text:
### A Pluto.jl notebook ###
# v0.17.4

using Markdown
using InteractiveUtils

# ╔═╡ 201f572b-9e68-4d80-b823-be41243382fd
md"""
## Interfacing with the [Polygon.io](https://polygon.io) Financial Data Application Programming Interface
"""

# ╔═╡ 48cce16c-31bc-438c-bef9-eee2c13fb220
md"""
### Introduction

Data is the key to making great financial decisions. [Polygon.io](https://polygon.io) is a financial data warehouse that provides an application programming interface (API) that can be used to download and analyze all kinds of financial information.

[Polygon.io](https://polygon.io) was founded by former Googlers in 2016. Since September of 2020, [Polygon.io](https://polygon.io) has provided both [free and paid access](https://polygon.io/pricing) to current and historical [stock](https://polygon.io/docs/stocks/getting-started), [option](https://polygon.io/docs/options/getting-started), [crypto](https://polygon.io/docs/crypto/getting-started) and [forex](https://polygon.io/docs/forex/getting-started) data.
Check out the [Polygon.io blog](https://polygon.io/blog/) for the latest updates and developments from [Polygon.io](https://polygon.io).

In this notebook, we'll discuss how to interface the [Serenity library](https://github.com/Pooksoft/Serenity) with the [Polygon.io REST API](https://polygon.io/docs/stocks). In particular, we'll:

* Download and analyze daily stock price data for an arbitrary ticker symbol and a specified date range
* Download and analyze crypto data for an arbitrary ticker and a specified date range
* Download and analyze option data for an arbitrary ticker and a specified date range

"""

# ╔═╡ d3884c8d-bd00-4059-a751-f29b211e9ef6
md"""
### Materials and Methods
"""

# ╔═╡ 4b4668f0-6894-4219-9de8-f2106fb70aca
md"""
##### What is my [Julia](https://julialang.org/downloads/) setup?
In this code block, we set up project paths and import external packages that we use to download and analyze the various types of financial data. In addition, we load a local copy of the [Serenity library](https://github.com/Pooksoft/Serenity).
"""

# ╔═╡ 6c3e5618-c689-4941-b540-e27734ee096b
md"""
### Results and Discussion
"""

# ╔═╡ 572cb255-04a5-402b-a10f-85f6a1a23f39
md"""
##### Stock price data, ticker news and other information
"""

# ╔═╡ 6589f632-2e09-4bc6-aa8e-30f6338d5729
md"""
##### Cryptocurrency price data and other crypto information
"""

# ╔═╡ feab4379-ce36-4a34-abf6-70a6a15dcfd2
md"""
##### Options data
"""

# ╔═╡ 05d72483-d219-4c4f-823c-c9ceb82caa3d

# ╔═╡ 8623a119-bf7f-4566-be78-9769c3a64042
md"""
### Conclusions
"""

# ╔═╡ 77473f9a-e5fd-41e6-89d4-acffcba38f45
md"""
### References
"""

# ╔═╡ bd0c9e41-fed0-46f9-bf2e-136a439692a2
begin

	# # what date range do we want to look at? (max: 2 years on free account)
	# Y = 2021 	# year
	# M = 12 		# month
	# D = 24 		# day

	# # uncomment if we want to refresh the data -
	# # current_date = now()
	# # Y = year(current_date)
	# # M = month(current_date)
	# # D = day(current_date)

	# # download data from the from -> to dates -
	# from = Date(Y - 2, M, D)
	# to = Date(Y,M,D)

	# # where are going to store the data?
	# dir_name = "daily-$(Y)-$(M)-$(D)"
	# savepath = joinpath(_PATH_TO_DATA,"polygon", "stock", dir_name)

	# # execute the call(s) to the data api
	# # NOTE: if we already have data in the from -> to range, we load that from disk
	# pd = download_ticker_data(psia_ticker_symbol_array, from, to; savepath);

	# show -
	nothing
end

# ╔═╡ 7ce9e2ec-a3a2-4b64-be6e-27618bd9f3f1
function ingredients(path::String)

	# this is from the Julia source code (evalfile in base/loading.jl)
	# but with the modification that it returns the module instead of the last object
	name = Symbol("Serenity")
	m = Module(name)
	Core.eval(m,
        Expr(:toplevel,
             :(eval(x) = $(Expr(:core, :eval))($name, x)),
             :(include(x) = $(Expr(:top, :include))($name, x)),
             :(include(mapexpr::Function, x) = $(Expr(:top, :include))(mapexpr, $name, x)),
             :(include($path))))
	m
end

# ╔═╡ 1461bfa4-633e-11ec-0ee6-a590181dc91b
begin

	# setup: we need to specify the paths where we can find our project resources
    _PATH_TO_ROOT = pwd()
    _PATH_TO_SRC = joinpath(_PATH_TO_ROOT, "src")
    _PATH_TO_DATA = joinpath(_PATH_TO_ROOT, "data")
    _PATH_TO_CONFIG = joinpath(_PATH_TO_ROOT, "configuration")

    # what packages are we going to use?
    using TOML
	using PlutoUI
	using HypothesisTests
	using PrettyTables
	using Colors
	using StatsPlots
	using Reexport

	# these packages are reexported by Serenity
	using DataFrames
    using CSV
    using HTTP
    using Dates
    using Statistics
    using Distributions
	using Optim
	using Convex
	using SCS
	using MathOptInterface
	using JSON
	using Colors

    # alias the Polygon.io URL -
    DATASTORE_URL_STRING = "https://api.polygon.io"

    # What we have here is a classic good news, bad news situation ...
    # Bad news: We don't check in our Polygon.io API key to GitHub (sorry).
    # Good news: Polygon.io API keys are free.
    # check out: https://polygon.io -
    configuration_dictionary = TOML.parsefile(joinpath(_PATH_TO_CONFIG, "Configuration.toml"))

