    end

    # Polynomial
    coeff = [3; -1; 2.0; 0.1; 1.2]
    p =
        z ->
            coeff[1] * one(z) +
            coeff[2] * z +
            coeff[3] * z^2 +
            coeff[4] * z^3 +
            coeff[5] * z^4
    discr = collect(1:5.0)

    (graph, cref) = graph_monomial(zeros(length(coeff)))
    opt_linear_fit!(
        graph,
        p,
        discr,
        cref,
        errtype = :relerr,
        linlsqr = :svd,
        droptol = 1e-15,
    )
    @test all(
        abs.(eval_graph(graph, discr) .- p.(discr)) .< 1e-14 * abs.(p.(discr)),
    )
    A = [3 4; 5 6.6]
    PA = p(A)
    @test norm(eval_graph(graph, A) - PA) < 1e-14 * norm(PA)

    new_input = :B
    (graph, cref) = graph_monomial(zeros(length(coeff)), input = new_input)
    opt_linear_fit!(
        graph,
        p,
        discr,
        cref,
        input = new_input,
        errtype = :relerr,
        linlsqr = :svd,
        droptol = 1e-15,
    )
    @test all(
        abs.(eval_graph(graph, discr, input = new_input) .- p.(discr)) .<
        1e-14 * abs.(p.(discr)),
    )
    A = [3 4; 5 6.6]
    PA = p(A)
    @test norm(eval_graph(graph, A, input = new_input) - PA) < 1e-14 * norm(PA)

    (graph, cref) = graph_monomial(zeros(length(coeff)))
    opt_linear_fit!(
        graph,
        p,
        discr,
        cref,
        errtype = :relerr,
        linlsqr = :svd,
        droptol = 1e-15,
    )
    @test all(
        abs.(eval_graph(graph, discr) .- p.(discr)) .< 1e-14 * abs.(p.(discr)),
    )
    A = [3 4; 5 6.6]
    PA = p(A)
    @test norm(eval_graph(graph, A) - PA) < 1e-14 * norm(PA)

    new_input = :Anew
    rename_node!(graph, :A, new_input, cref)
    opt_linear_fit!(
        graph,
        p,
        discr,
        cref,
        input = new_input,
        errtype = :relerr,
        linlsqr = :svd,
        droptol = 1e-15,
    )
    @test all(
        abs.(eval_graph(graph, discr, input = new_input) .- p.(discr)) .<
        1e-14 * abs.(p.(discr)),
    )
    A = [3 4; 5 6.6]
    PA = p(A)
    @test norm(eval_graph(graph, A, input = new_input) - PA) < 1e-14 * norm(PA)

    (graph, cref) = graph_monomial(zeros(length(coeff)))
    opt_linear_fit!(
        graph,
        p,
        complex(discr),
        cref,
        errtype = :relerr,
        linlsqr = :real_svd,
        droptol = 1e-15,
    )
    @test all(
        abs.(eval_graph(graph, discr) .- p.(discr)) .< 1e-14 * abs.(p.(discr)),
    )
    A = [3 4; 5 6.6]
    PA = p(A)
    @test norm(eval_graph(graph, A) - PA) < 1e-14 * norm(PA)
    @test isequal(eltype(graph), real(eltype(graph)))

    rename_node!(graph, :A, new_input)
    opt_linear_fit!(
        graph,
        p,
        complex(discr),
        cref,
        errtype = :relerr,
        input = new_input,
        linlsqr = :real_svd,
        droptol = 1e-15,
    )
    @test all(
        abs.(eval_graph(graph, discr, input = new_input) .- p.(discr)) .<
        1e-14 * abs.(p.(discr)),
    )
    A = [3 4; 5 6.6]
    PA = p(A)
    @test norm(eval_graph(graph, A, input = new_input) - PA) < 1e-14 * norm(PA)
    @test isequal(eltype(graph), real(eltype(graph)))

    # Other functions
    coeff = [3; -1; 2.0]
    p = z -> coeff[1] * one(z) + coeff[2] * z + coeff[3] * z^(-1)
    discr = [1; -1.1; 1.4]

    (graph, cref) = graph_test_graph()
    opt_linear_fit!(graph, p, discr, cref)
    @test all(abs.(eval_graph(graph, discr) .- p.(discr)) .< 1e-15)

