diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..aae07a8 --- /dev/null +++ b/.gitignore @@ -0,0 +1,162 @@ +# Byte-compiled / optimized / DLL files +__pycache__/ +*.py[cod] +*$py.class + +# C extensions +*.so + +# Distribution / packaging +.Python +build/ +develop-eggs/ +dist/ +downloads/ +eggs/ +.eggs/ +lib/ +lib64/ +parts/ +sdist/ +var/ +wheels/ +share/python-wheels/ +*.egg-info/ +.installed.cfg +*.egg +MANIFEST + +# PyInstaller +# Usually these files are written by a python script from a template +# before PyInstaller builds the exe, so as to inject date/other infos into it. +*.manifest +*.spec + +# Installer logs +pip-log.txt +pip-delete-this-directory.txt + +# Unit test / coverage reports +htmlcov/ +.tox/ +.nox/ +.coverage +.coverage.* +.cache +nosetests.xml +coverage.xml +*.cover +*.py,cover +.hypothesis/ +.pytest_cache/ +cover/ + +# Translations +*.mo +*.pot + +# Django stuff: +*.log +local_settings.py +db.sqlite3 +db.sqlite3-journal + +# Flask stuff: +instance/ +.webassets-cache + +# Scrapy stuff: +.scrapy + +# Sphinx documentation +docs/_build/ + +# PyBuilder +.pybuilder/ +target/ + +# Jupyter Notebook +.ipynb_checkpoints + +# IPython +profile_default/ +ipython_config.py + +# pyenv +# For a library or package, you might want to ignore these files since the code is +# intended to run in multiple environments; otherwise, check them in: +# .python-version + +# pipenv +# According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control. +# However, in case of collaboration, if having platform-specific dependencies or dependencies +# having no cross-platform support, pipenv may install dependencies that don't work, or not +# install all needed dependencies. +#Pipfile.lock + +# poetry +# Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control. +# This is especially recommended for binary packages to ensure reproducibility, and is more +# commonly ignored for libraries. +# https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control +#poetry.lock + +# pdm +# Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control. +#pdm.lock +# pdm stores project-wide configurations in .pdm.toml, but it is recommended to not include it +# in version control. +# https://pdm.fming.dev/#use-with-ide +.pdm.toml + +# PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm +__pypackages__/ + +# Celery stuff +celerybeat-schedule +celerybeat.pid + +# SageMath parsed files +*.sage.py + +# Environments +.env +.venv +env/ +venv/ +ENV/ +env.bak/ +venv.bak/ + +# Spyder project settings +.spyderproject +.spyproject + +# Rope project settings +.ropeproject + +# mkdocs documentation +/site + +# mypy +.mypy_cache/ +.dmypy.json +dmypy.json + +# Pyre type checker +.pyre/ + +# pytype static type analyzer +.pytype/ + +# Cython debug symbols +cython_debug/ + +# PyCharm +# JetBrains specific template is maintained in a separate JetBrains.gitignore that can +# be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore +# and can be added to the global gitignore or merged into this file. For a more nuclear +# option (not recommended) you can uncomment the following to ignore the entire idea folder. +#.idea/ + +default.profraw diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..0f0840c --- /dev/null +++ b/LICENSE @@ -0,0 +1,21 @@ +MIT License + +Copyright (c) 2024 AtelierArith + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all +copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. diff --git a/README.md b/README.md new file mode 100644 index 0000000..be7e379 --- /dev/null +++ b/README.md @@ -0,0 +1,26 @@ +# intro2software_with_julia_pluto_materials + +[計算物理春の学校 2024](https://atelierarith.github.io/intro2software_with_julia/) の Pluto Notebook 置き場 + +# 使い方 + +- Julia をインストールする．Git をインストールする +- 次のようにしてリポジトリをクローン，Pluto.jl をインストールする + +```console +$ git clone https://github.com/AtelierArith/intro2software_with_julia_pluto_materials +$ cd intro2software_with_julia_pluto_materials +$ julia -e 'using Pkg; Pkg.add("Pluto")' +``` + +Pluto.jl は Julia のためのリアクティブなノートブックを提供するパッケージである．次のコマンドを実行すると Pluto Notebook 環境が起動する: + +```console +$ julia -e 'using Pluto; Pluto.run()' +``` + +Pluto.jl 自体のドキュメントは https://github.com/fonsp/Pluto.jl/wiki にある． + +JuliaHub で動かす．何かしらの理由で手元で実行できない場合は JuliaHub を試すと良いかも. + +使い方は https://github.com/AtelierArith/intro2software_with_julia/issues/5 の手順を見ると良い． diff --git a/classical_gram_schmidt.html b/classical_gram_schmidt.html new file mode 100644 index 0000000..753d6e6 --- /dev/null +++ b/classical_gram_schmidt.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/classical_gram_schmidt.jl b/classical_gram_schmidt.jl new file mode 100644 index 0000000..90952f7 --- /dev/null +++ b/classical_gram_schmidt.jl @@ -0,0 +1,181 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 1142a7e1-6d83-4746-a602-c812512f2bc5 +begin + # Extras + using Test + using StableRNGs +end + +# ╔═╡ 6c0552e8-cb9c-11ee-3e7f-3be38a736c2f +md""" +# 古典的グラムシュミットの直交化法 +""" + +# ╔═╡ a5978b5d-e211-4f5d-9c55-6eb4e7af3fc5 +begin + function innerproduct(a::AbstractVector, b::AbstractVector) + s = zero(eltype(a)) + @inbounds for i in eachindex(a, b) + s += a[i] * b[i] + end + s + end + const ⋅ = innerproduct + norm(a::AbstractVector) = √(a ⋅ a) +end + +# ╔═╡ 539319ce-ca4d-4de7-8469-e896b589c690 +begin + @testset "innerproduct" begin + rng = StableRNG(999) + x = rand(rng, 2) + y = rand(rng, 2) + + @test innerproduct(x, y) == innerproduct(y, x) + @test x ⋅ y == y ⋅ x + + @test innerproduct([1, 0], [0, 1]) == 0 + @test innerproduct([0, 1], [1, 0]) == 0 + @test innerproduct([1, 1], [1, 1]) ≈ norm([1, 1])^2 ≈ 2.0 + end + + @testset "norm" begin + @test norm([0.0, 0.0]) == 0 + @test norm([1.0, 1.0]) == √2 + @test norm([1.0, 2.0]) == √5 + @test norm([2.0, 1.0]) == √5 + end + nothing +end + +# ╔═╡ 3ff41189-0b8f-459d-87af-2d927cba6d4c +struct ClassicalGramSchmidt end # Algorithm + +# ╔═╡ 3d7af0be-0cc0-4a04-89a7-2721b12d7e6c +md""" +`q⃗̃ₙ₊₁` というのを変数として利用でき，その入力支援も備わっているのが Julia の良いところ +""" + +# ╔═╡ 24a69ae7-2574-4e35-bb4b-9ccb72b42521 +function orthonormalize(::Type{ClassicalGramSchmidt}, A) + a⃗₁ = A[:, 1] + m = size(A, 2) + r₁₁ = norm(a⃗₁) + Q = similar(A) + R = similar(A, m, m) + q⃗₁ = a⃗₁ / r₁₁ + Q[:, 1] = q⃗₁ + R[1,1] = r₁₁ + for n = 1:(size(A, 2)-1) + a⃗ₙ₊₁ = @view A[:, n+1] + q⃗̃ₙ₊₁ = zero(a⃗ₙ₊₁) + for i = 1:n + q⃗ᵢ = @view Q[:, i] + rᵢ₍ₙ₊₁₎ = q⃗ᵢ ⋅ a⃗ₙ₊₁ + R[i, n+1] = rᵢ₍ₙ₊₁₎ + q⃗̃ₙ₊₁ -= rᵢ₍ₙ₊₁₎ * q⃗ᵢ + end + q⃗̃ₙ₊₁ += a⃗ₙ₊₁ + r₍ₙ₊₁₎₍ₙ₊₁₎ = norm(q⃗̃ₙ₊₁) + R[n+1,n+1] = r₍ₙ₊₁₎₍ₙ₊₁₎ + q⃗ₙ₊₁ = q⃗̃ₙ₊₁ / r₍ₙ₊₁₎₍ₙ₊₁₎ + Q[:, n+1] .