    # get some stuff from the configuration dictionary -
    POLYGON_API_KEY = configuration_dictionary["API"]["polygon_api_key"]

	# load the local version of the serenity library -
	Serenity = ingredients(joinpath(_PATH_TO_SRC, "Include.jl"));

    # show -
    nothing
end

# ╔═╡ 9d55aea4-ac0c-4c33-a161-80ddd34678c6
function option_contract_reference_api_endpoint(ticker_symbol::String, expiration::Date;
	sleeptime::Float64 = 20.0)::NamedTuple

	# save the file locally to avoid the API limit -
	savepath = joinpath(_PATH_TO_DATA,"polygon", "demo", "options","$(ticker_symbol)")

	# check: have we already made the savepath directory?
	if (ispath(savepath) == true)

		# yes: this means we already have downloaded this data - load the local version.

		# what are my local paths?
		local_path_to_data_file = joinpath(savepath,"C$(ticker_symbol).csv")
		local_path_to_header_file = joinpath(savepath,"header", "C$(ticker_symbol)-header.csv")

		# we already have this ticker downloaded. load the existing file from disk -
        data_table = CSV.read(local_path_to_data_file, DataFrame)
		header = CSV.read(local_path_to_header_file, Dict)

        # put the price DataFrame into the price_data_dictionary -
        return (header=header, data=data_table)
	else

		# no: download the options information for this ticker, both put and call -

		# make local paths -
		mkpath(savepath)
    	mkpath(joinpath(savepath, "header"))

		# call -
		call_endpoint_model = Serenity.PolygonOptionsContractReferenceEndpoint()
		call_endpoint_model.ticker = nothing
    	call_endpoint_model.underlying_ticker = ticker_symbol
    	call_endpoint_model.contract_type = Serenity.put
    	call_endpoint_model.expiration_date = expiration
    	call_endpoint_model.limit = 100
    	call_endpoint_model.order = "asc"
    	call_endpoint_model.sort = Serenity.strike_price
		call_endpoint_model.apikey = POLYGON_API_KEY

		# execute -
		(header,df) = Serenity.polygon(DATASTORE_URL_STRING, call_endpoint_model);

		# what are my local paths?
		local_path_to_data_file = joinpath(savepath,"C$(ticker_symbol).csv")
		local_path_to_header_file = joinpath(savepath,"header", "C$(ticker_symbol)-header.csv")

		# save locally -
		CSV.write(local_path_to_data_file, df)
		CSV.write(local_path_to_header_file, header)

		# sleep sometime before we return - helps avoid API limit issue -
		sleep(sleeptime)

        # put the price DataFrame into the price_data_dictionary -
        return (header=header, data=df)
	end
end

# ╔═╡ 1ae954ce-a4c7-4508-9ede-0fb59cd5d638
(h,d) = option_contract_reference_api_endpoint("SPY", Date(2022,1,3));

# ╔═╡ 9d5bb802-9857-4090-af31-11d8b0d5e333
d

# ╔═╡ 3e49d29f-c3d0-4824-9196-6030a73936df
# -------------------------------------------------------------------------------------------- #
# aggregates_api_endpoint(ticker_symbol::String) -> NamedTuple
#
# Downloads pricing data from the Polygon.io API for the Aggregate endpoint. Takes a string
# corresponding to the ticker symbol and downloads pricing data.
#
# The options for the API are hardcoded in this example,
# check out: https://polygon.io/docs/stocks/getting-started for more information.
#
# Arguements
# ticker_symbol::String String holding the ticker symbol that we are interested in
# -------------------------------------------------------------------------------------------- #
function aggregates_api_endpoint(ticker_symbol::String; sleeptime::Float64 = 20.0)::NamedTuple

	# save the file locally to avoid the API limit -
	savepath = joinpath(_PATH_TO_DATA,"polygon", "demo", "aggregates")
	local_path_to_data_file = joinpath(savepath, "$(ticker_symbol).csv")
	local_path_to_header_file = joinpath(savepath, "header", "$(ticker_symbol)-header.csv")

	# do we have a saved file already?
	if (ispath(local_path_to_data_file) == true && 	ispath(local_path_to_header_file) == true)

		# we already have this ticker downloaded. load the existing file from disk -
        data_table = CSV.read(local_path_to_data_file, DataFrame)
		header = CSV.read(local_path_to_header_file, Dict)

        # put the price DataFrame into the price_data_dictionary -
        return (header=header, data=data_table)

	else

		# make the dir to save the data (if we don't have it already) -
		mkpath(savepath)
    	mkpath(joinpath(savepath,"header"))

		# Build an API model for the aggregates endpoint -
		# see: https://polygon.io/docs/stocks/getting-started
		aggregates_api_model = Serenity.PolygonAggregatesEndpointModel()
		aggregates_api_model.adjusted = true
		aggregates_api_model.sortdirection = "asc"
		aggregates_api_model.apikey = POLYGON_API_KEY
		aggregates_api_model.limit = 5000
		aggregates_api_model.to = Date(2021, 12, 24)
		aggregates_api_model.from = Date(2019,12, 24)
		aggregates_api_model.multiplier = 1
		aggregates_api_model.timespan = "day"
		aggregates_api_model.ticker = ticker_symbol

		# execute the API call for these parameters -
		(header,df) = Serenity.polygon(DATASTORE_URL_STRING, aggregates_api_model);

