<p> Golden ratio </p>

35690f0b-d91c-49c6-b4e4-d286d6b6c17d 9 4 5 decimal

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Lagrange Iterated Product

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 :

r 2 - 1 - 012316 mat transpose :

data r r 0 * augment :

support data . 14 black plot :

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

support x L x L x diff 3 1 sys

x f x L :

Golden-Ratio search [min/max]

TOL 1014 -^:

<p> Golden ratio </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 :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

support x L 2 1 sys

X f 2 - 1 - y1 y2 ≤ TOL Extrema 1 el f 1 - 0 y1 y2 ≥ TOL Extrema 1 el f 01 y1 y2 ≤ TOL Extrema 1 el f 12 y1 y2 ≥ TOL Extrema 1 el f 23 y1 y2 ≤ TOL Extrema 1 el 51 mat :

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

X 1.66345 - 0.5737 - 0.5 1.5737 2.66345 51 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 2 1 sys

piecewise iterations iter f 2 - 1 - y1 y2 ≤ TOL Extrema 2 el f 1 - 0 y1 y2 ≥ TOL Extrema 2 el f 01 y1 y2 ≤ TOL Extrema 2 el f 12 y1 y2 ≥ TOL Extrema 2 el f 23 y1 y2 ≤ TOL Extrema 2 el 51 mat : 676868676651 mat

Transmit the coefficients ... NOT the function

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

vector polyroots c 2 - 07 - 2 - 115 mat transpose :

r c polyroots :

x Φ x r j el - (j 1 r rows product :

x c 1 2.5 - x ≤ (x 4 ≤ (& cases :

r 0.0352 0.5185 i * + 0.0352 0.5185 i * - 1.9225 - 3.8522 41 mat

x f x c * x Φ 2 1 sys