    (graph, cref) = graph_test_graph()
    discr = complex.(discr)
    opt_linear_fit!(graph, p, discr, cref, linlsqr = :real_backslash)
    @test all(abs.(eval_graph(graph, discr) .- p.(discr)) .< 1e-15)
    @test isequal(eltype(graph), real(eltype(graph)))

    # Overdetermined
    (graph, cref) = graph_test_graph()
    discr = collect(1:10.0)
    opt_linear_fit!(graph, p, discr, cref, errtype = :relerr, linlsqr = :nrmeq)
    PA = p(A)
    @test norm(eval_graph(graph, A) - PA) < 1e-13 * norm(PA)

    (graph, cref) = graph_test_graph()
    coeff = complex.(coeff)
    p = z -> coeff[1] * one(z) + coeff[2] * z + coeff[3] * z^(-1)
    opt_linear_fit!(
        graph,
        p,
        complex.(discr),
        cref,
        errtype = :relerr,
        linlsqr = :real_nrmeq,
    )
    @test norm(eval_graph(graph, A) - PA) < 1e-13 * norm(PA)
    @test isequal(eltype(graph), real(eltype(graph)))
end


export coordarray,
    rmsd,
    displacements,
    distance

"""
Get the atomic coordinates of a `StructuralElementOrList` as a 2D `Array` with
each column corresponding to one atom.
"""
function coordarray(element::StructuralElementOrList, selector_functions::Function...)
    atoms = collectatoms(element, selector_functions...)
    coords = zeros(3, length(atoms))
    for j in eachindex(atoms)
        coords[1,j] = x(atoms[j])
        coords[2,j] = y(atoms[j])
        coords[3,j] = z(atoms[j])
    end
    return coords
end

coordarray(coords::Array{Float64}, selector_functions::Function...) = coords

"""
Get the root-mean-square deviation (RMSD) between two `StructuralElementOrList`s
or two coordinate `Array`s of the same size. Assumes they are already
superimposed.
"""
function rmsd(coords_one::Array{Float64}, coords_two::Array{Float64})
    @assert size(coords_one) == size(coords_two) "Sizes of coordinate arrays are different - cannot calculate RMSD"
    difference = coords_one - coords_two
    return sqrt(sum(difference .* difference) / size(coords_one, 2))
end

rmsd(element_one::StructuralElementOrList, element_two::StructuralElementOrList, selector_functions::Function...) = rmsd(coordarray(element_one, selector_functions...), coordarray(element_two, selector_functions...))

"""
Get the displacements between atomic coordinates from two
`StructuralElementOrList`s or two coordinate `Array`s of the same size. Assumes
they are already superimposed.
"""
function displacements(coords_one::Array{Float64}, coords_two::Array{Float64})
    @assert size(coords_one) == size(coords_two) "Sizes of coordinate arrays are different - cannot calculate displacements"
    difference = coords_one - coords_two
    return sqrt(sum(difference .* difference, 1))[:]
end

displacements(element_one::StructuralElementOrList, element_two::StructuralElementOrList, selector_functions::Function...) = displacements(coordarray(element_one, selector_functions...), coordarray(element_two, selector_functions...))

"Get the minimum distance between two `StructuralElementOrList`s."
function distance(element_one::StructuralElementOrList, element_two::StructuralElementOrList, selector_functions::Function...)
    coords_one = coordarray(element_one, selector_functions...)
    coords_two = coordarray(element_two, selector_functions...)
    count_one = size(coords_one, 2)
    count_two = size(coords_two, 2)
    sq_dists = zeros(count_one, count_two)
    for i in 1:count_one, j in 1:count_two
        @inbounds sq_dists[i,j] = (coords_one[1,i] - coords_two[1,j]) ^ 2 + (coords_one[2,i] - coords_two[2,j]) ^ 2 + (coords_one[3,i] - coords_two[3,j]) ^ 2
    end
    return sqrt(minimum(sq_dists))
end

distance(atom_one::AbstractAtom, atom_two::AbstractAtom) = sqrt((x(atom_one) - x(atom_two)) ^ 2 + (y(atom_one) - y(atom_two)) ^ 2 + (z(atom_one) - z(atom_two)) ^ 2)


