Solve Quartic with Casio FX115MS

08302019, 02:02 PM
(This post was last modified: 03132020 06:23 PM by Albert Chan.)
Post: #5




RE: Solve Quartic with Casio FX115MS
An old algorithm for solving Quartics: Brown's method
(source: W.S. Brown, "Solution of Biquadratic Equations", Aircraft Engineering, 16,14 (1944) Quartics: x^4 + A x^3 + B x^2 + C x + D = 0 Step 1: get Y, the largest real root of resolvent cubics y^3  B y^2 + (AC4D) y + D(4BA²)C² = 0 it will work for any root of cubics, but later quadratics might involve complex numbers. Step 2: solve 2 quadratics c^2  A c + (BY) = 0 d^2  Y d + D = 0 Step 3: make sure roots, c1 d2 + c2 d1 = C. If not, swap d1, d2. This may required calculation of c1 d1 + c2 d2 as well, due to numerical imprecision. I have a slight improvement in this step, see example below Step 4: solve 2 more quadratics, the 4 roots is the roots of Quartics: x^2 + c1 x + d1 = 0 x^2 + c2 x + d2 = 0 Example: solve x^4 + 2 x^3 + 3 x^2 + 4 x + 5 = 0 A = 2 B = 3 C = 4 D = 5 E = AC4D = 12 F = D(4BA²)C² = 24 y^3  B y^2 + E y + F = 0 → Y = 4.482612919 c^2  A c + (BY) = 0 → [c1,c2] = [0.575630959, 2.575630959] d^2  Y d + D = 0 → [d1,d2] = [2.088157364, 2.394455555] if B and Y nearly equal, we may use another form for BY Y^2 (YB) + E Y + F = 0 → BY = (E Y + F) / Y² Calculate AY/2 If AY/2 > C, make sure roots of c,d sorted the same way If AY/2 < C, make sure roots of c,d sorted opposite way For this case, AY/2 = Y > C, thus c,d roots sorted the same way. x^2 + c1 x + d1 = 0 → x = 0.2878154795 ± 1.416093081 i x^2 + c2 x + d2 = 0 → x = 1.287815480 ± 0.8578967571 i 

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Messages In This Thread 
Solve Quartic with Casio FX115MS  Albert Chan  11292018, 05:28 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  11292018, 10:31 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  02042019, 01:39 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  08242019, 12:47 PM
RE: Solve Quartic with Casio FX115MS  SlideRule  03142020, 01:28 AM
RE: Solve Quartic with Casio FX115MS  Albert Chan  08302019 02:02 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  09022019, 02:35 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  02012020, 12:27 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  03132020, 06:30 PM

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