<content> <p> Golden ratio min/max </p> </content>

15ad3b5c-71c2-4f7f-a741-498d9a34e095 37 4 5 false false 0 0 decimal

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

r 2 - 1 - 012316 mat transpose :

r polyroots 0.8398 0.8099 - 0.4848 i * + 0.8099 - 0.4848 i * - 0.0567 0.9422 i * + 0.0567 0.9422 i * - 51 mat

r 0.8398 0.8099 - 0.4848 i * + 0.8099 - 0.4848 i * - 0.0567 0.9422 i * + 0.0567 0.9422 i * - 51 mat :

x L x r j el - (j 1 r rows product Re :

u 256 :

x L u - x 04 solve 2.8935

x f x L :

h 0.01 :

x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

x L u x h f D1 3 1 sys

4 - L 4 L 0 L 0.3134 L 41 mat 874.0074 - 1213.9712 0.6666 - 0.7514 - 41 mat

x ʃ ω f ω x - x int :

x ʃ x L 0.4565 2 0.4565 2 - 2 2 mat 3 1 sys

0.4565 2 0.4565 2 - 22 mat

x L

x f x L :

a 0.75 : b 1 : 12 mat xn b a + (2 / : 21 line

ε 1014 -^:

n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 21 line

.. sanity ..

xn 0.8398

xn f 1.6 1015 -^*

iterates n 45

Φ a b ε X Root Secant u 1 el 1015 -^: u 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : u i el u i 1 - el - abs ε ≥ u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : i i 1 + : 21 line while u 71 line :

x Φ x L 0.5 x ʃ * - eval :

a 0 : b 1 : ε 109 -^: 13 mat

sol Φ a b ε X :

ω sol sol rows row (: 0.4565 11 mat

ω L 0.7384 - 11 mat

ω L ω ʃ ≡

All roots ... accurate bloc scanner

The ROBUST 'zizag' root(s) Solve scanner f x0 x1 TOL Solve ρ x f x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : n 0 : xn f abs TOL > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 51 line X i el xn : 21 line for X 31 line :

compliant native fnct x exp x 2^+ 5 x * -

TOL 1014 -^:

x f x ʃ :

NOT compliant... ʃ\[REGION[4013995542464d9e9841a1049be44b7a]]\\[REGION[b99c938ec83d44e9af98105949b6f208]]\ f 2 - 1 - TOL Solve #

x f

rootsDichotomy local command line style f x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 31 line :

roots of ▼ \[REGION[9cffccdd88ec4fa3b7013bf4cee13e11]]\

x ʃ

TOL 1014 -^:

x f x ʃ :

f 2 - 1 - TOL D f 1 - 1 TOL D f 12 TOL D 31 sys 1.4954 - 0 1.4954 31 sys

Golden-Ratio search [min/max]

<content> <p> Golden ratio min/max </p> </content>

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 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x 4^2 x 3^* - 7 x 2^* - 2 - eval :

ε 1015 -^:

By setting f(x):=eval(,)by default, it will cope with all "_not defined"
As well, you may have to use local definition.
Some functions will ONLY accept higher 'ε' ... 10^{-14} ... 10^{-13}

t0 1 time :

X f 1.5 - 1 - y1 y2 ≤ ε Extrema f 0.5 - 0.5 y1 y2 ≥ ε Extrema f 2.5 3 y1 y2 ≤ ε Extrema 31 mat :

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 31 line :

1 time t0 - 0.5 s *

on failure , increase TOL TOL 1012 -^:

r 2.5 - 1.5 - TOL D 3.5 4 TOL D 21 mat : 1.922518473318 - 3.852153430325 21 mat

x f X 1 el X 1 el f + 20 red X 2 el X 2 el f + 20 red X 3 el X 3 el f + 20 red 3 5 mat r 1 el 0 . 14 black 1 5 mat r 2 el 0 . 14 black 1 5 mat 4 1 sys

X 1.26556446000827 - 2.81605475000001 108 -^* - 2.76556437660309 31 mat

Infinitesimal analysis ▼

Algo style Memo formula of incremental analysis ... h 0.01 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / : (1rst Derivative ≡ x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : (2nd Derivative ≡ x h D3 x 3 h * - f 8 x 2 h * - f * - 13 x h - f * + 13 x h + f * - 8 x 2 h * + f * + x 3 h * + f - 8 h 3^* / : x h D4 x 3 h * - f - 12 x 2 h * - f * + 39 x h - f * - 56 x f * + 39 x h + f * - 12 x 2 h * + f * + x 3 h * + f - 6 h 4^* / : x h f h_int h x h + f x f - (* 2 x f * + (x * : (Integral Curve [antiderivative] ≡ 71 line

x y axes x y / - :

Lagrange Polynomial ... user Quick

r 2 - 1 - 012316 mat transpose :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x L x r j el - (j 1 r rows product :

Lroots r r 0 * augment . 4 black plot :

Lroots Lroots :

x L Lroots axes 3 1 sys

x L Lroots axes 31 sys

n i Clear 1

n 08 range :

Factorial i i 1 n product vectorize :

native n Factorial augment :

native 00112236424512067207504084032092 mat

native

jmG: 20231119

Otherwise regular Quartic maple

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

r Clear (1

r x f x fsolve maple : 1.17 - 0.3701 i * + 0.2847 - 1.5209 i * - 0.2847 1.5209 i * + 1.17 0.3701 i * - 41 mat

x y axes x y / - :

x L x r j el - (j 1 r rows product Re :

x L 2 - axes 3 1 sys

x L 2 - x 1.5 - 1.5 solve 1.2496 - 1.2496 21 mat

maple observer x f op maple 2 - 3 i * + x 2^1 x 2^+ (* 21 mat