= q⃗ₙ₊₁ + end + Q, R +end + +# ╔═╡ dc268e9b-b2ca-4143-abe9-5beb87f4f723 +@testset "ClassicalGramSchmidt" begin + rng = StableRNG(123) + + A = rand(rng, 10, 10) + + Q, R = orthonormalize(ClassicalGramSchmidt, copy(A)) + @test A ≈ Q*R + q1 = Q[:, 1] + q2 = Q[:, 2] + q3 = Q[:, 3] + + @test innerproduct(q1, q1) ≈ 1.0 + @test innerproduct(q2, q2) ≈ 1.0 + @test innerproduct(q3, q3) ≈ 1.0 + + @test abs(innerproduct(q1, q2)) ≈ 0.0 atol = 1e-14 + @test abs(innerproduct(q2, q3)) ≈ 0.0 atol = 1e-14 + @test abs(innerproduct(q1, q3)) ≈ 0.0 atol = 1e-14 + + @test abs(innerproduct(q1, q2)) ≈ 0.0 atol = 1e-14 + @test abs(innerproduct(q2, q3)) ≈ 0.0 atol = 1e-14 + @test abs(innerproduct(q1, q3)) ≈ 0.0 atol = 1e-14 +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3" +Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" + +[compat] +StableRNGs = "~1.0.1" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "a4590486d7604ab3578478648d997eb50013d5ca" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.InteractiveUtils]] +deps = ["Markdown"] +uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240" + +[[deps.Logging]] +uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" + +[[deps.Markdown]] +deps = ["Base64"] +uuid = "d6f4376e-aef5-505a-96c1-9c027394607a" + +[[deps.Random]] +deps = ["SHA"] +uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[[deps.SHA]] +uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" +version = "0.7.0" + +[[deps.Serialization]] +uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" + +[[deps.StableRNGs]] +deps = ["Random", "Test"] +git-tree-sha1 = "ddc1a7b85e760b5285b50b882fa91e40c603be47" +uuid = "860ef19b-820b-49d6-a774-d7a799459cd3" +version = "1.0.1" + +[[deps.Test]] +deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] +uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" +""" + +# ╔═╡ Cell order: +# ╟─6c0552e8-cb9c-11ee-3e7f-3be38a736c2f +# ╠═1142a7e1-6d83-4746-a602-c812512f2bc5 +# ╠═a5978b5d-e211-4f5d-9c55-6eb4e7af3fc5 +# ╠═539319ce-ca4d-4de7-8469-e896b589c690 +# ╠═3ff41189-0b8f-459d-87af-2d927cba6d4c +# ╟─3d7af0be-0cc0-4a04-89a7-2721b12d7e6c +# ╠═24a69ae7-2574-4e35-bb4b-9ccb72b42521 +# ╠═dc268e9b-b2ca-4143-abe9-5beb87f4f723 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/classical_gram_schmidt.plutostate b/classical_gram_schmidt.plutostate new file mode 100644 index 0000000..0aa49fd Binary files /dev/null and b/classical_gram_schmidt.plutostate differ diff --git a/hello.html b/hello.html new file mode 100644 index 0000000..519bfca --- /dev/null +++ b/hello.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/hello.jl b/hello.jl new file mode 100644 index 0000000..7f4bb12 --- /dev/null +++ b/hello.jl @@ -0,0 +1,186 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 5be67619-2fa6-48e7-b8e4-492360b6c9c0 +# Example というパッケージをロードする +using Example + +# ╔═╡ bdb66039-1c23-4172-afef-ffab8452ddf6 +begin + # テストコードを記述するために必要なパッケージ + using Test +end + +# ╔═╡ d7f1cb15-6ab1-4fd6-8639-e20ded4f29ea +md""" +# Hello world +""" + +# ╔═╡ 408ea404-a897-4574-97e0-5cab52b9d154 +versioninfo() + +# ╔═╡ 04c275d6-ddd3-11ee-0102-4ff69f77eaac +println("Hello") + +# ╔═╡ aeed025d-ce2d-46ca-be64-3aea35e51932 +md""" +`using Example` によって `hello` という関数が利用できる +""" + +# ╔═╡ ee932197-1a89-4ada-b5eb-6e0514c2f47c +hello("YourName") # ここを変更して遊んでみよう + +# ╔═╡ 4c482f7b-0f75-4b77-ad79-62ce700d5834 +md""" +`Example.hello` のようにして `hello` が属するパッケージ名を明示しても良い +""" + +# ╔═╡ f853d44f-cb09-4fd2-8afa-ea89821c829b +md""" +`using Example: hello` のようにすると `hello` 関数をどこから持ってきたのかを他の人にもわかるように実装できる． + +Python ユーザのように名前空間を大事にしたい, 慎重に管理したい場合は下記のようにしてもよかろう: + +```julia +using Example: Example # Example から Example のみをロードする +Example.hello # このような方法でしか `hello` を使用できない +``` + +この場合 `hello("Name")` のように直接使うことはできない． また `using Example: Example` は `Example` を二回用いてるので記述がやや冗長になる． この冗長さを回避するために Julia は `import Example` という書き方も許している: + +```julia +import Example +Example.hello("Name") +``` +""" + +# ╔═╡ b5c5f82b-5cd6-4b71-8b9f-d4b9715d21b7 +begin + x = 2 + y = 3 +end + +# ╔═╡ a0f3c104-da2d-4540-ae62-84a9f2dee111 +z = 5 # 上のセルで定義された x, y の変数を変更すると z も更新されることに注意 + +# ╔═╡ ff5b2387-b22b-423d-8b4c-e99a085f5293 +""" + Point + +二次元空間の点を表すオブジェクト +""" +struct Point2D + x::Float64 + y::Float64 +end + +# ╔═╡ 44641fc0-7c8c-46f6-be51-438be0ed742d +""" + distance(p::Point2D, q::Poin2D) + +点 `p`, `q` の間の距離を計算する +""" +distance(p::Point2D, q::Point2D) = √((p.x - q.x) ^ 2 + (p.y - q.y) ^ 2) + +# ╔═╡ e2b487c7-7d56-4cb1-b50d-a7d58d5e5d5c +md""" +下記のようにテストを書くことができる． +""" + +# ╔═╡ e8a37180-4777-4087-9155-4a3c37530d54 +@testset "Point2D" begin + o = Point2D(0, 0) + @test o.x == 0. + @test o.y == 0. + p = Point2D(1, 2) + @test p.x == 1 + @test p.y == 2 + d1 = distance(p, o) + d2 = distance(o, p) + @test d1 == d2 ≈ √5 +end + +# ╔═╡ e10f9104-9905-484a-9144-18d9467ec743 +md""" +## 練習問題 + +- `distance` 関数があるセルを編集し敢えて間違った実装に改変してみよ． +- このとき，上記のテストが正常に動かなくなることを確認せよ +- Point2D の実装を元に3次元版を実装してみよ +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Example = "7876af07-990d-54b4-ab0e-23690620f79a" +Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" + +[compat] +Example = "~0.5.3" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "49929da40b6ade982f27b790233147f997c9e104" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.Example]] +git-tree-sha1 = "46e44e869b4d90b96bd8ed1fdcf32244fddfb6cc" +uuid = "7876af07-990d-54b4-ab0e-23690620f79a" +version = "0.5.3" + +[[deps.InteractiveUtils]] +deps = ["Markdown"] +uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240" + +[[deps.Logging]] +uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" + +[[deps.Markdown]] +deps = ["Base64"] +uuid = "d6f4376e-aef5-505a-96c1-9c027394607a" + +[[deps.Random]] +deps = ["SHA"] +uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[[deps.SHA]] +uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" +version = "0.7.0" + +[[deps.Serialization]] +uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" + +[[deps.Test]] +deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] +uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" +""" + +# ╔═╡ Cell order: +# ╟─d7f1cb15-6ab1-4fd6-8639-e20ded4f29ea +# ╠═5be67619-2fa6-48e7-b8e4-492360b6c9c0 +# ╠═bdb66039-1c23-4172-afef-ffab8452ddf6 +# ╠═408ea404-a897-4574-97e0-5cab52b9d154 +# ╠═04c275d6-ddd3-11ee-0102-4ff69f77eaac +# ╟─aeed025d-ce2d-46ca-be64-3aea35e51932 +# ╠═ee932197-1a89-4ada-b5eb-6e0514c2f47c +# ╟─4c482f7b-0f75-4b77-ad79-62ce700d5834 +# ╟─f853d44f-cb09-4fd2-8afa-ea89821c829b +# ╠═b5c5f82b-5cd6-4b71-8b9f-d4b9715d21b7 +# ╠═a0f3c104-da2d-4540-ae62-84a9f2dee111 +# ╠═ff5b2387-b22b-423d-8b4c-e99a085f5293 +# ╠═44641fc0-7c8c-46f6-be51-438be0ed742d +# ╟─e2b487c7-7d56-4cb1-b50d-a7d58d5e5d5c +# ╠═e8a37180-4777-4087-9155-4a3c37530d54 +# ╟─e10f9104-9905-484a-9144-18d9467ec743 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/hello.plutostate b/hello.plutostate new file mode 100644 index 0000000..58f98c7 Binary files /dev/null and b/hello.plutostate differ diff --git a/ifs_fern.html b/ifs_fern.html new file mode 100644 index 0000000..3576139 --- /dev/null +++ b/ifs_fern.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/ifs_fern.jl b/ifs_fern.jl new file mode 100644 index 0000000..7a029a6 --- /dev/null +++ b/ifs_fern.jl @@ -0,0 +1,1512 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 1575da66-69bc-404b-8b59-ef262f04da74 +begin + # パッケージのロード + using Random # 標準ライブラリ + + using Plots + using StaticArrays + using Distributions: Categorical +end + +# ╔═╡ a0722935-2638-4761-a3a3-7d48e79af51b +begin + # 開発用のためのパッケージ + using PlutoUI + using Test + using BenchmarkTools +end + +# ╔═╡ 1dcfa0dd-4763-42a5-8f42-bdc219f8dc12 +md""" +# 反復関数系に親しむ場 + +- [バーンズリーのシダ](https://ja.wikipedia.org/wiki/%E3%83%90%E3%83%BC%E3%83%B3%E3%82%BA%E3%83%AA%E3%83%BC%E3%81%AE%E3%82%B7%E3%83%80) を反復関数系を使って描画する．反復関数系を定義するアフィン変換は Wikipedia にて紹介されているパラメータを元に設定している． + +## 設定 + +次のようなアフィン変換 $f_1, f_2, f_3, f_4: \mathbb{R}^2 \to \mathbb{R}^2$ を考える: + +```math +\begin{aligned} + f_1: \begin{bmatrix} + x \\ y + \end{bmatrix} + &\mapsto + \begin{bmatrix} + 0 & 0 \\ + 0 & 0.16 \\ + \end{bmatrix} + \begin{bmatrix} + x \\ y + \end{bmatrix} \\ + f_2: \begin{bmatrix} + x \\ y + \end{bmatrix} + &\mapsto + \begin{bmatrix} + 0.85 & 0.04 \\ + -0.04 & 0.85 \\ + \end{bmatrix} + \begin{bmatrix} + x \\ y + \end{bmatrix} + + + \begin{bmatrix} + 0 \\ 1.6 + \end{bmatrix} \\ + f_3: \begin{bmatrix} + x \\ y + \end{bmatrix} + &\mapsto + \begin{bmatrix} + 0.20 & -0.26 \\ + 0.23 & 0.22 \\ + \end{bmatrix} + \begin{bmatrix} + x \\ y + \end{bmatrix} + + + \begin{bmatrix} + 0 \\ 1.6 + \end{bmatrix} \\ + f_4: \begin{bmatrix} + x \\ y + \end{bmatrix} + &\mapsto + \begin{bmatrix} + -0.15 & 0.28 \\ + 0.26 & 0.24 \\ + \end{bmatrix} + \begin{bmatrix} + x \\ y + \end{bmatrix} + + + \begin{bmatrix} + 0 \\ 0.44 + \end{bmatrix} +\end{aligned} +``` + +アフィン変換を特徴づける行列，ベクトルの値は Wikipedia の説明にある値を参考にしている. $p_1=(x_1, y_1)=(0, 0) \in \mathbb{R}^2$ と置く．