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implicit_function_theorem.jl
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implicit_function_theorem.jl
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### A Pluto.jl notebook ###
# v0.19.25
using Markdown
using InteractiveUtils
# ╔═╡ 5cf8e83f-6f01-4733-b13f-38976603301b
begin
using OrdinaryDiffEq
using DiffEqBase
using ImplicitEquations
using ForwardDiff
using Plots
end
# ╔═╡ 560977e3-09ed-48d1-9f3e-1adc999127ec
md"""
# 陰関数定理と平面曲線の可視化
[永井保成. 代数幾何学入門 代数学の基礎を出発点として. 森北出版](https://www.morikita.co.jp/books/mid/007841) の題材をもとに後述する陰関数定理から得られる微分方程式を解く. 解いて得られた解を描画して陰関数定理の気持ちを感じることにする.
"""
# ╔═╡ 6bb832d2-988f-4afc-ab95-2487d679cd52
md"""
２ 変数多項式 $f=f(x, y)$ を次で定める:
```math
f(x, y) = x^3 + x^2 - y^2
```
そして $f(x, y) = 0$ を満たす点 $(x, y)$ からなる集合を次で定める:
```math
C_f = \{(x,y) \mid f(x, y) = 0\}
```
この集合は代数幾何学の分野では平面曲線と呼ばれる. Julia だと [jverzani/ImplicitEquations.jl](https://github.com/jverzani/ImplicitEquations.jl) パッケージを使ってこの集合を描画もできる.
"""
# ╔═╡ 8140ad0c-26a1-42af-801e-fb6703151470
begin
f(x, y) = x^3 + x^2 - y^2
f(xy::AbstractVector) = f(xy[begin], xy[end])
end
# ╔═╡ 4d4f3e61-2e5f-4759-9b0f-0cac36441c60
plot(Eq(f, 0), aspect_ratio=:equal, xlim=[-1.5, 1], ylim=[-1, 1])
# ╔═╡ d5140e42-b735-4e4d-97ac-97ed8f97cd42
md"""
この方法 （単純にパッケージを使う） でももちろん良いが, 私が [SciML/OrdinaryDiffEq.jl](https://github.com/SciML/OrdinaryDiffEq.jl) を遊んでいる関係でここでは陰関数定理から定まる微分方程式を局所的に解いてそれら張り合わせて描画するアプローチをとる.
## 陰関数定理
平面曲線 $C_f$ 上の点 $(a, b) \in C_f$ の点で $f$ の $y$ に対する偏微分が $0$ でないとする:
```math
\left.\frac{\partial f}{\partial y} \right|_{(x, y) = (a, b)} \neq 0.
```
このとき陰関数定理によって, $a$ の近傍 $U = (a-\varepsilon, a + \varepsilon) \subset \mathbb{R}$ と $U$ 上の関数 $\varphi=\varphi(x)$ が存在し次が成り立つ:
```math
f(x, \varphi(x)) = 0 \textrm{ for all } x \in U.
```
これは簡単に言えば $f$ が局所的に $y = \varphi(x)$ と表記できることを述べている.
関数 $F(x) = f(x, \varphi(x))$ は $U$ 上で恒等的に $0$ であるので $F$ を微分した結果も $0$ になる:
```math
0 = \frac{dF}{dx}(x) = \frac{\partial f}{\partial x}(x, \varphi(x)) + \frac{\partial f}{\partial y}(x, \varphi(x)) \frac{d\varphi}{dx}(x)
```
ここから
```math
\varphi'(x) = \frac{d\varphi}{dx}(x) = - \frac{f_x(x, \varphi(x))}{f_y(x, \varphi(x))}
```
となる. ここで $f_x$, $f_y$ は各々 $x$, $y$ に関しての $f$ の偏導関数とした.
一方, 点 $(a, b)$ を通る傾き $m$ の直線は
```math
y = m (x - a) + b
```
と書けたので, $(a, b) \in C_f$ を通る $C_f$ の接線は $m=\varphi'(x)$ とおくことで
```math
\frac{\partial f}{\partial x}(a,b) (x - a) +
\frac{\partial f}{\partial y}(a,b) (y - b) = 0
```
となることがわかる. これはいわゆる接線の方程式である.
## 今回の例では
多項式函数 $f=f(x, y) = x^3 + x^2 - y^2$ の $y$ に関する偏導関数を考える:
```math
\frac{\partial f}{\partial y} = 2y
```
陰関数定理によって $b = 0$ 以外の点 $(a, b) \in C_f$ では局所的に $y = \varphi(x)$ と表示できるはずだ.
