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classical_gram_schmidt.jl
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classical_gram_schmidt.jl
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### 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