次の手続きによって点列 $p_2, p_3, \dots ,p_n, \dots$ を構成する: + +写像 $f$ として $f_1, f_2, f_3, f_4$ からランダムに選ぶ. ここで選び方は一様ではなく次の重みで選ぶ. $f_1$ は 1% の確率, $f_2$ は 85% の確率, $f_3$ は 7% の確率, $f_4$ は 7% の確率. $n$ ステップ目の点 $p_n$ が与えられたとき， + +```math +p_{n+1} = f(p_n) +``` + +によって $n+1$ 番目の点 $p_{n+1}$ を定義する． + +これを繰り返す． 得られた点を可視化ライブラリで表示してみよう． +""" + +# ╔═╡ 0c0e55c3-50d4-4555-bd38-4d0f717207a2 +md""" +Python が好きな人は↓をみると良い +https://gist.github.com/terasakisatoshi/29c9f0c468e95db0e392d49c3dedffa8 +""" + +# ╔═╡ c8a0a22c-e855-4465-a6e2-291574ca0e33 +md""" +## 実装 +""" + +# ╔═╡ 460d9fa8-4c9e-4900-8768-3529db774ed4 +md""" +## Affine 変換の実装 +""" + +# ╔═╡ 4ace389a-173a-11ee-0955-c1350a6f0427 +begin + struct Affine + W + b + end + + # このようにすると `aff::Affine` に対して `aff(x)` というように関数と似たような振る舞いをすることができる + # Callable object と呼ぶ + # https://docs.julialang.org/en/v1/manual/methods/#Function-like-objects + (aff::Affine)(x⃗::AbstractVector) = aff.W * x⃗ + aff.b + # 後でわかる（はず)だが， フィールド `W`, `b` に対して型アノテーションがないので型不安定な実装になっている． +end + +# ╔═╡ a565d903-8792-4643-ad30-d286e183a63b +md""" + +Python だと次の実装に対応する + +```python +import numpy as np +import sympy + +class Affine(): + def __init__(self, W, b): + self.W = W + self.b = b + def __call__(self, x): + return self.W @ x + self.b + +(W11, W12, W21, W22) = sympy.symbols("W11 W12 W21 W22") +(b1, b2) = sympy.symbols("b1 b2") +(x1, x2) = sympy.symbols("x1 x2") + +W = np.array([ + [W11, W12], + [W21, W22] +]) + +b = np.array([b1, b2]) +x = np.array([x1, x2]) + +aff = Affine(W, b) +y = aff(x) + +assert y[0] == W11 * x1 + W12 * x2 + b1 +assert y[1] == W21 * x1 + W22 * x2 + b2 +``` +""" + +# ╔═╡ 911d8378-df85-4652-90d3-388e36aea42b +begin + transforms = ( + Affine([0. 0.; 0 0.16], [0., 0.]), + Affine([0.85 0.04; -0.04 0.85], [0., 1.6]), + Affine([0.2 -0.26; 0.23 0.22], [0., 1.6]), + Affine([-0.15 0.28; 0.26 0.24], [0., 0.44]), + ) +end + +# ╔═╡ 8aa7498a-552a-44e4-8a62-01457b46594c +md""" +## カテゴリカル分布の作成 + +与えられた確率に従ってランダムに $f_1, f_2, f_3, f_4$ を選ぶ必要がある. これはカテゴリカル分布そのものである. +""" + +# ╔═╡ a9940972-7ad0-4317-89fc-2f36f7e2fe2f +catdist = Categorical([0.01,0.85,0.07,0.07]) + +# ╔═╡ 061db183-d77d-4bd0-aa16-7c74a760d97c +md""" +試しに `catdist` オブジェクトが所望の分布を作るかを確認してみよう. 10000 個のサンプルを生成する: +""" + +# ╔═╡ 6c290663-03a7-440a-8659-f97f73c86921 +let + nsamples = 100000 + rng = Xoshiro(123) + indices = rand(rng, catdist, nsamples) + histogram(indices, normalize=:pdf, nbins=4, ylabel="probabiity") + for i in sort(unique(indices)) + p = sum(indices .== i)/nsamples + println("$i が選ばれる確率は $(100p)% です") + end + #= + 下記のような結果を得る + 1 が選ばれる確率は 1.035% です + 2 が選ばれる確率は 84.90299999999999% です + 3 が選ばれる確率は 7.013% です + 4 が選ばれる確率は 7.0489999999999995% です + =# +end + +# ╔═╡ 720f2315-6402-4020-8701-8605477407ca +md""" +## 点列の作成 + +```math +p_{n+1} = f(p_n) +``` + +の箇所を実装してみよう. 素直に書くと次のようになる: +""" + +# ╔═╡ d3533d36-5f4e-4d70-8503-ebf590516d43 +function generate_points(tfms, catdist) + N = 100000 + xs = [] + ys = [] + + p = [0, 0] # p_1 + for i in 1:N + aff = tfms[rand(catdist)] + p = aff(p) + x = p[1] + y = p[2] + push!(xs, x) + push!(ys, y) + end + return xs, ys +end + +# ╔═╡ 927ce37f-53f3-4002-aef8-c4b9d7237baa +md""" +## 描画しよう + +`scatter` で点列を二次元平面に描画できる +""" + +# ╔═╡ c3e59e97-0ef7-409b-afe9-e123cc92770a +let + xs, ys = generate_points(transforms, catdist) + scatter( + xs, ys, + aspect_ratio=:equal, label=:none, xlim=[-3, 3], + axis=false, ticks=false + ) +end + +# ╔═╡ 0a42bdf5-7c14-470d-b772-8b4bcbf9b9e0 +md""" +## パフォーマンス計測 + +`generate_points(tfms, d)` にかかる時間を計測する `tfms` や `d` をセットアップする時間は含めないように `setup = begin ... end` の部分で初期化する処理を書いている． + +```julia +generate_points($transforms, $catdist) +``` + +のようにして `$` で補間しても良いが，ついつい忘れがちになるので `setup=begin ... end` の方が良いように思える． +""" + +# ╔═╡ 22dd99ac-4b53-4664-8ec7-9a7ce852ff9b +@benchmark generate_points(tfms, d) samples=100 setup = begin + tfms = transforms + d = catdist +end + +# ╔═╡ 8de58d64-534a-4691-885f-0f2c9a418bee +md""" +手元の環境では概ね 40 ms 程度かかっている. 実は `Affine` や `generate_points` の実装が効率の良い書き方になっていないので Julia が本来持っているパフォーマンスを引き出せていない． 「Julia は思ったよりもパフォーマンスがでない」というのはこのような状況に陥っている場合がほとんどである． + +高速な実装を求める(計算物理の文脈だとほとんどがそうだと思うが)場合は Julia 公式の [Performance Tips](https://docs.julialang.org/en/v1/manual/performance-tips/#man-performance-tips) を読むべきである. + +とはいえ全部読むのは大変なので + +- [Avoid untyped global variables](https://docs.julialang.org/en/v1/manual/performance-tips/#Avoid-untyped-global-variables) +- [Avoid fields with abstract type](https://docs.julialang.org/en/v1/manual/performance-tips/#Avoid-fields-with-abstract-type) +- [Pre-allocating outputs](https://docs.julialang.org/en/v1/manual/performance-tips/#Pre-allocating-outputs) +- [Measure performance with @time and pay attention to memory allocation](https://docs.julialang.org/en/v1/manual/performance-tips/#Measure-performance-with-[@time](@ref)-and-pay-attention-to-memory-allocation) + +を眺めながら色々試行錯誤して欲しい． 今回のような場合には固定長配列をサポートする [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl) を用いることを検討するべきである． StaticArrays.jl は日本人も含めた多くの人によってメンテナンスされている主要なパッケージの一つである． +""" + +# ╔═╡ bbf33a1f-42f3-439c-be61-6815cc775a1c +md""" +## 発展的な内容 + +- 今回実装した Affine 変換の実行速度の改善する方法は [数学と物理におけるJuliaの活用](https://joint1.imi.kyushu-u.ac.jp/research_chooses/view/2023a015) で作成していた[スライド](https://atelierarith.github.io/julia_tutorial/slideshow/tutorial_part1/build/#55) にヒント・答えが書いてある． 読むのすら時間が惜しい方は [こちら](https://atelierarith.github.io/intro2software_with_julia_pluto_materials/ifs_revised.html) のノートブックを試すと良い. なんにせよ．読者の演習問題とする． + +- Affine 変換パラメータを変えるとそれに応じて模様が変化する． [Lévy C curve](https://atelierarith.github.io/intro2software_with_julia_pluto_materials/ifs_l%C3%A9vy.html) も有名な例である． +- さらに機械的に多様な幾何学模様を生成するアルゴリズムも知られている． [random_logos](https://atelierarith.github.io/intro2software_with_julia_pluto_materials/random_logos.html) +""" + +# ╔═╡ 9ef49885-6c83-4334-94ab-2a1c18a9e39d + + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +StaticArrays = "90137ffa-7385-5640-81b9-e52037218182" +Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" + +[compat] +BenchmarkTools = "~1.3.2" +Distributions = "~0.25.97" +Plots = "~1.38.16" +PlutoUI = "~0.7.58" +StaticArrays = "~1.5.26" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "b01717bc4dd68ed52355957e98178982a6f055b6" + +[[deps.AbstractPlutoDingetjes]] +deps = ["Pkg"] +git-tree-sha1 = "0f748c81756f2e5e6854298f11ad8b2dfae6911a" +uuid = "6e696c72-6542-2067-7265-42206c756150" +version = "1.3.0" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + 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\ No newline at end of file diff --git "a/ifs_l\303\251vy.jl" "b/ifs_l\303\251vy.jl" new file mode 100644 index 0000000..c2a5911 --- /dev/null +++ "b/ifs_l\303\251vy.jl" @@ -0,0 +1,1261 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 1575da66-69bc-404b-8b59-ef262f04da74 +begin + using Plots + using StaticArrays + using ConcreteStructs + using Distributions +end + +# ╔═╡ a0722935-2638-4761-a3a3-7d48e79af51b +begin + using BenchmarkTools + using ProfileSVG +end + +# ╔═╡ 1dcfa0dd-4763-42a5-8f42-bdc219f8dc12 +md""" +# Lévy C curve + +[Lévy C curve](https://en.wikipedia.org/wiki/L%C3%A9vy_C_curve) を反復関数系 (iterated function system +) で作成する. +""" + +# ╔═╡ 4ace389a-173a-11ee-0955-c1350a6f0427 +begin + @concrete struct Affine + W + b + end + + (aff::Affine)(x) = aff.W * x + aff.b + (aff::Affine)(x, y) = aff(@SVector [x, y]) +end + +# ╔═╡ 8abb0481-e408-4a47-b33f-6686871a45cd +begin + transforms = ( + Affine([0.5 -0.5; 0.5 0.5], [0., 0.]), + Affine([0.5 0.5; -0.5 0.5], [0.5, 0.5]), + ) + catdist = Categorical([0.5, 0.5]) +end + +# ╔═╡ d3533d36-5f4e-4d70-8503-ebf590516d43 +function generate_points(tfms, d) + xs = Float64[] + ys = Float64[] + x = 0. + y = 0. + for i in 1:10000 + aff = tfms[rand(d)] + x, y = aff([x, y]) + push!