5_vector_spaces_(optional).jl

### A Pluto.jl notebook ###
# v0.19.40

using Markdown
using InteractiveUtils

# ╔═╡ 32fbc94f-710f-435f-91a8-13c07b9671f0
begin
	using LinearAlgebra
	using StaticArrays
	using SymPy
	using Plots
	using Images
	using PlutoUI
	using Symbolics
	using PartialFunctions
	using ForwardDiff
	using Markdown
end

# ╔═╡ c85e44f0-600a-4491-ba95-bf5b38d83578
begin
	"""`D` dimensional `F` (real/complex) vector space of vector type `T`."""
	abstract type VectorSpace{T, D, F<:Number} end

	basis(s::VectorSpace) = s.basis
	basis(s::VectorSpace, i::Integer) = basis(s)[i]

	struct Vect{D, F, V<:VectorSpace{<:Any,D,F}}
		coefs::SVector{D,<:F}
		space::V

		function Vect(x::AbstractVector, s::VectorSpace{T,D,F}) where {T,D,F}
			new{D,F,typeof(s)}(SVector{D,F}(x), s)
		end
	end

	Base.length(v::Vect) = length(v.coefs)
	vec(v::Vect) = sum(c * b for (b, c) in zip(basis(v.space), v.coefs))

	abstract type InnerProdSpace{T, D, F} <: VectorSpace{T, D, F} end

	struct EuclideanSpace{D, F} <: InnerProdSpace{SVector{D,F}, D, F}
		basis::SMatrix{D,D,F}

		function EuclideanSpace{D,F}(basis=I) where {D,F}
			new{D,F}(SMatrix{D,D,F}(basis))
		end
	end

	basis(s::EuclideanSpace) = eachcol(s.basis)

	function space_type(x::AbstractArray{T,N}) where {T<:Number,N}
		EuclideanSpace{size(x, 1), T <: Real ? Float64 : ComplexF64}
	end

	const ℝ{N} = EuclideanSpace{N, Float64}
	const ℂ{N} = EuclideanSpace{N, ComplexF64}

	Vect(coefs::AbstractVector; basis=I) = let
		T = space_type(basis isa AbstractArray ? basis : coefs)
		Vect(coefs, T(basis))
	end
	
	VectorSpace(basis::AbstractMatrix) = space_type(basis)(basis)

	Base.in(x::Vect, V::Type{<:VectorSpace}) = x.space isa V

	function Base.getindex(s::VectorSpace, x::T...) where {T}
		Vect(T <: AbstractVector && length(x) == 1 ? x[1] : collect(x), s)
	end

	function Base.:+(u::Vect{D,<:Number,V}, v::Vect{D,<:Number,V}) where {D,V}
		if u.space == v.space
			Vect(u.coefs .+ v.coefs, u.space)
		else
			vec(u) + vec(v)
		end
	end
	
	Base.:-(v::Vect) = Vect(-v.coefs, v.space)
	Base.:*(a::Number, v::Vect) = Vect(a.*v.coefs, v.space)
	Base.:*(x::Vect, y::Number) = y * x
	Base.:-(x, y) = x + (-y)
	Base.:/(x, y::Number) = (1/y) * x
	
	proj(v::Vect, s::VectorSpace) = proj(vec(v), s)

	function proj(x::T, s::VectorSpace{T,D,F}) where {T,D,F}
		B = basis(s)
		if !(x isa AbstractVector) && s isa InnerProdSpace
			x = [x ⋅ v for v in B]
			B = [u ⋅ v for u in B, v in B]
		end
		Vect(B \ x, s)
	end
end

# ╔═╡ e96c765f-5f5f-41db-8191-1a0d444bcde2
begin
	struct Func{L, H}
		exp::Sym
		var::Sym
	end

	struct FuncSpace{D, F, L, H} <: InnerProdSpace{Sym, D, F}
		basis::SVector{D,Sym}
		var::Sym

		function FuncSpace(basis::Vector{<:Sym}, (lo, hi), field=Real)
			var = only(free_symbols(basis))
			new{length(basis),field,lo,hi}(basis, var)
		end
	
		function FuncSpace(basis::Vector{Func{L,H}}, field=Real) where {L,H}
			var = only(Set(f.var for f in basis))
			new{length(basis),field,L,H}([f.exp for f in basis], var)
		end
	end

