<p> Extrema(f,a,b,test,TOL) </p>

a3e6d93e-b85c-449f-b38d-9e5e6cb4482f 22 4 5 decimal

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

Bracket wisely "roots" single iterative values ...

Part 2 ... construct the code ... export

4 appVersion 0.98.6179.21440 ≡

x y f 3 x 2^* x y * + 2 y 2^* + x - 4 y * - :

v 0.8340576562283 :

ε 0.0000001 :

bound the hi/lo limits lo v - ε + : hi v ε - : N 200 : 13 mat

t0 1 time :

populate solutions xd lo hi lo hi lo - N / + (range :

The lo/hi limits from preparation via maple and Golden ratio. The bounded 'xd' sets "real roots"

Circular code "roots" solve contour Code Typical Roots range ± 25 j xd length 1 range rho j el sol xd j el y f y roots at : for C xd rho augment : C1 xd - C 2 col - 10 f + augment : Code C C1 stack : 51 line :

Code 1 3 / 0 + 40 red 1 5 mat 2 1 sys f

v - xd ≤ (xd v ≤ (&

1 time t0 - 10 s *

Part 1 ... prepare the system

x F x y f 0 ≡ x solve maple :

Golden ratio u 0.791485578347766 : v 0.834057656228299 : 21 sys

x F 1 y - 123 y * 2 y - (* + sqrt + 6 / 1 - y + 123 y * 2 y - (* + sqrt + 6 / - 21 mat

sanity some pairs 10 f 00 f 13 / 0 f v u f 41 mat 2031016 -^* - 11015 -^* - 41 mat

x Q 1 x - 123 x * 2 x - (* + sqrt + 6 / :

0 Q 13 /

x F 1 x - 123 x * 2 x - (* + sqrt + 6 / ± :

x f x Q :

Golden-Ratio search [min/max]

TOL 1012 -^:

<p> Extrema(f,a,b,test,TOL) </p>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

X f 01 y1 y2 ≥ TOL Extrema 1 el : iterate f 01 y1 y2 ≥ TOL Extrema 2 el : 21 sys

x F X X f + 40 black 1 5 mat 2 1 sys

X X f iterate 31 mat 0.791485578347766 0.834057656228299 5831 mat

Code Code eval : sub Code 1375 range 12 range el : 21 sys

Code rows 402

Code 199 row Code 200 row Code 201 row Code 202 row Code 203 row 51 mat 0.817376 0.596658 12 mat 0.825717 0.652503 12 mat 0.834058 0.790996 12 mat 0.834058 0.791975 12 mat 0.825717 0.934639 12 mat 51 mat

xx vx RowIdx ct 1 : i 1 vx rows range xx vx i el ≡ idx ct el i : continue if ct ct 1 + : 21 line for idx 31 line :

Collect some interesting ellipse values

sub 0.17 - 2.022 + 100 red 1 5 mat 1 3 / 0 + 50 red 1 5 mat 0.834 - 1.208 + 25 red 1 5 mat 0.834 0.791 + 25 red 1 5 mat 5 1 sys

vx Code 2 col :

vx Max vx RowIdx 32211 mat

Code 322 row Code 1 row Code 201 row 31 mat 0.1668 - 2.0215 12 mat 0.8341 - 1.208 12 mat 0.8341 0.791 12 mat 31 mat

About these two Smath documents . . .

Each one gets the solvability limits differently:

1. Circular Code Ellipse . . . maple ► Golden ratio.

2. Circular Code Electron . . . al_nleqsolve

Each one is imperatively slightly resctrited within limits.

[1] is solved single values roots.

[2] is solved pairwise solve.

Both documents . . . set Options Roots (range) ± 5

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Ellipse Code :

vx Ellipse 1 col : vy Ellipse 2 col : 12 mat

Area Polygon wrt vertices coordinates A 12 / vx i el vy i 1 + el * vx i 1 + el vy i el * - (i 1 vx rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P vx i 1 + el vx i el - (2^vy i 1 + el vy i el - (2^+ sqrt i 1 vx rows 1 - sum :

Center of gravity Xo 16 A * / vx i el vx i 1 + el + (vx i el vy i 1 + el * vx i 1 + el vy i el * - (* i 1 vx rows 1 - sum * : Yo 16 A * / vy i el vy i 1 + el + (vx i el vy i 1 + el * vx i 1 + el vy i el * - (* i 1 vx rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 2.619 5.828 21 mat

Center of gravity Xo abs Yo abs 21 mat 0.000 1.000 21 mat