		# save locally -
		CSV.write(local_path_to_data_file, df)
		CSV.write(local_path_to_header_file, header)

		# sleep sometime before we return - helps avoid API limit issue -
		sleep(sleeptime)

        # put the price DataFrame into the price_data_dictionary -
        return (header=header, data=df)
	end
end

# ╔═╡ 3bd81185-9277-4903-bc0b-b41f5e3e7685
begin

	# initialize some space -
	stock_data_dictionary = Dict{String,NamedTuple}()

	# what crypto currencies are we interested in?
	ticker_stock_symbol_array = [
		"SPY", "GILD", "AAPL", "PFE", "MRNA", "PG", "JNJ", "AMD"
	];

	for ticker_stock ∈ ticker_stock_symbol_array
		# make the API call -> store the result in stock_data_dictionary -
		stock_data_dictionary[ticker_stock] = aggregates_api_endpoint(ticker_stock);
	end

	# show -
	nothing
end

# ╔═╡ cabe68be-fad4-4a60-8234-546643c09770
begin

	# initialize some space -
	crypto_data_dictionary = Dict{String,NamedTuple}()

	# what crypto currencies are we interested in?
	ticker_crypto_symbol_array = [
		"X:BTCUSD", "X:ETHUSD"
	];

	for ticker_crypto ∈ ticker_crypto_symbol_array

		# make the API call -> store the result in crypto_data_dictionary -
		crypto_data_dictionary[ticker_crypto] = aggregates_api_endpoint(ticker_crypto);
	end

	# show -
	nothing
end

# ╔═╡ e7ff2aa0-fceb-4fde-8ba6-7438ad1961d5
plot(crypto_data_dictionary["X:ETHUSD"].data[!,:close])

# ╔═╡ feb87b95-c7c7-4bb3-925e-0b7600c89954
begin

	# List of colors -
	WHITE = RGB(1.0, 1.0, 1.0)
	BACKGROUND = RGB(0.99, 0.98, 0.96)
	BLUE = RGB(68 / 255, 119 / 255, 170 / 255)
	LBLUE = RGB(102 / 255, 204 / 255, 238 / 255)
	GRAY = RGB(187 / 255, 187 / 255, 187 / 255)
	RED = RGB(238 / 255, 102 / 255, 119 / 255)

	# show -
	nothing
end

# ╔═╡ 02f7555b-3569-4364-b8d8-4859a36faeac
begin

	# initialize -
	N = length(stock_data_dictionary["SPY"].data[!,:close])
	𝒫 = length(ticker_stock_symbol_array)
	state_table = Array{Float64,2}(undef,N,𝒫)

	for (ticker_index, ticker_stock) ∈ enumerate(ticker_stock_symbol_array)

		# get value for this ticker -
		data_array = stock_data_dictionary[ticker_stock].data
		P₀ = data_array[1,:close]

		for time_index ∈ 1:N
			state_table[time_index,ticker_index] = (1/P₀)*data_array[time_index,:close]
		end
	end

	# which index is MRNA?
	idx_plot = findall(x-> x !="MRNA", ticker_stock_symbol_array)

	# plot -
	plot(state_table[:, idx_plot], legend=false, c=GRAY)

	# plot SPY -
	P_spy = stock_data_dictionary["SPY"].data[1,:close]
	P_spy_scaled = (1/P_spy)*stock_data_dictionary["SPY"].data[!,:close]
	plot!(P_spy_scaled,c=RED,lw=4)
end

# ╔═╡ b4100007-b259-4fe3-a88c-be88928fe423
html"""
<script>
	// initialize -
	var section = 0;
	var subsection = 0;
	var headers = document.querySelectorAll('h3, h5');

	// main loop -
	for (var i=0; i < headers.length; i++) {

		var header = headers[i];
	    var text = header.innerText;
	    var original = header.getAttribute("text-original");
	    if (original === null) {

			// Save original header text
	        header.setAttribute("text-original", text);
	    } else {

			// Replace with original text before adding section number
	        text = header.getAttribute("text-original");
	    }

	    var numbering = "";
	    switch (header.tagName) {
	        case 'H3':
	            section += 1;
	            numbering = section + ".";
	            subsection = 0;
	            break;
	        case 'H5':
	            subsection += 1;
	            numbering = section + "." + subsection;
	            break;
	    }
		// update the header text
		header.innerText = numbering + " " + text;
	};
</script>"""

# ╔═╡ 5596d565-04f3-44a6-9b72-05cf2b64a1ca
html"""
<style>
main {
    max-width: 1200px;
    width: 85%;
    margin: auto;
    font-family: "Roboto, monospace";
}

a {
    color: blue;
    text-decoration: none;
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.H1 {
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}
</style>"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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deps = ["Dates", "Random"]
git-tree-sha1 = "244586bc07462d22aed0113af9c731f2a518c93e"
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version = "1.3.10"

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

[[deps.Setfield]]
deps = ["ConstructionBase", "Future", "MacroTools", "Requires"]
git-tree-sha1 = "0afd9e6c623e379f593da01f20590bacc26d1d14"
uuid = "efcf1570-3423-57d1-acb7-fd33fddbac46"
version = "0.8.1"

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

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

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

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
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[[deps.SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "e08890d19787ec25029113e88c34ec20cac1c91e"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.0.0"

[[deps.Static]]
deps = ["IfElse"]
git-tree-sha1 = "7f5a513baec6f122401abfc8e9c074fdac54f6c1"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.4.1"

[[deps.StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

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

[[deps.StatsAPI]]
git-tree-sha1 = "0f2aa8e32d511f758a2ce49208181f7733a0936a"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.1.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "2bb0cb32026a66037360606510fca5984ccc6b75"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.13"

[[deps.StatsFuns]]
deps = ["ChainRulesCore", "InverseFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "bedb3e17cc1d94ce0e6e66d3afa47157978ba404"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.14"

