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cubic solver - Printable Version

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cubic solver - Albert Chan - 09-08-2023 02:26 PM

This program solve x^3 = a*x + b, for x

Code:
cubic_ab(a,b) := 
BEGIN
LOCAL d;
a, b := a/3, b/2;    /* x^3 = (3a)*x + (2b) */
d := sqrt(b^2-a^3) * (-1)^(sign(re(b))==-1);
b := surd(b+d, 3);
return [a/b, b];     /* alpha, beta */
END;
Code:
cubic(a,b) :=
BEGIN
LOCAL w := exp(2*pi/3*i);
[[1,1],[w,conj(w)],[conj(w),w]] * cubic_ab(a,b);
END;

Michael Penn's example




Let x = sin(θ) + cos(θ), it reduced to cubic: x^3 = 3*x - 11/8

Cas> r := cubic(3, -11/8)
Cas> simplify(r)                 → [1/2,    (3*√5-1)/4,       (-3*√5-1)/4]
Cas> float(Ans)                  → [0.5, 1.42705098312, −1.92705098312]
Cas> float(r)                      → [1.42705098312, −1.92705098312, 0.5]

We have Cas simplify bug here! symbolic and approximate numbers don't match.
Luckily, the algorithm doesn't care which cube roots is used (no need for principle root).

But, this bug should be fixed.

Cas> b := cubic_ab(3, -11/8) [2]     → (1/256*(-√15*48*i-176))^(1/3)
Cas> polar(float(b))                         → 1., −0.77627903074
Cas> polar(simplify(b))                    → 1 , atan(√15)

Simplify bug had phase angle off by 2*pi/3:

Cas> atan(√15) - 2*pi/3.0               → −0.77627903074

Update: cubic_ab(a,b) code explained here


RE: cubic solver - Albert Chan - 09-08-2023 02:44 PM

Root simplify returned wrong (not principle) root bug existed for XCas 1.5.0

But, it is worse on XCas 1.9.0-31 (mingw32 win32)
Result does not match phase *and* magnitude.

XCas> b := (1/256*(-√15*48*i-176))^(1/3)
XCas> polar(float(b))

1.0, -0.77627903074

XCas> simplify(polar(b))                     // good

\(\displaystyle 1\;,\;\frac{-\pi +\arctan \left(\frac{3}{11} \sqrt{15}\right)}{3}\)

XCas> simplify(polar(simplify(b)))       // bad

\(256\;,\;-\pi +\arctan \left(\frac{3}{11} \sqrt{15}\right)\)


RE: cubic solver - parisse - 09-08-2023 06:56 PM

Yes, there is an additional check in recent versions of Xcas, and here it failed because an argument was not reduced mod 2*pi.
https://github.com/geogebra/giac/commit/2f228406dbe099aaf7abf8d6b3fdb6ea8fd94c9d