(xs, x) + push!(ys, y) + end + xs, ys +end + +# ╔═╡ c3e59e97-0ef7-409b-afe9-e123cc92770a +begin + xs, ys = generate_points(transforms, catdist) + scatter(xs, ys, aspect_ratio=:equal, label=:none) +end + +# ╔═╡ 22dd99ac-4b53-4664-8ec7-9a7ce852ff9b +@benchmark generate_points($transforms, $catdist) + +# ╔═╡ 8623c05b-484b-41e1-9ce8-a82621ec7d95 + + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" +ConcreteStructs = "2569d6c7-a4a2-43d3-a901-331e8e4be471" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +ProfileSVG = "132c30aa-f267-4189-9183-c8a63c7e05e6" 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["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 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.7+1" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.52.0+1" + +[[deps.p7zip_jll]] +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 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+0" +""" + +# ╔═╡ Cell order: +# ╟─1dcfa0dd-4763-42a5-8f42-bdc219f8dc12 +# ╠═1575da66-69bc-404b-8b59-ef262f04da74 +# ╠═a0722935-2638-4761-a3a3-7d48e79af51b +# ╠═4ace389a-173a-11ee-0955-c1350a6f0427 +# ╠═8abb0481-e408-4a47-b33f-6686871a45cd +# ╠═d3533d36-5f4e-4d70-8503-ebf590516d43 +# ╠═c3e59e97-0ef7-409b-afe9-e123cc92770a +# ╠═22dd99ac-4b53-4664-8ec7-9a7ce852ff9b +# ╠═8623c05b-484b-41e1-9ce8-a82621ec7d95 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git "a/ifs_l\303\251vy.plutostate" "b/ifs_l\303\251vy.plutostate" new file mode 100644 index 0000000..1757d0e Binary files /dev/null and "b/ifs_l\303\251vy.plutostate" differ diff --git a/ifs_revised.html b/ifs_revised.html new file mode 100644 index 0000000..8ef11b8 --- /dev/null +++ b/ifs_revised.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/ifs_revised.jl b/ifs_revised.jl new file mode 100644 index 0000000..60ee608 --- /dev/null +++ b/ifs_revised.jl @@ -0,0 +1,1387 @@ +### A Pluto.jl notebook ### +# v0.19.38 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═╡ 1575da66-69bc-404b-8b59-ef262f04da74 +begin + using Plots + using StaticArrays + using Distributions +end + +# ╔═╡ 79a3dd1f-8800-4a9c-98c3-fa61ee9c1489 +using PlutoUI + +# ╔═╡ a0722935-2638-4761-a3a3-7d48e79af51b +begin + using BenchmarkTools + using ProfileSVG +end + +# ╔═╡ 1dcfa0dd-4763-42a5-8f42-bdc219f8dc12 +md""" +# 反復関数系に親しむ場 + +下記の項目を反復関数系 (iterated function system +) によって作成する. + +- [バーンズリーのシダ](https://ja.wikipedia.org/wiki/%E3%83%90%E3%83%BC%E3%83%B3%E3%82%BA%E3%83%AA%E3%83%BC%E3%81%AE%E3%82%B7%E3%83%80) + +反復関数系を定義するアフィン変換は Wikipedia にて紹介されているパラメータを元に設定している +""" + +# ╔═╡ ed8b151a-7f16-4528-93d3-55790a13f672 +begin + ui_use_typestable = @bind use_typestable Select([false, true], default=false) + ui_use_staticarrays = @bind use_staticarrays Select([false, true], default=false) + ui_pattern = @bind pattern Select([:Barnsley]) + nothing +end + +# ╔═╡ 4ace389a-173a-11ee-0955-c1350a6f0427 +begin + if use_typestable + struct Affine{T1, T2} + W::T1 + b::T2 + end + else + struct Affine + W + b + end + end + (aff::Affine)(x⃗::AbstractVector) = aff.W * x⃗ + aff.b + if use_staticarrays + (aff::Affine)(x, y) = aff(@SVector [x, y]) + else + (aff::Affine)(x, y) = aff([x, y]) + end +end + +# ╔═╡ 911d8378-df85-4652-90d3-388e36aea42b +pattern2ifs = Dict( + :Barnsley => let + transforms = ( + Affine([0. 0.; 0 0.16], [0., 0.]), + Affine([0.85 0.04; -0.04 0.85], [0., 1.6]), + Affine([0.2 -0.26; 0.23 0.22], [0., 1.6]), + Affine([-0.15 0.28; 0.26 0.24], [0., 0.44]), + ) + catdist = Categorical([0.01,0.85,0.07,0.07]) + (transforms, catdist) + end +) + +# ╔═╡ 8abb0481-e408-4a47-b33f-6686871a45cd +(transforms, catdist) = let + _transforms, _catdist = pattern2ifs[pattern] + if use_staticarrays + transforms = map(_transforms) do t + W = SMatrix{2, 2}(t.W) + b = SVector{2}(t.b) + Affine(W, b) + end + p = SVector{length(_catdist.p)}(_catdist.p) + catdist = Categorical(p) + (transforms, catdist) + else + transforms = _transforms + catdist = _catdist + (transforms, catdist) + end +end + +# ╔═╡ b47ba401-32de-49c5-af0e-24cdc2cef8d5 +typeof(transforms) + +# ╔═╡ d3533d36-5f4e-4d70-8503-ebf590516d43 +begin + function generate_points!(xs, ys, tfms, d) + x = 0. + y = 0. + for i in eachindex(xs, ys) + aff = tfms[rand(d)] + x, y = aff(x, y) + xs[i] = x + ys[i] = y + end + end + + function generate_points(tfms, d) + N = 100000 + xs = Vector{Float64}(undef, N) + ys = Vector{Float64}(undef, N) + generate_points!(xs, ys, tfms, d) + xs, ys + end +end + +# ╔═╡ c3e59e97-0ef7-409b-afe9-e123cc92770a +let + xs, ys = generate_points(transforms, catdist) + scatter( + xs, ys, + aspect_ratio=:equal, label=:none, xlim=[-3, 3], + axis=false, ticks=false + ) +end + +# ╔═╡ 2b5de833-5512-400c-93d5-2d40dcaca7ab +begin + [ + PlutoUI.ExperimentalLayout.hbox(["IFS パターン: ", ui_pattern]), + PlutoUI.ExperimentalLayout.hbox(["型安定: ", ui_use_typestable]), + PlutoUI.ExperimentalLayout.hbox(["StaticArrays を使う: ", ui_use_staticarrays]) + ] |> PlutoUI.ExperimentalLayout.vbox +end + +# ╔═╡ 22dd99ac-4b53-4664-8ec7-9a7ce852ff9b +@benchmark generate_points($transforms, $catdist) samples=100 + +# ╔═╡ 4bdb58fb-e71a-466b-bd99-77b0fda6d6cd +@benchmark generate_points!(xs, ys, tfm, d) samples=100 setup = begin + xs = Vector{Float64}(undef, 100000) + ys = Vector{Float64}(undef, 100000) + tfm = transforms + d = catdist +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +ProfileSVG = "132c30aa-f267-4189-9183-c8a63c7e05e6" +StaticArrays = "90137ffa-7385-5640-81b9-e52037218182" + +[compat] +BenchmarkTools = "~1.3.2" +Distributions = "~0.25.97" +Plots = "~1.38.16" +PlutoUI = "~0.7.51" +ProfileSVG = "~0.2.1" +StaticArrays = "~1.5.26" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 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"Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+0" +""" + +# ╔═╡ Cell order: +# ╟─1dcfa0dd-4763-42a5-8f42-bdc219f8dc12 +# ╠═1575da66-69bc-404b-8b59-ef262f04da74 +# ╠═79a3dd1f-8800-4a9c-98c3-fa61ee9c1489 +# ╠═a0722935-2638-4761-a3a3-7d48e79af51b +# ╠═ed8b151a-7f16-4528-93d3-55790a13f672 +# ╠═4ace389a-173a-11ee-0955-c1350a6f0427 +# ╠═911d8378-df85-4652-90d3-388e36aea42b +# ╠═8abb0481-e408-4a47-b33f-6686871a45cd +# ╠═b47ba401-32de-49c5-af0e-24cdc2cef8d5 +# ╠═d3533d36-5f4e-4d70-8503-ebf590516d43 +# ╠═c3e59e97-0ef7-409b-afe9-e123cc92770a +# ╠═2b5de833-5512-400c-93d5-2d40dcaca7ab +# ╠═22dd99ac-4b53-4664-8ec7-9a7ce852ff9b +# ╠═4bdb58fb-e71a-466b-bd99-77b0fda6d6cd +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/ifs_revised.plutostate b/ifs_revised.plutostate new file mode 100644 index 0000000..d304994 Binary files /dev/null and b/ifs_revised.plutostate differ diff --git a/implicit_function_theorem.html b/implicit_function_theorem.html new file mode 100644 index 0000000..68dee05 --- /dev/null +++ b/implicit_function_theorem.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/implicit_function_theorem.jl b/implicit_function_theorem.jl new file mode 100644 index 0000000..7793621 --- /dev/null +++ b/implicit_function_theorem.jl @@ -0,0 +1,2006 @@ +### 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 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +DiffEqBase = "2b5f629d-d688-5b77-993f-72d75c75574e" +ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" +ImplicitEquations = "95701278-4526-5785-aba3-513cca398f19" +OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" + +[compat] +DiffEqBase = "~6.125.0" +ForwardDiff = "~0.10.35" +ImplicitEquations = "~1.0.9" +OrdinaryDiffEq = "~6.51.2" +Plots = "~1.38.12" +""" + +# ╔═╡ 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Notebooks