[[deps.StatsPlots]]
deps = ["Clustering", "DataStructures", "DataValues", "Distributions", "Interpolations", "KernelDensity", "LinearAlgebra", "MultivariateStats", "Observables", "Plots", "RecipesBase", "RecipesPipeline", "Reexport", "StatsBase", "TableOperations", "Tables", "Widgets"]
git-tree-sha1 = "e1e5ed9669d5521d4bbdd4fab9f0945a0ffceba2"
uuid = "f3b207a7-027a-5e70-b257-86293d7955fd"
version = "0.14.30"

[[deps.StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

[[deps.SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

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

[[deps.TableOperations]]
deps = ["SentinelArrays", "Tables", "Test"]
git-tree-sha1 = "e383c87cf2a1dc41fa30c093b2a19877c83e1bc1"
uuid = "ab02a1b2-a7df-11e8-156e-fb1833f50b87"
version = "1.2.0"

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

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "bb1064c9a84c52e277f1096cf41434b675cd368b"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.1"

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

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

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "216b95ea110b5972db65aa90f88d8d89dcb8851c"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.6"

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

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

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

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

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.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"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "66d72dc6fcc86352f01676e8f0f698562e60510f"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.23.0+0"

[[deps.WeakRefStrings]]
deps = ["DataAPI", "InlineStrings", "Parsers"]
git-tree-sha1 = "c69f9da3ff2f4f02e811c3323c22e5dfcb584cfa"
uuid = "ea10d353-3f73-51f8-a26c-33c1cb351aa5"
version = "1.4.1"

[[deps.Widgets]]
deps = ["Colors", "Dates", "Observables", "OrderedCollections"]
git-tree-sha1 = "80661f59d28714632132c73779f8becc19a113f2"
uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62"
version = "0.6.4"

[[deps.WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.5"

[[deps.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"

[[deps.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"

[[deps.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"

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

[[deps.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"

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

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

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

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

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

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

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

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

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

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.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:
# ╟─201f572b-9e68-4d80-b823-be41243382fd
# ╟─48cce16c-31bc-438c-bef9-eee2c13fb220
# ╟─d3884c8d-bd00-4059-a751-f29b211e9ef6
# ╟─4b4668f0-6894-4219-9de8-f2106fb70aca
# ╠═1461bfa4-633e-11ec-0ee6-a590181dc91b
# ╟─6c3e5618-c689-4941-b540-e27734ee096b
# ╟─572cb255-04a5-402b-a10f-85f6a1a23f39
# ╠═3bd81185-9277-4903-bc0b-b41f5e3e7685
# ╠═02f7555b-3569-4364-b8d8-4859a36faeac
# ╟─6589f632-2e09-4bc6-aa8e-30f6338d5729
# ╠═cabe68be-fad4-4a60-8234-546643c09770
# ╠═e7ff2aa0-fceb-4fde-8ba6-7438ad1961d5
# ╟─feab4379-ce36-4a34-abf6-70a6a15dcfd2
# ╠═1ae954ce-a4c7-4508-9ede-0fb59cd5d638
# ╠═9d5bb802-9857-4090-af31-11d8b0d5e333
# ╠═05d72483-d219-4c4f-823c-c9ceb82caa3d
# ╠═9d55aea4-ac0c-4c33-a161-80ddd34678c6
# ╠═3e49d29f-c3d0-4824-9196-6030a73936df
# ╟─8623a119-bf7f-4566-be78-9769c3a64042
# ╟─77473f9a-e5fd-41e6-89d4-acffcba38f45
# ╠═feb87b95-c7c7-4bb3-925e-0b7600c89954
# ╠═bd0c9e41-fed0-46f9-bf2e-136a439692a2
# ╟─7ce9e2ec-a3a2-4b64-be6e-27618bd9f3f1
# ╟─b4100007-b259-4fe3-a88c-be88928fe423
# ╟─5596d565-04f3-44a6-9b72-05cf2b64a1ca
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002

max_stars_repo_name: Pooksoft/Serenity

max_issues_repo_name: Alba-Intelligence/Serenity

max_forks_repo_name: Alba-Intelligence/Serenity


text:
module IndirectShooting

using NLsolve

# Includes
include("ShootingDataManager.jl")
include("ShootingSolver.jl")
include("FSSSolver.jl")

# Exports
export FSSSolver
export solve!

export initializeData!
export GetInitialGuessConverged
export GetHomotopyConverged
export GetSolution
export GetHomotopySolutionVector
export GetHomotopyParams
export GetHomotopyConvergenceFlags

end

ext: jl


text:
println(zero(Float32))
println(zero(Float64))
println(zero(Int128))
println(one(Float32))
println(one(Float64))

println(zero(BigInt))
println(zero(BigFloat))

#: 0.0
#: 0.0
#: 0
#: 1.0
#: 1.0

#: 0
#: 0.0


text:
const BIBPATHS = String[]

function add_bibpath(path::String, paths::String...)
    push!(BIBPATHS, abspath(path))
    push!(BIBPATHS, abspath.(paths)...)
    unique!(BIBPATHS)
end

bibpaths() = BIBPATHS


text:
using XGrad
using CuArrays
using BenchmarkTools

include("functions.jl")

# fuonction perf_test(f; _ctx=Dict(), inputs...)
#     # default eltype for CPU is Float64
#     ctx = copy(_ctx)
#     vals = ([val for (name, val) in inputs]...)
#     println("Compiling derivatives for CPU")
#     @time df = xdiff(f; ctx=ctx, inputs...)
#     mem = Dict()
#     println("Testing on CPU...")
#     r1 = @benchmark $df($vals...; mem=$mem)
#     show(STDOUT, MIME{Symbol("text/plain")}(), r1)
#     println("\n")

