09e20894-42a4-493d-9678-347665785a6e 59 6 5 auto

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a 1 el a 2 el x * + a 3 el x 2^* + x 3^+ canonic format ≡

a 204 - 613 mat transpose :

# 204 - 6114 mat transpose ≡

for polyroots, x^3 is silent

In Mathematics, polynomials are represented in Lagrange format, thus wrt their roots, which results in the canonical format. In the cubic example, the polynomial does cross the X only once, by inspection it will have complex roots. This cubic solver is extracted from Mathcad 11/Maple. This cubic solver is most useful in modeling the the PID control system.

S1 36 a 2 el * a 3 el * 108 a 1 el * - 8 a 3 el (3^* - 1212 a 2 el (3^* 3 a 2 el (2^* a 3 el (2^* - 54 a 1 el * a 2 el * a 3 el * - 81 a 1 el (2^* + 12 a 1 el * a 3 el (3^* + (12 /^* + (13 /^:

q 13 / a 2 el * 19 / a 3 el (2^* - :

q 5.3333 -

S1 9.3391 -

Z 16 / S1 * 6 q S1 / * - 13 / a 3 el * - 1 - 12 / S1 * 3 q S1 / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S1 * 6 q S1 / * + (* + 1 - 12 / S1 * 3 q S1 / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S1 * 6 q S1 / * + (* - 31 mat :

a Z 12 mat 204 - 631 mat 6.983 - 0.4915 1.6194 i * + 0.4915 1.6194 i * - 31 mat 12 mat

Z1 Z 1 el 20 - Z 1 el 2022 mat :

Cubic roots invariants

Z 1 el Z 2 el * Z 3 el * abs 20

q 3^r 2^+ 0 > q 3^r 2^+ 0 ≡ q 3^r 2^+ 0 < 31 mat 1 real pair complex conjugate all real roots, at lest TwoEqual all real roots 31 mat ≡

[1]

Z 1 el Z 2 el * Z 1 el Z 3 el * + Z 2 el Z 3 el * + 4 -

Z 1 el Z 2 el + Z 3 el + (- 6

Notes: I was not enable to convince Smath/Maple to collect the Lagrange canonical form. In [1] Smath does not threshold "residuals" imaginaries.

x Lagrange x Z 1 el - (x Z 2 el - (* x Z 3 el - (* :

Z 6.982977 - 0.491489 1.619428 i * + 0.491489 1.619428 i * - 31 mat

Lagrange complex cubic plot

x Lagrange Z1 20 3 1 sys

204 - 6141 mat polyroots 0.491 1.619 i * + 0.491 1.619 i * - 6.983 - 31 mat

20 4 x * - 6 x 2 ^ * + x 3 ^ +

x Lagrange 20 ≡ x 10 - 2 solve Does NOT solve complex ≡

x sanity a 1 el a 2 el x * + a 3 el x 2^* + x 3^+ :

Smath "solve' does not like the Lagrange complex ! 'Lagrange & canonic' are inseparable companions. 1. Lagrange enables to manufacture polynomial from the roots you supply to shape some stuff. 2. 'canonic' will solve your stuff.

Datum 30 :

Sol x sanity Datum ≡ x 10 - 2 solve :

i 2 :

Collect x sanity Datum Sol i el Datum 10 + Sol i el Datum 10 - 22 mat 31 sys :

Sol 6.3813 - 1.0756 - 1.4569 31 mat

Title 5 - 24 Some cubic Datum solutions 14 black 15 mat :

Collect

Smath reads math regions from left => right, genius to construct the plotting argumemts ! as you may want the construct UP or RIGHT

Datum solution(s)

The region below got corrupted. Visit Cubic solver [Lagrange canonic]. The suite below is correct

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T P char size color plotG n T length : plot T 1 el P 1 el char size color augment : i 2 n range plot plot T i el P i el char size color augment stack : for plot 41 line :

r 1371041 mat :

r 1 3.5 7.5 1041 mat :

<= observe the Smath numerical parsing

x f x r n el - (n 14 product : 1 - x + (7 - 2 x * + (* 15 - 2 x * + (* 10 - x + (* 4 /

i 1 r rows range u i el 0 : for

roots r u o 7 red plotG :

Lagrange root(s) construction

x f roots 2 1 sys

jmGiraud: 2015:07:25

Resume this document The 'canonic cubic solver' is complete by itself. It does not suffers the obsolete Cardan. It solves wrt to complex/real roots to feed the more mathematical Lagrange, which is an important notion in advanced algebra and symbolic subuse.

a 204 - 6114 mat transpose :

a 1 el a 2 el x * + a 3 el x 2^* + x 3^+

r a polyroots :

r 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

polyroots("vector") : Returns the roots of the polynomial whose coefficients are in "vector"