+ + + + diff --git a/logitic_ivp.html b/logitic_ivp.html new file mode 100644 index 0000000..e862375 --- /dev/null +++ b/logitic_ivp.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/logitic_ivp.jl b/logitic_ivp.jl new file mode 100644 index 0000000..667c994 --- /dev/null +++ b/logitic_ivp.jl @@ -0,0 +1,1351 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 521181ff-67eb-42dc-8b84-8caf148dcd7e +using Plots + +# ╔═╡ 470c6265-f881-4bbb-8e90-ff281e97445e +md""" +# Julia と型， 多重ディスパッチ + +このノートブックでは [大槻純也, Pythonによる計算物理, 森北出版](https://www.morikita.co.jp/books/mid/017081) をテキストと呼び，同書の第2章を題材として取り上げています． ナイーブな微分方程式ソルバーを実装しロジスティック方程式 + +```math +\frac{dy}{dt} = f(y) \coloneqq y(1 - y) +``` + +を解いてみましょう． テキストではオイラー法, 修正オイラー法, ルンゲ・クッタ法などいくつか微分方程式を解くためのアルゴリズムが紹介されています． これらのアルゴリズムは時刻 $t$ の時における $y$ が与えられた時に $t + \Delta t$ での $y$ を計算するという意味では共通しています． つまり + +```julia +y_next = step(t, y, Δt） +``` + +という量を計算するという意味では共通しています． この類似性に着目しこれらのアルゴリズムを系統的に実装するにはどうすれば良いでしょうか？ + +例えば Python に慣れている方であれば `IVPSolver` というざっくりとした親クラスを用意し各アルゴリズムに対応する子クラスを実装すれば良さそうだと考えるでしょう． + +```python +# 初期値問題を取り扱う基底クラス +# なお，　このPython実装は文字通り適当に書いているので動作する保証は全くない +# Python が得意な受講者は実際に動くように修正し修正するプルリクエストを作ると面白いだろう +class IVPSolver(): + def solve(self, tspan: list, yinit: float) + # 微分方程式を解く + dt = tspan[1] - tspan[0] + ys = [yinit,] + + for i in len(tspan): + if i == len(ys) + break + y_next = self.step(ys[i], dt) + ys.append(y_next) + end + # plt.plot(ts, ys) によって解を描画したいなぁという気持ち + return ys + +# オイラー法 +class Euler(IVPSolver): + def __init__(self, f): + self.f = f + + def step(self, y, dt): + y_next = y + self.f(y) * dt + return y_next + +# 修正オイラー法の実装 +class ModifiedEuler(IVPSolver): + pass + +# ルンゲ・クッタ法の実装 +class RK(IVPSolver): + pass + +# ロジスティック方程式の右辺 +def f(y): + return y * (1 - y) + +import numpy as np +from matplotlib import pyplot as plt +tmax = 20 +nt = 101 +tspan = linspace(0, tmax, nt) +dt = tspan[begin+1]-tspan[begin] +yinit = 1e-3 +# オイラー法による解放を提供するコンストラクタを作成 +solver = Euler(f) +# 微分方程式を解く +ys = solver.solve(tspan, yinit) +plt.scatter(tspan, ys) +``` + +上記の実装のように基底クラスを継承し `step` メソッドを実装すれば微分方程式を解くコードや可視化する箇所はつかい回せそうだ． このように基底クラスを出発点とし各事情に合わせてどんどん実装を育てていくやり方がよく行われている． Julia の場合は"クラスの継承"の代わりに抽象型と多重ディスパッチを組み合わせて類似のことを実現できる. +""" + +# ╔═╡ 9fc9f5fa-3634-4777-9fda-9a4dda3a5e8c +md""" +## パッケージのロード +""" + +# ╔═╡ bb4089b6-2d05-49f0-901a-9a977eb954d4 +md""" +ロジスティック方程式の右辺は次のように書ける: +""" + +# ╔═╡ e8bc3d4e-cb4f-11ee-0d29-0307845be82c +f(y) = y*(1 - y) # y * (one(y) - y) とも書ける + +# ╔═╡ 7e3585ac-bab5-408a-b581-79a60004b372 +md""" +`solve` 関数の中でループをぐるぐる回し `y=ys[i]` が与えられた時に `ynext = step(solver, y)` を計算するロジックを実装している． `step` の処理は未実装であり，これは各アルゴリズムに依存する． `solver` オブジェクトは `IVPSolver` のサブタイプとして実装する. 例えばオイラー法の場合は次のようにして実装できる． +""" + +# ╔═╡ 2a718086-6cb7-477b-a20d-4a30f885ede4 +md""" +このようにすると `solver = Euler(f, dt)` によってソルバーのインスタンスを作れば + +```julia +solve(solver, , <初期値>) +``` + +と実行することで微分方程式を解くことができる. 実際に次のセルのようにして実行できる. +""" + +# ╔═╡ 3bd87598-fed2-4f0b-a7e3-14978bc3a356 +md""" +実行結果は厳密解と少しずれている． これはソフトウェアのバグではなく，オイラー法の精度に起因するものである. +""" + +# ╔═╡ 9bbdabff-2e71-4576-9ad1-a017af56764a +md""" +さて `solve` 関数には + +```julia +solve(solver::IVPSolver, ... +``` + +のように型アノテーションが付いていた. これは `T <: IVPSolver` を満たす型 `T` のオブジェクトを受け付けると理解すれば良い. +""" + +# ╔═╡ 950de090-69f3-47f4-a7e2-73067140f078 +md""" +同様に `IVPSolver` のサブタイプとなるように修正オイラー法，ルンゲ・クッタ法に対応する型を実装する. +""" + +# ╔═╡ e7dd8651-3f0d-4a32-885c-d0b102a04df3 +begin + struct RK{Fn, T<:AbstractFloat} <: IVPSolver + f::Fn + dt::T + end + + function step(solver::RK, y) + f = solver.f + dt = solver.dt + + k1 = f(y) + k2 = f(y + k1 * dt / 2) + k3 = f(y + k2 * dt / 2) + k4 = f(y + k3 * dt) + return y + (k1 + 2k2 + 2k3 + k4) * dt / 6 + end +end + +# ╔═╡ 7d9183e9-9173-465e-a82f-d1826db0a6b2 +begin + """ + IVPSolver + + 初期値問題を解く微分方程式ソルバーを取りまとめる抽象型 + """ + abstract type IVPSolver end + + """ + solve(solver::IVPSolver, tspan, yinit) + + `yinit` を初期値とする常微分方程式 \$\\frac{dy}{dt} = f(y)\$ を \$t \\in\$ `tspan` の範囲で解く. + """ + function solve(solver::IVPSolver, tspan, yinit) + ys = similar(tspan) + ys[begin] = yinit + + for i in eachindex(tspan, ys) + i == length(ys) && break + y = ys[i] + ynext = step(solver, y) + ys[i+1] = ynext + end + + return ys + end +end + +# ╔═╡ 2596fc08-02f7-4a8b-a8d8-291ffa5372bb +begin + struct Euler{Fn, T<:AbstractFloat} <: IVPSolver + f::Fn + dt::T + end + + function step(sf::Euler, y) + return y + sf.f(y) * sf.dt + end +end + +# ╔═╡ 71286f7d-9f82-4293-a90f-b8d46976d5be +let + tmax = 20 + nt = 101 + tspan = range(0, tmax, length=nt) + dt = tspan[begin+1]-tspan[begin] + solver = Euler(f, dt) + yinit = 1e-3 + sol = solve(solver, tspan, yinit) + scatter(range(0, tmax, length=nt), sol, label="$(nameof(typeof(solver)))") + plot!(t->(yinit/(yinit + (1 - yinit) * exp(-t))), label="exact") +end + +# ╔═╡ 6fe74fa6-1c65-4db9-806d-3cf9f176afe8 +begin + struct ModifiedEuler{Fn, T<:AbstractFloat} <: IVPSolver + f::Fn + dt::T + end + + function step(solver::ModifiedEuler, y) + f = solver.f + dt = solver.dt + + k1 = f(y) + k3 = f(y + k1 * dt) + return y + (k1 + k3) * dt / 2 + end +end + +# ╔═╡ d2aecd4e-2c68-4ceb-af94-626bdb0f5743 +let + tmax = 20 + nt = 101 + tspan = range(0, tmax, length=nt) + dt = tspan[begin+1]-tspan[begin] + solver = ModifiedEuler(f, dt) + yinit = 1e-3 + sol = solve(solver, tspan, yinit) + scatter(range(0, tmax, length=nt), sol, label="$(nameof(typeof(solver)))") + plot!(t->(yinit/(yinit + (1 - yinit) * exp(-t))), label="exact") +end + +# ╔═╡ 533f0d38-be86-4bfe-8eea-32c1c8b4e8f5 +let + tmax = 20 + nt = 101 + tspan = range(0, tmax, length=nt) + dt = tspan[begin+1]-tspan[begin] + solver = RK(f, dt) + yinit = 1e-3 + sol = solve(solver, tspan, yinit) + scatter(range(0, tmax, length=nt), sol, label="$(nameof(typeof(solver)))", alpha=0.5) + plot!(t->(yinit/(yinit + (1 - yinit) * exp(-t))), label="exact", alpha=0.5, lw=3) +end + +# ╔═╡ b157df51-efef-49ef-8c50-0810dd012993 +md""" +さて， 実装を進めるにつれて `step (generic function with N methods)` の `N` の部分がどんどん増えてることに気づいてるだろうか？ Julia は入力の引数の数，型に応じて関数の振る舞いを制御できる．この仕組みを Julia では多重ディスパッチと呼ぶ． 今回は 3 種類の具体的なソルバーを用意し対応する `step` メソッドを実装しているので `N methods` の N として 3 になっている. +""" + +# ╔═╡ 00ae1557-9d2c-401e-83df-010f9af89021 +md""" + +実際には Python であれば SciPy, Julia であれば DifferentialEquations.jl が提供しているソルバーを利用するべき + +- [scipy_solve_ivp](https://atelierarith.github.io/intro2software_with_julia_pluto_materials/scipy_solve_ivp.html) +- [ode](https://atelierarith.github.io/intro2software_with_julia_pluto_materials/ode.html) +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" + +[compat] +Plots = "~1.40.1" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "f34c4afa9052a065b20f2322868ec5c65692889c" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.Artifacts]] +uuid = 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"Pkg"] +git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22" +uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc" +version = "1.11.0+0" + +[[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.libinput_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"] +git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce" +uuid = "36db933b-70db-51c0-b978-0f229ee0e533" +version = "1.18.0+0" + +[[deps.libpng_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"] +git-tree-sha1 = "93284c28274d9e75218a416c65ec49d0e0fcdf3d" +uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" +version = "1.6.40+0" + +[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] +git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.7+1" + +[[deps.mtdev_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11" +uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e" +version = "1.1.6+0" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.52.0+1" + +[[deps.p7zip_jll]] +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" +""" + +# ╔═╡ Cell order: +# ╟─470c6265-f881-4bbb-8e90-ff281e97445e +# ╟─9fc9f5fa-3634-4777-9fda-9a4dda3a5e8c +# ╠═521181ff-67eb-42dc-8b84-8caf148dcd7e +# ╟─bb4089b6-2d05-49f0-901a-9a977eb954d4 +# ╠═e8bc3d4e-cb4f-11ee-0d29-0307845be82c +# ╠═7d9183e9-9173-465e-a82f-d1826db0a6b2 +# ╟─7e3585ac-bab5-408a-b581-79a60004b372 +# ╠═2596fc08-02f7-4a8b-a8d8-291ffa5372bb +# ╟─2a718086-6cb7-477b-a20d-4a30f885ede4 +# ╠═71286f7d-9f82-4293-a90f-b8d46976d5be +# ╟─3bd87598-fed2-4f0b-a7e3-14978bc3a356 +# ╟─9bbdabff-2e71-4576-9ad1-a017af56764a +# ╟─950de090-69f3-47f4-a7e2-73067140f078 +# ╠═6fe74fa6-1c65-4db9-806d-3cf9f176afe8 +# ╠═e7dd8651-3f0d-4a32-885c-d0b102a04df3 +# ╠═d2aecd4e-2c68-4ceb-af94-626bdb0f5743 +# ╠═533f0d38-be86-4bfe-8eea-32c1c8b4e8f5 +# ╟─b157df51-efef-49ef-8c50-0810dd012993 +# ╟─00ae1557-9d2c-401e-83df-010f9af89021 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/logitic_ivp.plutostate b/logitic_ivp.plutostate new file mode 100644 index 0000000..9a7ca36 Binary files /dev/null and b/logitic_ivp.plutostate differ diff --git a/lu_decomposition.html b/lu_decomposition.html new file mode 100644 index 0000000..2834b45 --- /dev/null +++ b/lu_decomposition.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/lu_decomposition.jl b/lu_decomposition.jl new file mode 100644 index 0000000..37732b3 --- /dev/null +++ b/lu_decomposition.jl @@ -0,0 +1,98 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 39c948c0-d2b8-11ee-2147-5fd144b963fe +using Random + +# ╔═╡ 1c9983f8-b302-40c6-a1e4-c4ba89e738fb +begin + function decomposeLU!(A) + n = size(A, 1) + + for k in 1:n-1 + a_kk = A[k, k] + w = inv(a_kk) + for i in (k+1):n + A[i, k] = A[i, k] * w + for j in (k+1):n + A[i, j] = A[i, j] - A[i, k] * A[k, j] + end + end + end + + L = zero(A) + U = zero(A) + for i in 1:size(A, 1) + L[i, i] = one(eltype(A)) + for j in 1:(i-1) + L[i, j] = A[i, j] + end + for j in i:size(A, 2) + U[i, j] = A[i, j] + end + end + L, U + end + + decomposeLU(A) = decomposeLU!(copy(A)) +end + +# ╔═╡ 012cb058-b91b-4c24-bc4b-fef115f97d58 +begin + rng = Xoshiro(123) + A = rand(rng, 5, 5) + L, U = decomposeLU(A) +end + +# ╔═╡ dfec4cf8-c864-48b8-adbf-0eb96059134a +L + +# ╔═╡ 95fed2cf-60ce-4fd6-a599-107d55dec1a1 +U + +# ╔═╡ ef3d07dd-7cb5-4c89-9c8c-5ab563d4fbb2 +L * U + +# ╔═╡ 9b26bea2-98ee-4125-a1bd-f58cac980c3d +L * U + +# ╔═╡ e8307483-95e4-4213-be4b-7c54ef1847f4 +A ≈ L * U + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "fa3e19418881bf344f5796e1504923a7c80ab1ed" + +[[deps.Random]] +deps = ["SHA"] +uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[[deps.SHA]] +uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" +version = "0.7.0" +""" + +# ╔═╡ Cell order: +# ╠═39c948c0-d2b8-11ee-2147-5fd144b963fe +# ╠═1c9983f8-b302-40c6-a1e4-c4ba89e738fb +# ╠═012cb058-b91b-4c24-bc4b-fef115f97d58 +# ╠═dfec4cf8-c864-48b8-adbf-0eb96059134a +# ╠═95fed2cf-60ce-4fd6-a599-107d55dec1a1 +# ╠═ef3d07dd-7cb5-4c89-9c8c-5ab563d4fbb2 +# ╠═9b26bea2-98ee-4125-a1bd-f58cac980c3d +# ╠═e8307483-95e4-4213-be4b-7c54ef1847f4 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/lu_decomposition.plutostate b/lu_decomposition.plutostate new file mode 100644 index 0000000..4729f6d Binary files /dev/null and b/lu_decomposition.plutostate differ diff --git a/modified_gram_schdmit.html b/modified_gram_schdmit.html new file mode 100644 index 0000000..f00fa31 --- /dev/null +++ b/modified_gram_schdmit.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/modified_gram_schdmit.jl b/modified_gram_schdmit.jl new file mode 100644 index 0000000..4a58f46 --- /dev/null +++ b/modified_gram_schdmit.jl @@ -0,0 +1,172 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ f1d2f741-7774-460c-87e7-9b2f2cfa3d9f +begin + # Extras + using Test + using StableRNGs +end + +# ╔═╡ f786af1c-bb14-482b-8d6c-cf204f2d71da +md""" +# 修正グラムシュミットの直交化 +""" + +# ╔═╡ 18c479b4-c714-472b-abad-b03cc6afb206 +begin + function innerproduct(a::AbstractVector, b::AbstractVector) + s = zero(eltype(a)) + @inbounds for i in eachindex(a, b) + s += a[i] * b[i] + end + s + end + const ⋅ = innerproduct + norm(a::AbstractVector) = √(a ⋅ a) +end + +# ╔═╡ 393e5b2d-c126-4827-9d91-b0539dfc02f1 +begin + @testset "innerproduct" begin + rng = StableRNG(999) + x = rand(rng, 2) + y = rand(rng, 2) + + @test innerproduct(x, y) == innerproduct(y, x) + @test x ⋅ y == y ⋅ x + + @test innerproduct([1, 0], [0, 1]) == 0 + @test innerproduct([0, 1], [1, 0]) == 0 + @test innerproduct([1, 1], [1, 1]) ≈ norm([1, 1])^2 ≈ 2.0 + end + + @testset "norm" begin + @test norm([0.0, 0.0]) == 0 + @test norm([1.0, 1.0]) == √2 + @test norm([1.0, 2.0]) == √5 + @test norm([2.0, 1.0]) == √5 + end + nothing +end + +# ╔═╡ 532c14fe-bede-46b5-8888-a7045fd8d93c +struct ModifiedGramSchmidt end # Algorithm + +# ╔═╡ 40e2ae0c-d2af-11ee-31c9-dfa8f7798362 +function orthonormalize(::Type{ModifiedGramSchmidt}, A) + a₁ = A[:, 1] + m = size(A, 2) + r₁₁ = norm(a₁) + q₁ = a₁ / r₁₁ + Q = similar(A) + R = similar(A, m, m) + Q[:, 1] .= q₁ + R[1,1] = r₁₁ + for n in 1:(m-1) + aₙ₊₁ = @view A[:, n+1] + for i in 1:n + qᵢ = @view Q[:, i] + rᵢ₍ₙ₊₁₎ = qᵢ ⋅ aₙ₊₁ + R[i, n+1] = rᵢ₍ₙ₊₁₎ + @. aₙ₊₁ = aₙ₊₁ - rᵢ₍ₙ₊₁₎ * qᵢ + end + r₍ₙ₊₁₎₍ₙ₊₁₎ = norm(aₙ₊₁) + R[n+1,n+1] = r₍ₙ₊₁₎₍ₙ₊₁₎ + qₙ₊₁ = aₙ₊₁ / r₍ₙ₊₁₎₍ₙ₊₁₎ + Q[:, n+1] .= qₙ₊₁ + end + Q, R +end + +# ╔═╡ bee5f314-e6dd-48bb-8a57-d12b4d289032 +@testset "ModifiedGramSchmidt" begin + rng = StableRNG(123) + A = rand(rng, 10, 10) + + Q, R = orthonormalize(ModifiedGramSchmidt, copy(A)) + @test A ≈ Q*R + q1 = Q[:, 1] + q2 = Q[:, 2] + q3 = Q[:, 3] + + @test innerproduct(q1, q1) ≈ 1.0 + @test innerproduct(q2, q2) ≈ 1.0 + @test innerproduct(q3, q3) ≈ 1.0 + + @test abs(innerproduct(q1, q2)) ≈ 0.0 atol = 1e-14 + @test abs(innerproduct(q2, q3)) ≈ 0.0 atol = 1e-14 + @test abs(innerproduct(q1, q3)) ≈ 0.0 atol = 1e-14 + + @test abs(innerproduct(q1, q2)) ≈ 0.0 atol = 1e-16 + @test_broken abs(innerproduct(q2, q3)) ≈ 0.0 atol = 1e-16 + @test_broken abs(innerproduct(q1, q3)) ≈ 0.0 atol = 1e-16 +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3" +Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" + +[compat] +StableRNGs = "~1.0.1" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.1" +manifest_format = "2.0" +project_hash = "a4590486d7604ab3578478648d997eb50013d5ca" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.InteractiveUtils]] +deps = ["Markdown"] +uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240" + +[[deps.Logging]] +uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" + +[[deps.Markdown]] +deps = ["Base64"] +uuid = "d6f4376e-aef5-505a-96c1-9c027394607a" + +[[deps.Random]] +deps = ["SHA"] +uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[[deps.SHA]] +uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" +version = "0.7.0" + +[[deps.Serialization]] +uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" + +[[deps.StableRNGs]] +deps = ["Random", "Test"] +git-tree-sha1 = "ddc1a7b85e760b5285b50b882fa91e40c603be47" +uuid = "860ef19b-820b-49d6-a774-d7a799459cd3" +version = "1.0.1" + +[[deps.Test]] +deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] +uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" +""" + +# ╔═╡ Cell order: +# ╟─f786af1c-bb14-482b-8d6c-cf204f2d71da +# ╠═f1d2f741-7774-460c-87e7-9b2f2cfa3d9f +# ╠═18c479b4-c714-472b-abad-b03cc6afb206 +# ╠═393e5b2d-c126-4827-9d91-b0539dfc02f1 +# ╠═532c14fe-bede-46b5-8888-a7045fd8d93c +# ╠═40e2ae0c-d2af-11ee-31c9-dfa8f7798362 +# ╠═bee5f314-e6dd-48bb-8a57-d12b4d289032 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/modified_gram_schdmit.plutostate b/modified_gram_schdmit.plutostate new file mode 100644 index 0000000..22d7e04 Binary files /dev/null and b/modified_gram_schdmit.plutostate differ diff --git a/ode.html b/ode.html new file mode 100644 index 0000000..1fb6cd3 --- /dev/null +++ b/ode.