using Quadhps
using ForwardDiff, QuadGK, Test

@testset "Quadhps tests" begin

  @testset "Basic" begin
    a = Quadhps.quadhp(x->1.0, 0.0, 2.0, rtol=2*eps())
    @test a[1] ≈ 2.0 atol=10 * eps()
    @test a[2] ≈ 0.0 atol=10 * eps()
    b = Quadhps.quadhp(x->x, 1.0, 2.0, rtol=2*eps())
    @test b[1] ≈ 1.5 atol=10 * eps()
    @test b[2] ≈ 0.0 atol=10 * eps()
    c = Quadhps.quadhp(x->3 .* x.^2 .+ 1.0, 2.0, 5.0, rtol=120*eps())
    @test c[1] ≈ 120.0 atol=1000 * eps()
    @test c[2] ≈ 0.0 atol=1000 * eps()
  end

  @testset "Versus QuadGK" begin
    f(x) = 1 ./ (1 .- x.^2)
    lower = -0.999
    upper = 0.999
    a = Quadhps.quadhp(x->f(x), lower, upper, rtol=2*eps())[1]
    b = QuadGK.quadgk(x->f(x), lower, upper, rtol=2*eps())[1]
    @test a ≈ b atol=1000 * eps()
  end

  @testset "Types" begin
    f(x) = x
    lower = BigFloat(0.1)
    upper = 10.0
    a = Quadhps.quadhp(x->f(x), lower, upper)[1]
    b = QuadGK.quadgk(x->f(x), lower, upper)[1]
    @test a ≈ b rtol=eps()
  end

  @testset "Vectorised" begin
    a = Quadhps.quadhp(x->[1, 2], 0.0, 1.0)[1]
    @test a[1] ≈ 1 atol=2 * eps()
    @test a[2] ≈ 2 atol=2 * eps()
  end

  @testset "Intervals" begin
    a = Quadhps.quadhp(x->3 .* x.^2 .+ 1.0, 0.0, 5.0, rtol=120*eps())
    b = Quadhps.quadhp(x->3 .* x.^2 .+ 1.0, 0.0, 2.0, 5.0, rtol=120*eps())
    c = Quadhps.quadhp(x->3 .* x.^2 .+ 1.0, 0.0, 2.0, 4.0, 5.0, rtol=120*eps())
    @test a[1] ≈ b[1]
    @test a[1] ≈ c[1]
  end

  @testset "ForwadDiff" begin
    integrand = sin
    differentiand(x) = Quadhps.quadhp(y->integrand(y[1]), 0.0, x[1])[1]
    b = π/2
    @test integrand(b) ≈ ForwardDiff.gradient(differentiand, [b])[1]
    a = ForwardDiff.gradient(x->
      Quadhps.quadhp(y->sin(y[1]*x[1]), 0.0, 1.0)[1],
          [0.5])[1]
    @test a ≈ 0.469181324769897
  end
end


# utility functions

tofloat(array::AbstractArray{T}) where T = map(Float32, array)
tofloat(array::AbstractArray{Float32}) = array

fft_frequencies(samplerate::Real, nfft::Int) = collect(linspace(0f0, samplerate / 2f0, (nfft >> 1) + 1))

"""returns the number of frames when the signal is partitioned into overlapping frames"""
nframes(length::Int, framesize::Int, hopsize::Int) = div(length - framesize, hopsize) + 1

"""Provides a view of a signal partitioned into overlapping frames, similar to MATLAB's buffer"""
struct FrameView{T<:AbstractVector} <: AbstractVector{Vector}
    original::T
    framesize::Int
    hopsize::Int
    nframes::Int

    function FrameView(original, framesize, hopsize)
        # n = noverlap is a problem - the algorithm will not terminate.
        #@boundscheck((0 < hopsize <= framesize), error("hopsize must be between zero and framesize"))
        new{T}(original, framesize, hopsize, nframes(length(original), framesize, hopsize))
    end
end

function Base.getindex(x::FrameView, i::Int)
    #@boundscheck((i >= 1 && i <= x.k), Base.throw_boundserror(x, i))
    slice(x.original, (i-1) * x.hopsize + (1:x.framesize))
end

Base.size(x::FrameView) = (x.nframes,)

function Base.iterate(x::FrameView, i::Int)
   if i > x.nframes
       nothing
   else
       (x[i], i+1)
   end
end

