<p> Cubic Solver </p>

d6744263-50e3-40ff-9eb5-c91949832fa2 27 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

from cubic solver(s) r 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat :

Lagrange roots style x L x r n el - (n 13 product :

a 204 - 6114 mat transpose :

a polyroots 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

Lagrange canonic x canonic a 1 el a 2 el x * + a 3 el x 2^* + a 4 el x 3^* + :

x c 16 - x ≤ (x 2 ≤ (& cases :

x L x c * x canonic 2 1 sys

Otherwise encountered ...
canonic style convention x f 204 x * - 6 x 2^* + x 3^+ :

x expr x f op maple :

x 1 :

coeffs x expr :

coeffs 204 - 6141 mat

coeffs polyroots 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

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

x Clear 1

===== Lagrange roots styles =====

Lagrange roots style x L x r n el - (n 13 product :

Lagrange roots style x Labs x r n el abs - (n 13 product :

Lagrange roots style x Lre x r n el Re - (n 13 product :

Lagrange roots style x Lim x r n el Im - (n 13 product :

x L x Labs x Lre 10 x Lim * 4 1 sys

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

cubic vector a 204 - 613 mat transpose :

polyroots vector # 204 - 6114 mat transpose ≡

for cubic solver, x^3 is made "unary" Divide else coeffs by coeff of ♦ x^3

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 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 / * + (* - 16 / S1 * 6 q S1 / * - 13 / a 3 el * - 31 mat :

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

Cubic roots invariants have application in Process Control.

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 least 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

max/min abscissea X 4.3094 - 0.3094 21 mat ≡

a 204 - 6114 mat transpose :

x Clear 1

r a polyroots : 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

Lagrange roots style x L x r n el - (n 13 product :

Possible triplet for project that may exist beween Im region. NO application recorded.

x L 0.4915 0.4915 L + 100 red 1 5 mat r 2 el Im r 2 el Im L + 20 black 1 5 mat 1.6194 33.5046 + 20 black 1 5 mat 4 1 sys

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

Lagrange roots style x L x r n el - (n 13 product :

x L 2 1 sys

r 2 el Re L 19.6021

r 2 el Im L 37.966

r 3 el Im 1.6194

r 3 el Im L abs 33.5046

Sort sparse complex roots ...

Companion Matrix E 000421002701013 - 001544 mat :

E dn_LinAlgEigenvalues 0.9429 - 0.9402 3.1749 i * + 0.9402 3.1749 i * - 4.0624 41 mat

L(x) sorted Lagrange Re [■ ] ≡

r 0.9429 - 0.9402 3.1749 i * + 0.9402 3.1749 i * - 4.0624 41 mat :

r 0.9402 3.1749 i * + 0.9402 3.1749 i * - 0.9429 - 4.0624 41 mat :

x Lagrange x r j el - j 14 product Re :

Lagrange roots style x L x r n el - (n 14 product :

0.943 - 0 . 14 black 1 5 mat x L 4.062 0 . 14 black 1 5 mat 3 1 sys x Lagrange

Convert/solve NON-Unary Cubic ...

a v Clear 1

polyroots vector v 204 - 6214 mat transpose :

cubic vector a 102 - 313 mat transpose :

r v polyroots :

<p> Cubic Solver </p>

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 1.6667 -

S1 4.1832 -

Z 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 / * + (* - 16 / S1 * 6 q S1 / * - 13 / a 3 el * - 31 mat :

Lagrange roots style x L x r n el - (n 13 product :

Difficult to verdict the most accurate ... None is known exactly, polyroots assumed better, assumed from less error propagation vs cubic ?

x L 2 1 sys

v r 12 mat 204 - 6241 mat 0.543864911137843 1.46647761371543 i * + 0.543864911137843 1.46647761371543 i * - 4.08772982227569 - 31 mat 12 mat

a Z 12 mat 102 - 331 mat 0.543864911137928 1.46647761371553 i * + 0.543864911137928 1.46647761371553 i * - 4.08772982227586 - 31 mat 12 mat