HP71B IBOUND fooled

[Albert Chan]

(08-10-2022 04:48 PM)Albert Chan Wrote:  For cubic(a,b), (sin,asin) → (cos,acos), we *still* get solutions of cubic, albeit in different order.

Let θ = asin(4b/sqrt(4*a/3)^3)

(0 - θ)/3 = pi/2 - (2*pi - (pi/2-θ))/3
(2*pi - θ)/3 = pi/2 - (0 - (pi/2-θ))/3

Swapping sin/asin to cos/acos, or vice versa, first 2 solutions order also swapped.

Again, Super Golden ratio example, for all roots.

XCAS> cubic(1/3, 29/27) .+ 1/3.

[-0.232786+0.792552*i, 1.46557+5.60374e-17*i, -0.232786-0.792552*i]

XCAS> cubic2(1/3, 29/27) .+ 1/3.

[1.46557, -0.232786+0.792552*i, -0.232786-0.792552*i]

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Explanation of algebraic way to solve cubic roots
https://www.hpmuseum.org/forum/thread-10...#pid150296

Kahan's method, for accurate numerical roots of cubic
https://math.stackexchange.com/a/3374897