天下りではあるが $x > -1$ に対して
```math
\begin{aligned}
\varphi_{+}(x) &= x\sqrt{x + 1}, \\
\varphi_{-}(x) &= -x\sqrt{x + 1}
\end{aligned}
```
と定義する. これは方程式 $f(x, y) = 0$ は
```math
y^2 = x^2(x + 1)
```
のように変形できるので $f(x, \varphi_{\pm}(x)) = 0$ であることに注意する.
いやいや, $\varphi_{\pm}$ があればほぼ解析は終わってるのでは？と思うかもしれない. まぁそれはおっしゃる通りなのだが， 微分方程式
```math
\varphi'(x) = \frac{d\varphi}{dx}(x) = - \frac{f_x(x, \varphi(x))}{f_y(x, \varphi(x))}
```
を解いて陰関数定理を確かめたいのである.
"""
# ╔═╡ c0073985-b610-404a-b4b3-fc59b72cbc3e
md"""
# コード
もちろん多項式 $f$ の導関数は計算できるのだが， 私は怠惰な人間なので微分計算は [JuliaDiff/ForwardDiff.jl](https://github.com/JuliaDiff/ForwardDiff.jl) に任せるとする.
"""
# ╔═╡ a5a9597d-9270-49ac-8824-d2bfb1012e24
begin
∇f(xy::AbstractVector) = ForwardDiff.gradient(f, xy)
∇f(x, y) = ∇f([x, y])
end
# ╔═╡ 62896202-2e3e-4921-ae4d-adabfd3cb740
md"""
上記のコードによって Julia の関数を与えることがわかる. `∇f` をの使い方は下記を参考にすると良い. また， 怠惰ではあるが私はちゃんと微分計算ができうことを読者は理解してくれるであろう.
"""
# ╔═╡ e2f69e55-e828-4d90-b175-d6b1ed583ea3
let
x = rand()
y = rand()
@assert all(∇f([x, y]) .≈ (3x^2 + 2x, -2y))
@assert all(∇f(x, y) .≈ (3x^2 + 2x, -2y))
end
# ╔═╡ 1290a6fa-978f-4b62-8370-1fac74d01114
md"""
陰関数定理から得られる微分方程式の初期値のために Julia としての関数を定義する.
"""
# ╔═╡ 4a5c4e81-51df-4872-82be-785039201ba8
begin
φ₊(x) = x√(x + 1)
φ₋(x) = -x√(x + 1)
end
# ╔═╡ 67fd6c41-1861-47c0-acdb-10a7b8ff3573
md"""
微分方程式
```math
\varphi'(x) = \frac{d\varphi}{dx}(x) = - \frac{f_x(x, \varphi(x))}{f_y(x, \varphi(x))}
```
は次のように Julia の関数として記述できる. 戻り値が $\varphi'(x)$ を計算していると思えば良い.
"""
# ╔═╡ a64a2b9c-743e-4f96-a0fb-7fdd313a8fd5
function update(φ, ∇f, x)
fx, fy = ∇f(x, φ)
-fx/fy
end
# ╔═╡ 624baeb4-33de-41aa-8003-1d3198e739ff
md"""
`update` 関数, $\varphi$ の初期値 `φ_init`, `x` に関しての微分方程式の解く範囲 `xspan` を指定する. `∇f` は関数オブジェクトであるが， を微分方程式のパラメータとしてみなし解くことができる.
```julia
prob = ODEProblem(update, φ_init, xspan, ∇f)
sol = solve(prob, Tsit5())
```
コードは次の通り:
"""
# ╔═╡ 58d1ad33-72a4-41ce-a509-648a3c236632
begin
p = plot(title="\$x^3 + x^2 - y^2 = 0\$", aspect_ratio=:equal)
x_init = -0.9999
x_final = -0.001
φ_init = φ₊(x_init)
xspan = (x_init, x_final)
prob = ODEProblem(update, φ_init, xspan, ∇f)
sol₊1 = solve(prob, Tsit5())
plot!(p, x_init:0.01:x_final, sol₊1.(x_init:0.01:x_final), label="sol₊1")
x_init = 0.001
x_final = 1.
φ_init = φ₊(x_init)
xspan = (x_init, x_final)
prob = ODEProblem(update, φ_init, xspan, ∇f)
sol₊2 = solve(prob, Tsit5())
plot!(p, x_init:0.01:x_final, sol₊2.(x_init:0.01:x_final), label="sol₊2")
x_init = -0.9999
x_final = -0.001
φ_init = φ₋(x_init)
xspan = (x_init, x_final)
prob = ODEProblem(update, φ_init, xspan, ∇f)
sol₋1 = solve(prob, Tsit5())
plot!(p, x_init:0.01:x_final, sol₋1.(x_init:0.01:x_final), label="sol₋1")
x_init = 0.001
x_final = 1.