#     # CUDA better works with Float32
#     inputs = [k => convert(Array{Float32}, v) for (k, v) in inputs]
#     gpu_vals = ([CuArray(val) for (name, val) in inputs]...)
#     println("Compiling derivatives for GPU")
#     ctx_gpu = merge(_ctx, Dict(:codegen => CuCodeGen()))
#     @time df_gpu = xdiff(f; ctx=ctx_gpu, inputs...)
#     mem = Dict()
#     println("Testing on GPU...")
#     r2 = @benchmark $df_gpu($gpu_vals...; mem=$mem)
#     # GPUArrays.gc()
#     show(STDOUT, MIME{Symbol("text/plain")}(), r2)
#     println("\n")
#     println("\n----------------------------------------\n")
#     return r1, r2
# end

function perf_test(f; _ctx=Dict(), inputs...)
    # default eltype for CPU is Float64
    inputs = [k => convert(Array{Float32}, v) for (k, v) in inputs]
    ctx = copy(_ctx)
    vals = ([val for (name, val) in inputs]...)
    println("Compiling derivatives for CPU")
    @time df = xdiff(f; ctx=ctx, inputs...)
    mem = Dict()
    println("Testing on CPU...")
    r1 = @benchmark $df($vals...; mem=$mem)
    show(STDOUT, MIME{Symbol("text/plain")}(), r1)
    println("\n")

    # CUDA better works with Float32
    inputs = [k => convert(CuArray{Float32}, v) for (k, v) in inputs]
    gpu_vals = ([val for (name, val) in inputs]...)
    println("Compiling derivatives for GPU")
    # ctx_gpu = merge(_ctx, Dict(:codegen => CuCodeGen()))
    @time df_gpu = xdiff(f; ctx=copy(_ctx), inputs...)
    mem = Dict()
    println("Testing on GPU...")
    r2 = @benchmark $df_gpu($gpu_vals...; mem=$mem)
    # GPUArrays.gc()
    show(STDOUT, MIME{Symbol("text/plain")}(), r2)
    println("\n")
    println("\n----------------------------------------\n")
    return r1, r2
end

function benchmark_autoencoder()
    f = autoencoder_cost
    println("\n## On larger data\n")
    We1 = rand(2000, 10_000); b1 = rand(2000); We2 = rand(1000, 2000); b2 = rand(1000);
    Wd = rand(10_000, 1000); x = rand(10_000, 100);
    inputs = [:We1 => We1, :We2 => We2, :Wd => Wd, :b1 => b1, :b2 => b2, :x => x];
    perf_test(f; inputs...)

    println("\n## On smaller data\n")
    We1 = rand(200, 1000); b1 = rand(200); We2 = rand(100, 200); b2 = rand(100);
    Wd = rand(1000, 100); x = rand(1000, 100);
    inputs = [:We1 => We1, :We2 => We2, :Wd => Wd, :b1 => b1, :b2 => b2, :x => x];
    perf_test(f; inputs...)
end

function benchmark_mlp1()
    f = mlp1
    println("\n## On larger data\n")
    w1=rand(2000, 10000); w2=rand(1000, 2000); w3=rand(1000, 1000); x1=rand(10000, 500);
    inputs = [:w1=>w1, :w2=>w2, :w3=>w3, :x1=>x1];
    perf_test(f; inputs...)

    println("\n## On smaller data\n")
    w1=rand(200, 1000); w2=rand(100, 200); w3=rand(100, 100); x1=rand(1000, 10);
    inputs = [:w1=>w1, :w2=>w2, :w3=>w3, :x1=>x1];
    perf_test(f; inputs...)
end

function benchmark_mlp2()
    f = mlp2
    println("\n## On larger data\n")
    w1 = randn(2000, 10000); w2 = randn(1000, 2000); w3 = randn(1000, 1000); x1 = randn(10000, 500);
    b1 =  randn(2000); b2 = randn(1000); b3 = randn(1000)
    inputs = [:w1=>w1, :w2=>w2, :w3=>w3, :b1 => b1, :b2 => b2, :b3 => b3, :x1=>x1];
    perf_test(f; inputs...)

    println("\n## On smaller data\n")
    w1=rand(200, 1000); w2=rand(100, 200); w3=rand(100, 100); x1=rand(1000, 10);
    b1 =  rand(200); b2 = rand(100); b3 = rand(100)
    inputs = [:w1=>w1, :w2=>w2, :w3=>w3, :b1 => b1, :b2 => b2, :b3 => b3, :x1=>x1];
    perf_test(f; inputs...)
end

# function benchmark_rnn()
#     f = rnn
#     println("\n## On larger data\n")
#     Wxh = randn(4096, 4096); Whh = randn(4096, 4096); Why = randn(128, 4096);
#     hprev = randn(4096); x = randn(4096); y = randn(128);
#     inputs = [:Wxh=>Wxh, :Whh=>Whh, :Why=>Why, :hprev => hprev, :x => x, :y=>y];
#     perf_test(f; ctx=Dict(:cost => :cost), inputs...)

