cubic solver

[Albert Chan]

(07-02-2024 02:34 PM)Albert Chan Wrote:  For cos version, replace sin/asin with cos/acos.

For numerical calculations, sin version perhaps is better. (is there exceptions?)
But there is a use for cosine, if we want to get real (α, β) from real (k, t)

cos(real or purely imaginary) = real

y^3 = a*y+b,      k = √(4a/3),      t = 4b/k³,      θ = acos(t)/3

y = k* cos(2n/3*pi - θ)
   = k*(cos(2n/3*pi)*cos(θ) + sin(2n/3*pi)*sin(θ))
   = [cos(2n/3*pi), i*sin(2n/3*pi)] • [k*cos(θ), k*sin(θ)/i]

Compare this to radical (α, β) version

y = cis(2m/3*pi) * α + cis(-2m/3*pi) * β
   = [cos(2m/3*pi), i*sin(2m/3*pi)] • [α+β, α−β]

Assuming m=n (it doesn't matter if they are not the same), we have:

α+β = k*cos(θ)
α−β = -i*k*sin(θ)

2α = (α+β) + (α−β) = k*cis(-θ)
2β = (α+β) − (α−β) = k*cis(θ)

LHS = one of RHS pairs. (we don't know if m=n, or sort order of (α, β))

k/2*cis(±acos(t)/3) = {α, β} | {α*ω, β/ω} |{α/ω, β*ω} | {β, α} | {β*ω, α/ω} | {β/ω, α*ω}

Example, x^3 = x + 1

XCas> a, b := cubic_ab(1.,1.)       → [0.337726750973, 0.986991206271]
XCas> k, t := cubic_kt(1.,1.)        → [1.15470053838, 2.59807621135]
XCas> θ := acos(t)/3                   → 0.536211995193*i
XCas> k/2 * [cis(θ), cis(-θ)]         → [0.337726750973, 0.986991206271]