φ_init = φ₋(x_init)
xspan = (x_init, x_final)
prob = ODEProblem(update, φ_init, xspan, ∇f)
sol₋2 = solve(prob, Tsit5())
plot!(p, x_init:0.01:x_final, sol₋2.(x_init:0.01:x_final), label="sol₋2")
scatter!(p, [0],[0], markershape=:+, label="singular point")
p
end
# ╔═╡ 6d7c4869-201d-497f-846c-80ea29a655ce
md"""
結果は次のようになる
"""
# ╔═╡ a6dce9bd-7cab-4318-902f-89086eab766e
p
# ╔═╡ 614a96b7-53ec-48d7-85d3-be9ee0b2e288
md"""
# 解のマージ
Julia の Discourse にある議論から `DiffEqBase.build_solution` を使えば良いらしい．
- https://discourse.julialang.org/t/joining-two-ode-solutions/63710/5
下記の構文に従って作成すれば良さそう.
```julia
DiffEqBase.build_solution(prob, alg, sol.t, sol.u)
```
prob は sol を構成する際に用いた ODEProblem オブジェクトであれば良い．これらは plot を用いた描画の際に行う interpolation のために使われるらしい.
- https://discourse.julialang.org/t/joining-two-ode-solutions/63710/5
"""
# ╔═╡ 4e497972-532a-4713-a28c-171db38cffcf
begin
plot()
sol1 = sol₊1
sol2 = sol₊2
DiffEqBase.build_solution(
let
x_init = -0.999
x_final = 1.
φ_init = φ₊(x_init)
xspan = (x_init, x_final)
prob = ODEProblem(update, φ_init, xspan, ∇f)
end,
Tsit5(),
vcat(sol1.t, sol2.t),
vcat(sol1.u, sol2.u)
) |> plot!
sol1 = sol₋1
sol2 = sol₋2
DiffEqBase.build_solution(
let
x_init = -0.999
x_final = 1.
φ_init = φ₋(x_init)
xspan = (x_init, x_final)
prob = ODEProblem(update, φ_init, xspan, ∇f)
end,
Tsit5(),
vcat(sol1.t, sol2.t),
vcat(sol1.u, sol2.u)
) |> plot!
end
# ╔═╡ 509ea044-c328-4faf-8bbd-3586d7e176c4
md"""
### 悩み.
４つの領域で得られた解オブジェクトを `vcat` なり `hcat` なりで統合できると嬉しいな...
"""
# ╔═╡ 3f0dbec2-bf3b-4802-98df-8d3b3b121724
md"""
### おまけ
点 $(a, b) = (0, 0)$ では $f_x(a, b) = f_y(a, b) = 0$ となるため各変数に対しての陰関数定理の議論が使えない. このような点 $(a, b)$ を`特異点 (singular point)`と呼ぶ. 今回の例では原点付近で $\varphi_{\pm}$ で定めた関数が交差する形になっているので原点での接線を定めることができない. その意味で "特異な点" になってそうだという気持ちはわかると思う.
"""
# ╔═╡ ab2dd0f7-7292-44dc-b1dd-34648236d84e
md"""
Twitter で教えていただいたので `Plots.countour` を用いた方法も紹介しておく.
https://twitter.com/genkuroki/status/1662913853712379905?s=20
"""
# ╔═╡ 11270a29-3b02-4644-a4f5-d8f215cf021d
let
# @genkuroki
# https://twitter.com/genkuroki/status/1662913853712379905?s=20
# からこのノートブック上で動かすように修正したもの
p = plot(legend=false, aspect_ratio=:equal)
f1(t) = t^2 - 1
g1(t) = t * f1(t)
ts = range(-√2, √2, 1000)
plot!(p, f1.(ts), g1.(ts))
end
# ╔═╡ d965213b-792d-49f3-be9b-c147363e5f61
let
# @genkuroki
# https://twitter.com/genkuroki/status/1662913853712379905?s=20
# からこのノートブック上で動かすように修正したもの
p = plot(legend=false, aspect_ratio=:equal)
xs, ys = -2:0.01:2, -1.4:0.01:1.4
contour(xs, ys, f, levels=[0,0])
end
# ╔═╡ b431cc90-7c0d-422b-9b43-cf7819e5d4b9
md"""
以上
"""
# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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