#     # println("\n## On smaller data\n")
#     # w1=rand(200, 1000); w2=rand(100, 200); w3=rand(100, 100); x1=rand(1000, 10);
#     # b1 =  rand(200); b2 = rand(100); b3 = rand(100)
#     # inputs = [:w1=>w1, :w2=>w2, :w3=>w3, :b1 => b1, :b2 => b2, :b3 => b3, :x1=>x1];
#     # perf_test(f; ctx=Dict(:cost => :cost), inputs...)
# end


text:
# Use baremodule to shave off a few KB from the serialized `.ji` file
baremodule libcgal_julia_jll
using Base
using Base: UUID
import JLLWrappers

JLLWrappers.@generate_main_file_header("libcgal_julia")
JLLWrappers.@generate_main_file("libcgal_julia", UUID("e9ad47b2-a301-5fb7-a0bd-6eece649b37c"))
end  # module libcgal_julia_jll


text:
global rs = nothing
@info "Precompiling extra Reduce methods (set `ENV[\"REDPRE\"]=\"0\"` to disable)"
Reduce.Load(); atexit(() -> kill(rs))

rcall(:((1+pi)^2)) == convert(Expr,RExpr(rcall("(1+pi)**2")))
try; "1/0" |> rcall; false; catch; true; end
RExpr("(x+i)^3")
Reduce._syme(Reduce.r_to_jl)
R"x+2" == R"2+x-1+1"
:((x+1+π)^2; int(1/(1+x^3),x)) |> RExpr |> Reduce.parse
string(R"x+1")
RExpr(:x) == R"x"
Algebra.:*(RExpr(:x),R"x") == R"x^2"
convert(RExpr,R"x").str == convert(Array{String,1},R"x")
load_package([:rlfi]) == load_package(:rlfi,:rlfi)
rcall(:(x^2+2x+1),off=[:factor])
rcall("x + 1","factor")
Expr(:function,:(fun(z)),:(return begin; x = 700; y = x; end)) |> RExpr |> Reduce.parse
Expr(:for,:(i=2:34),:(product(i))) |> rcall
try; Expr(:type,false,:x) |> RExpr; false; catch; true; end
try; :(@time f(x)) |> RExpr; false; catch; true; end
Expr(:function,:(fun(a,b)),:(return y=a^3+3*a^2*b+3*a*b^2+b^3)) |> factor |> expand
try; Expr(:for,:(i=2:34),:(product(i))) |> RExpr |> parse; false; catch; true; end
R"begin; 1:2; end" |> Reduce.parse |> RExpr |> string
latex(:(x+1))
#Algebra.length(:(x+y))
Algebra.log(:(ℯ^x))
#Algebra.nextprime(100)
#Algebra.ceiling(1.2)
Algebra.impart(:(1+2*im))
#Algebra.impart(2+1.7im)
#Algebra.bernoulli(2)
Sys.islinux() && Reduce.RSymReplace("!#03a9; *x**2 + !#03a9;")
Algebra.int(:(x^2+y),:x) |> RExpr == Algebra.int("x^2+y","x") |> RExpr
R"/(2,begin 2; +(7,4); return +(4,*(2,7))+9 end)" |> Reduce.parse
Algebra.df(Expr(:function,:(fun(z,c)),:(return begin; zn = z^2+c; nz = z^3-1; end))|>RExpr,:z)
:([1 2; 3 4]) |> RExpr |> Reduce.parse |> RExpr == [1 2; 3 4] |> RExpr

Algebra.nextprime("3")
expand("(x-2)^2") |> RExpr == R"(x-2)^2"
nat("x+1")
Algebra.:^(:x,2) == :(x^2)
Algebra.://(NaN,NaN)
join(split(R"x+1;x+2"))
Algebra.sub(:x=>7,Algebra.:+(:x,7)) == Algebra.sub([:x=>7,:z=>21],Algebra.:-(:z,:x))
#squash(Expr(:function,:(fun(x)),:(z=3;z+=:x))).args[2] == squash(:(y=:x;y+=3))
squash(:(sqrt(x)^2))
Expr(:block,:(x+1)) |> RExpr == R"1+x"
Algebra.limit(Algebra.:^(Algebra.:-(1,Algebra.:/(1,:n)),Algebra.:-(:n)),:n,Inf)
Algebra.log(Algebra.exp(:pi))
Algebra.://(2,Inf)
Algebra.://(Inf,2)

rcall("x")
Algebra.operator(:x); Algebra.clear(:x);
#try Algebra.det([:x :y]) catch; true end
join([R"1",R"1"]) == R"1;1"
list([R"1",R"x"]) == list((1,:x))
Reduce.lister(:x) == R"x"
!latex(false)
@rounded @factor x^2-2x+1
@rounded @off_factor @rcall x^2
Algebra.det([:a :b; :c :d])
Algebra.tp([:a;:b]) == [:a :b;]
transpose(:x)
adjoint(:x)
Algebra.operator(:cbrt)
Algebra.rlet(:(cbrt(~x))=>:(x^(1/3)))
Algebra.rlet([:(cbrt(~x))=>:(x^(1/3))])
Algebra.rlet(Dict(:(cbrt(~x))=>:(x^(1/3))))
Algebra.clear(:cbrt)
Algebra.:+(1,R"x")
Algebra.inv([:a :b; :c :d])
Algebra.inv(1)
Algebra.:\([1],[2])
#Algebra.://(1.0,1.0)
Algebra.:/([:a :b; :c :d],2)
Algebra.:+([:a :b; :c :d],1) == Algebra.:+(1,[:a :b; :c :d])
Algebra.:+([:x],1) == Algebra.:+(1,[:x])
Algebra.:+([:x,:y]',1) == Algebra.:+(1,[:x,:y]')
Algebra.solve(:(x-1),:x) == Algebra.solve((:(x-1),),:x)
Algebra.order(nothing); Algebra.korder(nothing)


text:
#=
Classical discrete time LQ optimal control and filtering problems. The
control problems we consider take the form

    \max \sum_{t = 0}^N \beta^t \{a_t y_t - h y^2_t / 2 - [d(L)y_t]^2 \}

subject to h > 0, 0 < \beta < 1 and

 * y_{-1},y_{-2}, \dots, y_{-m} (initial conditions)