"""returns the DCT filters"""
function dct(nfilters::Int, ninput::Int)
    basis = Array(Float32, nfilters, ninput)
    samples = (1f0:2f0:2ninput) * π / 2ninput
    for i = 1:nfilters
        basis[i, :] = cos(i * samples)
    end

    basis *= sqrt(2f0/ninput)
    basis
end

"""returns the number of times that the value changes its sign"""
function zero_crossings(array::AbstractVector{T}, size::Int = length(array), offset::Int = 0) where {T<:Real}
    result = 0
    previous = 0
    for i in offset + (1:size)
        number = array[i]
        sgn = number == 0 ? 0 : number > 0 ? 1 : -1
        if sgn != previous
            result += 1
        end
        previous = sgn
    end
    result
end


export degrees, degree, mindegree_vertex, maxdegree_vertex, minmaxdegree_vertex

"""Get degrees of all vertices in a graph."""
degrees(eg::EliminateGraph) = degree.(Ref(eg), vertices(eg))

"""
    neighbors(eg::EliminateGraph, i::Int)

Get degree of vertex `i`.
"""
@inline function degree(eg::EliminateGraph, vi::Int)
    sum(vj->isconnected(eg,vi,vj), vertices(eg))
end

"""find `(vertex, degree)` with minimum degree of a graph."""
function mindegree_vertex(eg::EliminateGraph)
    N = nv(eg)
    dmin = 999999
    vmin = 0
    for vj in vertices(eg)
        dj = 0
        for vi in vertices(eg)
            dj += isconnected(eg,vi,vj)
        end
        if dj < dmin
            dmin = dj
            vmin = vj
        end
    end
    return vmin, dmin
end

"""find `(min_degree_vertex, max_degree_vertex, min_degree, max_degree)` of a graph."""
function minmaxdegree_vertex(eg::EliminateGraph)
    N = nv(eg)
    dmin = 999999
    dmax = 0
    vmin = 0
    vmax = 0
    for vj in vertices(eg)
        dj = 0
        for vi in vertices(eg)
            dj += isconnected(eg,vi,vj)
        end
        if dj < dmin
            dmin = dj
            vmin = vj
        end
        if dj > dmax
            dmax = dj
            vmax = vj
        end
    end
    return vmin, vmax, dmin, dmax
end

"""find `(vertex, degree)` with maximum degree of a graph."""
function maxdegree_vertex(eg::EliminateGraph)
    N = nv(eg)
    dmax = 0
    vmax = 0
    for vj in vertices(eg)
        dj = 0
        for vi in vertices(eg)
            dj += isconnected(eg,vi,vj)
        end
        if dj > dmax
            dmax = dj
            vmax = vj
        end
    end
    return vmax, dmax
end


using LinearAlgebra

a = [1.0000000           sqrt(2.0)           2.0000000
    sqrt(2.0)           3.0000000           sqrt(2.0)
    2.0000000           sqrt(2.0)           1.0000000]

display(eigen!(a))


cd(@__DIR__); include("setups/path128.jl");
gr(dpi = 200)

## (a)
Uf = unitary_folding_operator(W, GP; ϵ = 0.38, J = 2)' # ϵa ≈ 0.38⋅∥𝛟₁∥_{∞}
plt = plot(size = (500, 400))
    heatmap!(Uf, c = :viridis, ratio = 1, yflip = true, xlim = [1, N], clims = (-1, 1))
    xticks!([1; 16:16:128], vcat(string(1), [string(k) for k in 16:16:128]))
    yticks!([1; 16:16:128], vcat(string(1), [string(k) for k in 16:16:128]))
savefig(plt, "../figs/Path128_Uf_j2.png")

## (b)
Uf = unitary_folding_operator(W, GP; ϵ = 0.38, J = 3)'
plt = plot(size = (500, 400))
    heatmap!(Uf, c = :viridis, ratio = 1, yflip = true, xlim = [1, N], clims = (-1, 1))
    xticks!([1; 16:16:128], vcat(string(1), [string(k) for k in 16:16:128]))
    yticks!([1; 16:16:128], vcat(string(1), [string(k) for k in 16:16:128]))
savefig(plt, "../figs/Path128_Uf_j3.png")