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/ode.jl b/ode.jl new file mode 100644 index 0000000..8e0d408 --- /dev/null +++ b/ode.jl @@ -0,0 +1,2080 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 88de6714-fd1c-11ed-3458-89fc42c5e7e1 +using DifferentialEquations + +# ╔═╡ 7aa3b70a-827a-482c-8052-7d34413922d4 +begin + using Plots + import PlotlyBase + import PlotlyKaleido + plotly() +end + +# ╔═╡ ed6cee9a-c325-4141-bffe-8715a59dfe5d +begin +p = 10 +r = 28 +b = 8/3 +end + +# ╔═╡ 75bbd970-3f7c-4d6c-b40f-7a857478d358 +function lorenz!(du, u, params, t) + p = params[1] + r = params[2] + b = params[3] + du[1] = -p * u[1] + p * u[2] + du[2] = -u[1] * u[3] + r * u[1] - u[2] + du[3] = u[1] * u[2] - b * u[3] + nothing +end + +# ╔═╡ bcc6c1bc-69b1-4e15-8988-a08249afaeb1 +begin +params = [p, r, b] +u0 = [1.0, 0.0, 0.0] +tspan = (0.0, 100.0) +end + +# ╔═╡ 169b59c2-d2c2-4c0a-8b05-e6ffec918e25 +prob = ODEProblem(lorenz!, u0, tspan, params) + +# ╔═╡ d46b95d0-97ab-408b-b3ef-9ab3ebeb55b6 +sol = solve(prob, Tsit5()); + +# ╔═╡ a2b8adc1-ba82-4a66-aac7-88bb8b717b72 +plot(sol) + +# ╔═╡ def2042c-5d56-4ddd-ab40-d36c1fca8970 +sol + +# ╔═╡ 67bd3b97-8dbe-41da-ae4c-dc92c326d8a1 +plot(sol, idxs=(1,2,3), lw=1) + +# ╔═╡ 9ddddc3a-bac3-446b-9602-7ff7a3e85a35 +md""" +[https://www.dynamicmath.xyz/strange-attractors/](https://www.dynamicmath.xyz/strange-attractors/) にはおしゃれな例がいっぱいある．興味のある方は遊んでみると良いかも． +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa" +PlotlyBase = "a03496cd-edff-5a9b-9e67-9cda94a718b5" +PlotlyKaleido = "f2990250-8cf9-495f-b13a-cce12b45703c" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" + +[compat] +DifferentialEquations = "~7.7.0" +PlotlyBase = "~0.8.19" +PlotlyKaleido = "~2.2.4" +Plots = "~1.40.1" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing 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+ +[[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" +""" + +# ╔═╡ Cell order: +# ╠═88de6714-fd1c-11ed-3458-89fc42c5e7e1 +# ╠═7aa3b70a-827a-482c-8052-7d34413922d4 +# ╠═ed6cee9a-c325-4141-bffe-8715a59dfe5d +# ╠═75bbd970-3f7c-4d6c-b40f-7a857478d358 +# ╠═bcc6c1bc-69b1-4e15-8988-a08249afaeb1 +# ╠═169b59c2-d2c2-4c0a-8b05-e6ffec918e25 +# ╠═d46b95d0-97ab-408b-b3ef-9ab3ebeb55b6 +# ╠═a2b8adc1-ba82-4a66-aac7-88bb8b717b72 +# ╠═def2042c-5d56-4ddd-ab40-d36c1fca8970 +# ╠═67bd3b97-8dbe-41da-ae4c-dc92c326d8a1 +# ╟─9ddddc3a-bac3-446b-9602-7ff7a3e85a35 +# 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\ No newline at end of file diff --git a/random_logos.jl b/random_logos.jl new file mode 100644 index 0000000..770d169 --- /dev/null +++ b/random_logos.jl @@ -0,0 +1,67 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ 8c13dea2-1a84-11ee-117d-d5937495696f +begin + using Pkg + Pkg.activate(; temp=true) + Pkg.add(Pkg.PackageSpec( + url="https://github.com/AtelierArith/RandomLogos.jl", + rev="f794a1b", + )) + Pkg.add("ImageShow") + Pkg.add("ImageCore") + + using Random + using ImageCore + using ImageShow + + import RandomLogos + using RandomLogos: render + using RandomLogos: Config +end + +# ╔═╡ f99fbad8-5165-4083-9aab-cde62eaf3ec3 +md""" + +# RandomLogos.jl で遊んでみる + +野良パッケージを Pluto.jl の上で動かす例として RandomLogos.jl を用いることにする． これは [Connor Anderson と Ryan Farrell による特異値分解による反復関数系のサンプリングアルゴリズム](https://catalys1.github.io/fractal-pretraining/) の Julia 実装を提供している． + +- ソースコード: https://github.com/AtelierArith/RandomLogos.jl/tree/main +- ドキュメント: https://atelierarith.github.io/RandomLogos.jl/dev/ + +""" + +# ╔═╡ fe1047ef-d512-4db7-bc49-bb5bc5e8bba9 +let + canvas = RandomLogos.render( + joinpath(pkgdir(RandomLogos), "examples", "config_mt.toml") + ) + canvas +end + +# ╔═╡ a900e785-7780-496d-8426-d0e668e9531c +let + configpath = joinpath(pkgdir(RandomLogos), "examples", "config_mt.toml") + config = Config(configpath) + logos = Matrix{RGB{N0f8}}[] + for s in 1:25 + rng = MersenneTwister(999 + s) + ifs = rand(rng, RandomLogos.SigmaFactorIFS{2}) + rng = MersenneTwister(999 + 2s) + canvas = render(rng, ifs, config) + push!(logos, canvas) + end + + reshape(logos, 5, 5) +end + +# ╔═╡ Cell order: +# ╟─f99fbad8-5165-4083-9aab-cde62eaf3ec3 +# ╠═8c13dea2-1a84-11ee-117d-d5937495696f +# ╠═fe1047ef-d512-4db7-bc49-bb5bc5e8bba9 +# ╠═a900e785-7780-496d-8426-d0e668e9531c diff --git a/random_logos.plutostate b/random_logos.plutostate new file mode 100644 index 0000000..3d50b37 Binary files /dev/null and b/random_logos.plutostate differ diff --git a/schrodinger_eq_scipy.html b/schrodinger_eq_scipy.html new file mode 100644 index 0000000..2e714ff --- /dev/null +++ b/schrodinger_eq_scipy.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/schrodinger_eq_scipy.jl b/schrodinger_eq_scipy.jl new file mode 100644 index 0000000..4215486 --- /dev/null +++ b/schrodinger_eq_scipy.jl @@ -0,0 +1,1330 @@ +### A Pluto.jl notebook ### +# v0.19.43 + +using Markdown +using InteractiveUtils + +# ╔═╡ 17750a7c-4273-11ef-2bea-9b56de603b65 +begin + using Printf + using Markdown + + using CondaPkg + using Plots + using PythonCall + CondaPkg.add_pip("numpy") + CondaPkg.add_pip("scipy") +end + +# ╔═╡ f3ec2781-9ba9-4acf-863e-2f2c693951da +md""" +# SciPy を利用したシュレディンガー方程式の数値計算結果を Pluto ノートブックから利用する + +これは [大槻純也, Pythonによる計算物理, 森北出版](https://j-otsuki.github.io/comput-phys-book/) の題材にある [時間発展シュレディンガー方程式](https://www.physics.okayama-u.ac.jp/~otsuki/lecture/CompPhys2/pde/schro.html) +のトピックと対応する実装 [schr_wall.py](https://github.com/j-otsuki/comput-phys-book/blob/main/python/schr_wall.py) を学ぶために用いたノートである． + +理論的な背景は[大槻氏のウェブサイト](https://www.physics.okayama-u.ac.jp/~otsuki/lecture/CompPhys2/index.html) を，具体的な実装は [j-otsuki/comput-phys-book](https://github.com/j-otsuki/comput-phys-book) を参考にすると良い． + +このノートブックはパッケージ化されていない１画面に収まる程度の小規模な Python の実装を Pluto　ノートブックから利用するサンプルになっている． +""" + +# ╔═╡ be3a7a86-efda-4bdd-9876-653bb504d340 +md""" + +Pluto のセルに下記のような変数を定義する + +```julia +my_python_script = raw"# write Python script here>" +``` + +次に上記で定義した `my_python_script` を一度 mylib.py のようにしてファイルに落とし込む． (もしかしたら Python の `eval` を使えば楽にできるかもしれないけれど) + +```julia +mylib = mktempdir() do d + rm(joinpath(d, "mylib.py"), force=true) + open(joinpath(d, "mylib.py"), "w") do io + print(io, my_python_script) + end + + pyimport("sys").path.append(d) + mylib = pyimport("mylib") + importlib = pyimport("importlib") + importlib.reload(mylib) + end +``` + +`mylib.py` の格納場所を Python の `sys.path` に `append` によって追加する．