Threaded Mode | Linear Mode

Albert Chan · (This post was last modified: 09-02-2019 07:42 PM by Albert Chan.)

Even if no crashing, the results are not very good.

if X = 100, X³ + X = 1000100

Solve X³ + X - 1000100 = 0 for X, should get back 100

Cubic solver gives back 100.0033332 ?

I thought maybe it is due to rounding errors, trying to deflate cubic to quadratic.
But, no. X = 100 is the only real root, the rest are complex: 50 ± √(7501) i

The equation were *already* a depressed cubic, with r = 1/3, s = -1000100/2
From Thomas Klemm cubic formula thread, using the same calculator (to be fair):

X = -2 √(r) sinh(1/3 * asinh(s/r^1.5)) = 100

Do this another way, with X = u - r/u, where u³ = -(s + sign(s) √(s² + r³))
With r this small, and s negative, u³ ~ -2s = 1000100

u ~ 100.0033332
X = u - r/u = 100

Edit: if discriminant s² + r³ < 0, X = u - r/u imaginary part cancelled out:

θ = acos(-s/(-r)^1.5) / 3
k = 2√(-r)
X = k*cos(θ), k*cos(θ + 120°), k*cos(θ + 240°), all real roots