Legendre変換のお勉強: (x, y)ではなく(p, w)で関数を表す

example_legendre.jl

using Plots

# 適当な2次関数，導関数，接線の方程式
f(x; b=-3, d=2) = (b .+ x) .^2 .+ d
df(x; b=-3, d=2) = 2 .* (b .+ x)
fₜₐₙ(x, xₚ; b=-3, d=2) = df(xₚ) .* (x .- xₚ) .+ f(xₚ, b=b, d=d)


# 関数
x = 1:0.1:5;
y = f(x);

# 導関数
x₁ = 4
y₁ = fₜₐₙ(x, x₁)
x₂ = 3
y₂ = fₜₐₙ(x, x₂)
x₃ = 2
y₃ = fₜₐₙ(x, x₃)

plot(x, y, ylims=(0,6), lw=3, linecolor=:red)
plot!(x, y₁, lw=1.5, linecolor=:blue, label="tan(x=$x₁)")
plot!(x, y₂, lw=1.5, linecolor=:blue, label="tan(x=$x₂)")
plot!(x, y₃, lw=1.5, linecolor=:blue, label="tan(x=$x₃)")

# legendre

x = -1:0.1:7;
y = f(x);

# 導関数
x₁ = 4
y₁ = fₜₐₙ(x, x₁)
y₁₀ = fₜₐₙ(0, x₁)
x₂ = 1
y₂ = fₜₐₙ(x, x₂)
y₂₀ = fₜₐₙ(0, x₂)


plot(x, y, lw=3, linecolor=:red)
plot!(x, y₁, lw=1.5, linecolor=:blue, label="tan(x=$x₁)")
plot!(x, y₂, lw=1.5, linecolor=:green, label="tan(x=$x₂)")
plot!([0], seriestype=:vline, linecolor=:black, lw=0.5, label="")
plot!([0], [y₁₀], seriestype=:scatter, color=:blue, label="")
plot!([0], [y₂₀], seriestype=:scatter, color=:green, label="")
# vline!(x=0, lw=3, linecolor=:red)