	basis(s::FuncSpace{D,F,L,H}) where {D,F,L,H} = Func{L,H}.(s.basis, s.var)
	vec(v::Vect{D,F,<:FuncSpace}) where {D,F} = sum(v.coefs .* basis(v.space)).exp
	(v::Vect{D,F,FuncSpace})(x) where {D,F} = vec(v)(x)

	function PolySpace(deg, range=(0, 1), field=Real, var=:x)
		basis = symbols(var, real=true) .^ (0:deg)
		FuncSpace(basis, range, field)
	end

	function FourierSpace(deg, range=(0, 2π), field=Real, var=:x)
		a, b = range
		T = b - a
		t = symbols(var, real=true) - a
		basis = [exp(-im*PI*k*t/T) for k = 0:deg]
		basis = field <: Real ? real.(basis) : basis
		FuncSpace(basis, range, field)
	end

	Base.:*(a, f::T) where {T<:Func} = T(a * f.exp, f.var)
	Base.:+(f::T, g::T) where {T<:Func} = T(f.exp + g.exp, f.var)
	Base.:-(f::T) where {T<:Func} = T(-f.exp, f.var)

	proj(f::Func, s::FuncSpace) = proj(f.exp, s)
	proj(f::Function, s::FuncSpace) = proj(f(s.var), s)
end

# ╔═╡ 40ae0bd6-0fe5-4537-9cee-9485a8b741f3
methodswith(Vect)

# ╔═╡ 248c304c-de63-4523-b334-e201ed5f07f2
begin
	u = Vect([3, 4])
	v = Vect([3, 4], basis=[1 2; 3 4])
	@show v ∈ ℝ{2}
	u + v
end

# ╔═╡ 2a5009df-5e84-4b50-b819-c19333eb4a8b
let
	f(x) = cos(2x^2)
	g = @show vec(proj(f, p1))
	h = @show vec(proj(f, p2))
	xs = -2.2:0.001:2.2
	plot(f, xs, label="function", legend = :outertopright)
	plot!(g, xs, label="polynomial")
	plot!(h, xs, label="fourier")
end

# ╔═╡ 3f880785-c0ef-447c-b5ee-db258fc9db16
"Linear recurrence of the form ``x_{n+D} = \\sum_{k=0}^{D-1} c_k x_{n+k}``"
struct LinearRecurrence{D} <: VectorSpace{Vector, D, Float64}
	coefs::SVector{D, <:Number}
	roots::SVector{D, <:Number}  # roots of characteristic polynomial

	function LinearRecurrence(coefs...)
		D = length(coefs)
		v = collect(coefs)
		m = vcat(v', I(D)[1:end-1, :])
		new{D}(v, eigvals(m))
	end
end

# ╔═╡ 6b847bf5-87df-4e87-8b1c-fb9da5da0d54
basis(lr::LinearRecurrence{D}) where {D} = hcat([r .^ (1:D) for r in lr.roots]...)

# ╔═╡ 8e4eb2c7-20d3-4aa0-8c09-34fdb8293a9c
begin
	import LinearAlgebra: dot  # to overload the '⋅' operator

	dot(A::StaticMatrix, x::StaticVector) = A * x
	dot(u::Vect, v::Vect) = 
		sum(a * (u ⋅ x) for (a, x) in zip(v.coefs, basis(v.space)))
	dot(u::Vect, x) = sum(a * (x ⋅ y) for (a, y) in zip(u.coefs, basis(u.space)))
	dot(f::Func, g::Func) = dot(f, g.exp)
	dot(f::Func{L,H}, g) where {L,H} =
		integrate(SymPy.simplify(real(f.exp * conj(g))), (f.var, L, H))
	dot(x, u::Union{Vect, Func}) = dot(u, x)

	norm(v) = sqrt(v ⋅ v)
	proj(u, v) = (u ⋅ v) / (v ⋅ v) * v

	function orthogonalize(s::V) where {T,D,F,V<:InnerProdSpace{T,D,F}}
		new_basis = [basis(s, 1)]
		for i in 2:D
			u = basis(s, i)
			push!(new_basis, u - sum(proj(u, v) for v in new_basis))
		end
		new_basis ./ norm.(new_basis)
	end
end

