HP-45 Application Book (Not getting Correct Answer))

[Albert Chan]

Hi, Namir

I have not read the book, nor your code, but this is easy to explain.

³√(-8) = -2, but (-8)^(1/3) = -2*ω, where ω = cis(2*pi/3) = (-1 + i√3) / 2

Because of this ambiguity, below is a *bad* cubic formula.
x might not be a root al all! One term might require ω factor correction.



Instead of doing 2 cube roots, do only 1, and derive the other, guaranteed correct one.
Bonus: we can get the one with least cancellation errors, and have both accurate terms.

x^3 - 4*x^2 + 8*x - 8 = 0            // let x = y + 4/3
y^3 + 8/3*y - 56/27 = 0               // c = 8/3, d = 56/27, to match above formula

--> (y - (α + β)) * (y - (α*ω + β/ω)) * (y - (α/ω + β*ω)) = 0

α = ³√(d/2 + √((d/2)^2 + (c/3)^3)) = ³√(28/27 + 4/3) = ³√(64/27) = 4/3

β = (-c/3) / α = (-8/9) / (4/3) = -2/3

β is correctly paired with α --> all generated roots are correct.

x = 4/3 + y

x1 = 4/3 + (4/3)     + (-2/3)     = 2
x2 = 4/3 + (4/3)*ω + (-2/3)/ω = 1 + i√3
x3 = 4/3 + (4/3)/ω + (-2/3)*ω = -1/3 − i√3

Had we use cube root for β, we may get (28/27 - 4/3)^(1/3) = -2/3/ω

XCas> ω := exp(i*2*pi/3)
XCas> float(normal([[1,1],[ω,1/ω],[1/ω,ω]] * [4/3, -2/3/ω] .+ 4/3))

[3.0+0.57735026919*i, 1.0+0.57735026919*i, -1.15470053838*i]