これにより + +```python +# Python +import mylib +``` + +相当の実装ができる． + +試行錯誤する(=`my_python_script` を編集する) 際に `mylib.py` を更新する必要がある． どうやら mylib.py を格納する一時ディレクトリ `d` は使いまわされるようなので `rm` 関数で一度消すことにしている． また更新した mylib モジュールを再ロードするために Python 標準の `importlib` パッケージを利用して `importlib.reload(mylib)` を実行している． +""" + + +# ╔═╡ c2e4b721-10fb-4e2d-8300-58808d179d26 +begin +# Edit Python script +my_python_script = raw""" +import numpy as np +from scipy.sparse import csr_matrix as sparse_matrix +from scipy.integrate import solve_ivp + +def make_differential_op(nx, dx): + f0 = np.identity(nx) + f1 = np.roll(f0, 1, axis=1) + f2 = np.roll(f0, 2, axis=1) + f_1 = f1.transpose() + f_2 = f2.transpose() + + D1 = sparse_matrix(f1 - f_1) / (2 * dx) + D2 = sparse_matrix(f1 - 2 * f0 + f_1) / (dx * dx) + D3 = sparse_matrix(f2 - 2 * f1 + 2 * f_1 - f_2) / (2 * dx * dx * dx) + + return (D1, D2, D3) + +def schrodinger(t, u, D2, v): + return -1.0j * (-0.5 * (D2 @ u) + v * u) + +def solve(x, t, u0, v): + dx = x[1] - x[0] + _, op_df2, _ = make_differential_op(x.size, dx) + tspan = (t[0], t[-1]) + args = (op_df2, v) + sol = solve_ivp( + schrodinger, tspan, u0, dense_output=True, args=args, rtol=1e-8, + ) + return sol + +def main(): + xmin, xmax = -50, 50 + nx = 1000 + x = np.linspace(xmin, xmax, nx, endpoint=False) + + k = 4 + x0 = -20 + delta = 4 + u0 = np.exp(1.0j * k * x - 0.5 * ((x - x0) / delta) ** 2) + dx = x[1] - x[0] + integ = np.sum(np.absolute(u0) ** 2) * dx + u0 /= np.sqrt(integ) + + v_height = 10.0 + width = 0.2 + v = np.where(np.abs(x) < width, v_height, 0) + tmax = 16 + nt = 81 + t = np.linspace(0, tmax, nt) + sol = solve(x, t, u0, v) + return x, t, sol +""" +end; + +# ╔═╡ 3e692a8b-0aa4-4941-bad4-7cef12fc4032 +Markdown.parse(raw"```python\n" * my_python_script * raw"```") + +# ╔═╡ 21e28ce0-6a8b-438e-a89e-c13cc081dd5f +begin + mylib = mktempdir() do d + rm(joinpath(d, "mylib.py"), force=true) + open(joinpath(d, "mylib.py"), "w") do io + print(io, my_python_script) + end + + pyimport("sys").path.append(d) + mylib = pyimport("mylib") + importlib = pyimport("importlib") + importlib.reload(mylib) + end + np = pyimport("numpy") + x, t, sol = mylib.main() + println(sol.t.size) + println(np.diff(sol.t).min()) + println(np.diff(sol.t).max()) +end + +# ╔═╡ fff48d97-69b1-42b6-8222-a3679520ed95 +md""" +トンネル効果の様子を可視化できる． +""" + +# ╔═╡ 46f0f644-bdc0-441a-b76b-8df0fb816995 +begin + u = np.absolute(sol.sol(t)) ^ 2 + u = u.T +end + +# ╔═╡ 65d2c521-8b84-46db-8646-fc0ccebceca9 +begin + anim = @animate for i in pyrange(u.shape[0]) + f = pyconvert(Float64, t[i]) + s = @sprintf "%.2f" f + plot(PyVector(x), PyVector(u[i]), xlim=[-30, 30], ylim=[0, 0.4], title="t=$(s)") + end + + gif(anim, "wall.gif") +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +CondaPkg = "992eb4ea-22a4-4c89-a5bb-47a3300528ab" +Markdown = "d6f4376e-aef5-505a-96c1-9c027394607a" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7" +PythonCall = "6099a3de-0909-46bc-b1f4-468b9a2dfc0d" + +[compat] +CondaPkg = "~0.2.22" +Plots = "~1.40.5" +PythonCall = "~0.9.20" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# 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"1.4.1+1" +""" + +# ╔═╡ Cell order: +# ╟─f3ec2781-9ba9-4acf-863e-2f2c693951da +# ╟─be3a7a86-efda-4bdd-9876-653bb504d340 +# ╠═17750a7c-4273-11ef-2bea-9b56de603b65 +# ╟─3e692a8b-0aa4-4941-bad4-7cef12fc4032 +# ╠═c2e4b721-10fb-4e2d-8300-58808d179d26 +# ╠═21e28ce0-6a8b-438e-a89e-c13cc081dd5f +# ╟─fff48d97-69b1-42b6-8222-a3679520ed95 +# ╠═46f0f644-bdc0-441a-b76b-8df0fb816995 +# ╠═65d2c521-8b84-46db-8646-fc0ccebceca9 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/schrodinger_eq_scipy.plutostate b/schrodinger_eq_scipy.plutostate new file mode 100644 index 0000000..979ccb1 Binary files /dev/null and b/schrodinger_eq_scipy.plutostate differ diff --git a/scipy_solve_ivp.html b/scipy_solve_ivp.html new file mode 100644 index 0000000..27df722 --- /dev/null +++ b/scipy_solve_ivp.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/scipy_solve_ivp.jl b/scipy_solve_ivp.jl new file mode 100644 index 0000000..2d858ad --- /dev/null +++ b/scipy_solve_ivp.jl @@ -0,0 +1,1203 @@ +### A Pluto.jl notebook ### +# v0.19.36 + +using Markdown +using InteractiveUtils + +# ╔═╡ ea57eaa4-ccb5-11ee-3e7a-63d2e6a789bf +begin + using PythonCall + using CondaPkg + using Plots + + CondaPkg.add_pip("scipy") +end + +# ╔═╡ 8fb849af-5578-45d6-a809-07d7772b2003 +scipy = pyimport("scipy") + +# ╔═╡ 06ab026a-ca8f-4c33-8fbe-efcf0086a36f +function f(t, u) + return @. u * (one(u) - u) +end + +# ╔═╡ 05522ea3-e8a2-47e3-b846-dd79ad23f6e7 +begin + u_0 = [1e-3] + tspan = (0, 20.0) + sol = scipy.integrate.solve_ivp(f, tspan, u_0, dense_output=true) + sol.message +end + +# ╔═╡ 33cdccfa-4cd7-4646-b23b-34c277812873 +begin + t = range(0, 20.0, length=101) + logistic(t, u_0 = 1e-3) = u_0 / (u_0 + (one(u_0) - u_0) * exp(-t)) + + plot(title="du/dt = f(u(t))") + plot!(t, logistic.(t), lw=3) + + scatter!( + t, PyVector(sol.sol(t).flatten()), + label="dense output" + ) + scatter!( + PyVector(sol.t.flatten()), PyVector(sol.y.flatten()), + label="selected points" + ) +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +CondaPkg = "992eb4ea-22a4-4c89-a5bb-47a3300528ab" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PythonCall = "6099a3de-0909-46bc-b1f4-468b9a2dfc0d" + +[compat] +CondaPkg = "~0.2.22" +Plots = "~1.40.1" +PythonCall = "~0.9.15" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.2" +manifest_format = "2.0" +project_hash = "d2a7b76b06cafdbc04ab49e3bc2d85bca739e87a" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.BitFlags]] 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"0.1.1+0" + +[[deps.Xorg_libxcb_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] +git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d" +uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" +version = "1.15.0+0" + +[[deps.Xorg_libxkbfile_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"] +git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4" +uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a" +version = "1.1.2+0" + +[[deps.Xorg_xcb_util_cursor_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"] +git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd" +uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43" +version = "0.1.4+0" + +[[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 = 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+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"] +git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba" +uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06" +version = "3.2.9+0" + +[[deps.fzf_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8" +uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09" +version = "0.43.0+0" + +[[deps.gperf_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033" +uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70" +version = "3.1.1+0" + +[[deps.libaom_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" +uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" +version = "3.4.0+0" + +[[deps.libass_jll]] +deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] +git-tree-sha1 = 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+git-tree-sha1 = "873b4f805771d3e4bafe63af759a26ea8ca84d14" +uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" +version = "1.6.42+0" + +[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] +git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.7+1" + +[[deps.micromamba_jll]] +deps = ["Artifacts", "JLLWrappers", "LazyArtifacts", "Libdl"] +git-tree-sha1 = "66d07957bcf7e4930d933195aed484078dd8cbb5" +uuid = "f8abcde7-e9b7-5caa-b8af-a437887ae8e4" +version = "1.4.9+0" + +[[deps.mtdev_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11" +uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e" +version = "1.1.6+0" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.52.0+1" + +[[deps.p7zip_jll]] +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" +""" + +# ╔═╡ Cell order: +# ╠═ea57eaa4-ccb5-11ee-3e7a-63d2e6a789bf +# ╠═8fb849af-5578-45d6-a809-07d7772b2003 +# ╠═06ab026a-ca8f-4c33-8fbe-efcf0086a36f +# ╠═05522ea3-e8a2-47e3-b846-dd79ad23f6e7 +# ╠═33cdccfa-4cd7-4646-b23b-34c277812873 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/scipy_solve_ivp.plutostate b/scipy_solve_ivp.plutostate new file mode 100644 index 0000000..975ea59 Binary files /dev/null and b/scipy_solve_ivp.plutostate differ diff --git a/wall.gif b/wall.gif new file mode 100644 index 0000000..aaba8de Binary files /dev/null and b/wall.gif differ