# ╔═╡ 0462c099-18ee-415d-b06a-7bb8eafa360f
let
	m = hcat(orthogonalize(v.space)...)
	w = Vect([3, -4], basis=@show m)
	norm(w)
end

# ╔═╡ 331ceecb-53e4-41f9-b8de-2aece6d1eea8
begin
	p1 = @show PolySpace(6, (-2, 2))
	v1 = p1[1:7]
	s1 = FuncSpace(orthogonalize(p1))
	u1 = proj(v1, s1)
	@show vec(u1).evalf()
	u1
end

# ╔═╡ 25319673-99ff-436d-bf5d-34c540f4faf1
begin
	p2 = @show FourierSpace(6, (-2, 2))
	v2 = p2[0, 1, -1, 2, -2, 3, -3]
	s2 = FuncSpace(orthogonalize(p2))
	u2 = proj(v2, s2)
	@show vec(u2).evalf()
	u2
end

# ╔═╡ 9510e0c5-c658-4358-871f-3c433a8f183d
begin
	Base.getindex(x::Vect{<:Any,<:Number,<:LinearRecurrence}, i::Integer) = 
		(x.space.roots .^ i) ⋅ x.coefs
	Base.getindex(x::Vect{<:Any,<:Number,<:LinearRecurrence}, i) = (j -> x[j]).(i)
end

# ╔═╡ 907a813d-945d-40b2-9262-cc2975b8a436
let R = LinearRecurrence(1, 1)  # x[n+2] = x[n+1] + x[n]
	@show basis(R)
	x = R[1, 1]
	@show x[1:5]
	fib = proj([1, 1], R)
	@show fib.coefs
	@time fib[1:32]
end

# ╔═╡ 2eded938-c85c-4bf4-95aa-8d8b02c6cbbb
"Solutions of homogeneous ODE of the form ``\\sum_{k=0}^Dc_k\\frac{d^k}{dt^k}x(t)=0``"
struct ODESolutions{D} <: VectorSpace{Vector, D, Float64}
	eq::Vector
	basis::Vector
	var::Sym

	function ODESolutions(coefs...; var=symbols(:t, real=true))
		v = collect(coefs)
		ord = length(v) - 1
		eq = dot(v, var .^ (0:ord))
		new{ord}(v, exp.(solve(eq, var) .* var), var)
	end
end

# ╔═╡ fb1ad4f2-5b02-4080-bb0f-94a1bf918562
function Base.getindex(ode::ODESolutions, inivals::Pair...)
	ts, xs = collect.(zip(inivals...))
	A = [subs(b, ode.var => t) for t in ts, b in basis(ode)]
	ode[A \ xs]
end

# ╔═╡ b3ab790a-e748-4634-816a-ed7acd3f31d3
md"# Type Hierarchy of Vector Spaces"

# ╔═╡ cd139cca-0b06-40da-b2b7-f49e3b3bcd52
let ode = ODESolutions(1, 2, -3)  # x + 2x' - 3x'' = 0
	x = vec(ode[0=>2, 1=>1])
	D = SymPy.Differential(ode.var)
	@show x + 2D(x) - 3(D^2)(x)
	x(0).evalf(), x(1).evalf()
end

# ╔═╡ 9e4e4f8d-1d34-4c35-b814-b8d6708bf2ab
macro show_all(block)
	if block.head == Symbol("let")
		lines = block.args[2].args
	elseif block.head == Symbol("block")
		lines = block.args
	else
		return block
	end
	foreach(enumerate(lines)) do (i, ex)
		if ex isa Union{Symbol, Expr, Number}
			lines[i] = :(@show $ex)
		end
	end
	return block
end

# ╔═╡ 672fb2ff-5782-4411-85d0-ca83506372c8
begin
	almost(text) = Markdown.MD(Markdown.Admonition("warning", "Almost there!", [text]))
	still_missing(text=md"Replace `missing` with your answer.") = Markdown.MD(Markdown.Admonition("warning", "Here we go!", [text]))
	keep_working(text=md"The answer is not quite right.") = Markdown.MD(Markdown.Admonition("danger", "Keep working on it!", [text]))
	yays = [md"Fantastic!", md"Splendid!", md"Great!", md"Yay ❤", md"Great! 🎉", md"Well done!", md"Keep it up!", md"Good job!", md"Awesome!", md"You got the right answer!", md"Let's move on to the next section."]
	correct(text=rand(yays)) = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]))
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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