39287ce2-c45b-4186-9bc3-cb37df6bdd2a 11 15 12 decimal

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

Collapse "Newton-Raphson" in utilities f q init ε Zero i 1 : x 1 el init : x 2 el x i el x i el f x i el q / - : x i 1 + el x i el - abs ε > i i 1 + : x i 1 + el x i el x i el f x i el q / - : 21 line while x 51 line :

The 'while loop' automates the prescribed accuracy of the root(s).

Newton-Raphson

The function x f 4 x 2^- exp * 1 - 2 x 2^* + (* π sqrt / :

... the derivative x q 8 x * x 2^- exp * 32 x 2^* - (* π sqrt / :

Zero f q 0.7 - 1014 -^Zero :

X Zero Zero rows el : 0.707107 -

x f x q X 2 X 2 - 2 2 mat 3 1 sys

u 1 el 0.7 : i 14 range u i 1 + el u i el u i el f u i el q / - : for 21 line

Zero 0.7 - 0.707072135785007 - 0.707106780337926 - 0.707106781186548 - 0.707106781186549 - 51 mat

u 0.7 0.707072135785007 0.707106780337926 0.707106781186548 0.707106781186549 51 mat

x f 5 x + x 2^- x 3^+ : x q x f x diff : 21 sys

x f x q 2 1 sys

Zero f q 1 - 1014 -^Zero :

X Zero Zero rows el : 1.27816307279815 -

x f 5 x + x 2^- x 3^+ 021 mat : r x f 1 el x f 2 el - x 1.5 - roots : 21 sys

r 1.27816307279815 -

r f 1 el r f 2 el - 11014 -^* -

x f 1 el x f 2 el 2 1 sys

511 - 141 mat polyroots 1.27816307279815 - 1.13908153639908 1.61689738910596 i * + 1.13908153639908 1.61689738910596 i * - 31 mat

x f 1 el x 3.8207922517468 ≡ at 50

x f 1 el x r ≡ at 11014 -^* -

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

a 511 - 13 mat transpose :

# 511 - 114 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 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 /^:

ω 13 / a 2 el * 19 / a 3 el (2^* - :

ω 0.2222

S1 0.7666

i 1 - sqrt :

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

a z 12 mat 511 - 31 mat 1.2782 - 1.1391 1.6169 i * + 1.1391 1.6169 i * - 31 mat 12 mat

Z1 z 1 el 10 - z 1 el 1022 mat :

Cubic roots invariants

z 1 el z 2 el * z 3 el * abs 5

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 * + 1

z 1 el z 2 el + z 3 el + (- 1 -

z 1.2782 - 1.1391 1.6169 i * + 1.1391 1.6169 i * - 31 mat

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

z 2 el abs 1.9778

Lagrange complex cubic plot

x Lagrange Z1 2 1 sys

x f 5 x + x 2^- x 3^+ 021 mat : r x f 1 el x f 2 el - x 1.5 - roots : 21 sys

r 1.27816307279815 -

r f 1 el r f 2 el - 11014 -^* -

root r 10 r 10 - 22 mat :

x f 1 el x f 2 el root 3 1 sys