 * d(L) = d_0 + d_1L + d_2L^2 + \dots + d_mL^m

The sequence {y_t} is scalar

Authors: Shunsuke Hori
=#

using Polynomials

struct LQFilter{TR<:Real, TI<:Integer, TF<:AbstractFloat}
    d::Vector{TF}
    h::TR
    y_m::Vector{TF}
    m::TI
    phi::Vector{TF}
    beta::TR
    phi_r::Union{Vector{TF},Void}
    k::Union{TI,Void}
end

"""
Parameters
----------
d : Vector
        The order of the coefficients: [d_0, d_1, ..., d_m]
h : Real
        Parameter of the objective function (corresponding to the
        quadratic term)
y_m : Vector
        Initial conditions for y
r : Vector
        The order of the coefficients: [r_0, r_1, ..., r_k]
        (optional, if not defined -> deterministic problem)
beta : Real or nothing
        Discount factor (optional, default value is one)
h_eps :
"""
function LQFilter{TR<:Real}(d::Vector{TR},
                   h::TR,
                   y_m::Vector{TR};
                   r::Union{Vector{TR},Void}=nothing,
                   beta::Union{TR,Void}=nothing,
                   h_eps::Union{TR,Void}=nothing,
                   )

    m = length(d) - 1

    m == length(y_m) ||
        throw(ArgumentError("y_m and d must be of same length = $m"))

    #---------------------------------------------
    # Define the coefficients of phi up front
    #---------------------------------------------

    phi = Vector{TR}(2m + 1)
    for i in -m:m
        phi[m-i+1] = sum(diag(d*d', -i))
    end
    phi[m+1] = phi[m+1] + h

    #-----------------------------------------------------
    # If r is given calculate the vector phi_r
    #-----------------------------------------------------

    if r == nothing
        k=nothing
        phi_r = nothing
    else
        k = size(r,1) - 1
        phi_r = Vector{TR}(2k + 1)

        for i = -k:k
            phi_r[k-i+1] = sum(diag(r*r', -i))
        end

        if h_eps != nothing
            phi_r[k+1] = phi_r[k+1] + h_eps
        end
    end

    #-----------------------------------------------------
    # If beta is given, define the transformed variables
    #-----------------------------------------------------
    if beta == nothing
        beta = 1.0
    else
        d = beta.^(collect(0:m)/2) * d
        y_m = y_m * beta.^(- collect(1:m)/2)
    end

    return LQFilter(d,h,y_m,m,phi,beta,phi_r,k)
end

"""
This constructs the matrices W and W_m for a given number of periods N
"""
function construct_W_and_Wm(lqf::LQFilter, N::Integer)

    d, m = lqf.d, lqf.m

    W = zeros(N + 1, N + 1)
    W_m = zeros(N + 1, m)

    #---------------------------------------
    # Terminal conditions
    #---------------------------------------

    D_m1 = zeros(m + 1, m + 1)
    M = zeros(m + 1, m)

    # (1) Constuct the D_{m+1} matrix using the formula

    for j in 1:(m+1)
        for k in j:(m+1)
            D_m1[j, k] = dot(d[1:j,1], d[k-j+1:k,1])
        end
    end

    # Make the matrix symmetric
    D_m1 = D_m1 + D_m1' - diagm(diag(D_m1))

    # (2) Construct the M matrix using the entries of D_m1

    for j in 1:m
        for i in (j + 1):(m + 1)
            M[i, j] = D_m1[i-j, m+1]
        end
    end
    M

    #----------------------------------------------
    # Euler equations for t = 0, 1, ..., N-(m+1)
    #----------------------------------------------
    phi, h = lqf.phi, lqf.h

    W[1:(m + 1), 1:(m + 1)] = D_m1 + h * eye(m + 1)
    W[1:(m + 1), (m + 2):(2m + 1)] = M

    for (i, row) in enumerate((m + 2):(N + 1 - m))
        W[row, (i + 1):(2m + 1 + i)] = phi'
    end

    for i in 1:m
        W[N - m + i + 1 , end-(2m + 1 - i)+1:end] = phi[1:end-i]
    end

    for i in 1:m
        W_m[N - i + 2, 1:(m - i)+1] = phi[(m + 1 + i):end]
    end

    return W, W_m
end

"""
This function calculates z_0 and the 2m roots of the characteristic equation
associated with the Euler equation (1.7)
Note:
------
`poly(roots)` from Polynomial.jll defines a polynomial using its roots that can be
evaluated at any point by `polyval(Poly,x)`. If x_1, x_2, ... , x_m are the roots then
    polyval(poly(roots),x) = (x - x_1)(x - x_2)...(x - x_m)
"""
function roots_of_characteristic(lqf::LQFilter)
    m, phi = lqf.m, lqf.phi

    # Calculate the roots of the 2m-polynomial
    phi_poly=Poly(phi[end:-1:1])
    proots = roots(phi_poly)
    # sort the roots according to their length (in descending order)
    roots_sorted = sort(proots, by=abs)[end:-1:1]
    z_0 = sum(phi) / polyval(poly(proots), 1.0)
    z_1_to_m = roots_sorted[1:m]     # we need only those outside the unit circle
    lambdas = 1 ./ z_1_to_m
    return z_1_to_m, z_0, lambdas
end

"""
This function computes the coefficients {c_j, j = 0, 1, ..., m} for
    c(z) = sum_{j = 0}^{m} c_j z^j
Based on the expression (1.9). The order is
    c_coeffs = [c_0, c_1, ..., c_{m-1}, c_m]
"""
function coeffs_of_c(lqf::LQFilter)
    m = lqf.m
    z_1_to_m, z_0, lambdas = roots_of_characteristic(lqf)
    c_0 = (z_0 * prod(z_1_to_m) * (-1.0)^m)^(0.5)
    c_coeffs = coeffs(poly(z_1_to_m)) * z_0 / c_0
    return c_coeffs
end

"""
This function calculates {lambda_j, j=1,...,m} and {A_j, j=1,...,m}
of the expression (1.15)
"""
function solution(lqf::LQFilter)
    z_1_to_m, z_0, lambdas = roots_of_characteristic(lqf)
    c_0 = coeffs_of_c(lqf)[end]
    A = zeros(lqf.m)
    for j in 1:m
        denom = 1 - lambdas/lambdas[j]
        A[j] = c_0^(-2) / prod(denom[1:m .!= j])
    end
    return lambdas, A
end

"""
This function constructs the covariance matrix for x^N (see section 6)
for a given period N
"""
function construct_V(lqf::LQFilter; N::Integer=nothing)
    if N == nothing
        error("N must be provided!!")
    end
    if !(typeof(N) <: Integer)
        throw(ArgumentError("N must be Integer!"))
    end

    phi_r, k = lqf.phi_r, lqf.k
    V = zeros(N, N)
    for i in 1:N
        for j in 1:N
            if abs(i-j) <= k
                V[i, j] = phi_r[k + abs(i-j)+1]
            end
        end
    end
    return V
end

"""
Assuming that the u's are normal, this method draws a random path
for x^N
"""
function simulate_a(lqf::LQFilter, N::Integer)
    V = construct_V(N + 1)
    d = MVNSampler(zeros(N + 1), V)
    return rand(d)
end

"""
This function implements the prediction formula discussed is section 6 (1.59)
It takes a realization for a^N, and the period in which the prediciton is formed

Output:  E[abar | a_t, a_{t-1}, ..., a_1, a_0]
"""
function predict(lqf::LQFilter, a_hist::Vector, t::Integer)
    N = length(a_hist) - 1
    V = construct_V(N + 1)

    aux_matrix = zeros(N + 1, N + 1)
    aux_matrix[1:t+1 , 1:t+1 ] = eye(t + 1)
    L = chol(V)'
    Ea_hist = inv(L) * aux_matrix * L * a_hist

    return Ea_hist
end

"""
- if t is NOT given it takes a_hist (Vector or Array) as a deterministic a_t
- if t is given, it solves the combined control prediction problem (section 7)
　(by default, t == nothing -> deterministic)

for a given sequence of a_t (either determinstic or a particular realization),
it calculates the optimal y_t sequence using the method of the lecture

Note:
------
lufact normalizes L, U so that L has unit diagonal elements
To make things cosistent with the lecture, we need an auxiliary diagonal
matrix D which renormalizes L and U
"""

function optimal_y(lqf::LQFilter, a_hist::Vector, t = nothing)
    beta, y_m, m = lqf.beta, lqf.y_m, lqf.m

    N = length(a_hist) - 1
    W, W_m = construct_W_and_Wm(lqf, N)

    F = lufact(W, Val{true})

    L, U = F[:L], F[:U]
    D = diagm(1.0./diag(U))
    U = D * U
    L = L * diagm(1.0./diag(D))

    J = flipdim(eye(N + 1), 2)

    if t == nothing   # if the problem is deterministic
        a_hist = J * a_hist

        #--------------------------------------------
        # Transform the a sequence if beta is given
        #--------------------------------------------
        if beta != 1
            a_hist =  reshape(a_hist * (beta^(collect(N:0)/ 2)),N + 1, 1)
        end

        a_bar = a_hist - W_m * y_m           # a_bar from the lecutre
        Uy = \(L, a_bar)            # U @ y_bar = L^{-1}a_bar from the lecture
        y_bar = \(U, Uy)            # y_bar = U^{-1}L^{-1}a_bar
        # Reverse the order of y_bar with the matrix J
        J = flipdim(eye(N + m + 1), 2)
        y_hist = J * vcat(y_bar, y_m)     # y_hist : concatenated y_m and y_bar
        #--------------------------------------------
        # Transform the optimal sequence back if beta is given
        #--------------------------------------------
        if beta != 1
            y_hist = y_hist .* beta.^(- collect(-m:N)/2)
        end

    else           # if the problem is stochastic and we look at it
        Ea_hist = reshape(predict(a_hist, t), N + 1, 1)
        Ea_hist = J * Ea_hist

        a_bar = Ea_hist - W_m * y_m           # a_bar from the lecutre
        Uy = \(L, a_bar)            # U @ y_bar = L^{-1}a_bar from the lecture
        y_bar = \(U, Uy)            # y_bar = U^{-1}L^{-1}a_bar

        # Reverse the order of y_bar with the matrix J
        J = flipdim(eye(N + m + 1), 2)
        y_hist = J * vcat(y_bar, y_m)     # y_hist : concatenated y_m and y_bar
    end
    return y_hist, L, U, y_bar
end


text:
# This file was generated, do not modify it. # hide
describe(power, :mean, :nmissing)


text:
import DifferentialEquations: ODEProblem, EnsembleProblem, solve, remake, Tsit5

"""

    Apply the dynamical models to generate numerical forecasts.

"""
function (ssm::StateSpaceModel)(x::Array{Float64,2})

    nv, np = size(x)
    xf = similar(x)
    p = ssm.params
    tspan = (0.0, ssm.dt_integration)
    u0 = zeros(Float64, nv)

    prob = ODEProblem(ssm.model, u0, tspan, p)

    function prob_func(prob, i, repeat)
        remake(prob, u0 = vec(x[:, i]))
    end

    monte_prob = EnsembleProblem(prob, prob_func = prob_func)

    sim = solve(monte_prob, Tsit5(); trajectories = np, save_everystep = false)

    for i = 1:np
        xf[:, i] .= last(sim[i].u)
    end

    xf, xf

end