<p> Golden ratio </p>

1ee46c98-2a31-4635-89a1-ac1f2dd13ca9 3 4 5 decimal

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

b1 3.4445 - : b2 1.5 : β 4 - : 13 mat x spline1 β x * 2 β * b1 + (x 2^* - β b1 + (x 3^* + : x spline2 β x * 2 β * b2 + (x 2^* - β b2 + (x 3^* + : 31 sys

x λ 10 x ≤ (x 1 ≤ (& cases :

x arc1 1 x spline1 x diff (2^+ sqrt : x arc2 1 x spline2 x diff (2^+ sqrt : 21 sys

x spline1 x λ * x spline2 x λ * 2 1 sys

dorso x arc1 x λ * x 01 int : 1.82603

torso x arc2 x λ * x 01 int : 1.82604

1. Prepare the project

2. Points specification

N 1000 : (Density of random ≡

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

xmax 1 : xmin 0 : 21 line

ymax 0.5 : ymin 1 - : 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

x y1 x spline1 : x y2 x spline2 : 21 line

3. Fill critical region

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x y2 - 0 ≥ (y x y1 - 0 ≥ (- :

#A0A0A0

t0 1 time :

A routine that finds an array of N random 
pairs in the critical region. Pts the routine is a compressor it maintains density 'N' within the variable region i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 51 line :

a b c spec . 6 #A0A0A0 13 mat :

Pts Pts :

pts Pts a b c spec pts :

1 time t0 - 35 s *

x y1 0 ≡ x 0.5 0.6 solve 0.53731

Elastic Dolphin [jmG, Sophie Germain]

x y1 x λ * x y2 x λ * pts 0.875 0.125 . 15 black 1 5 mat 4 1 sys

dorso torso 21 mat 1.826 1.826 21 mat

Golden-Ratio search [min/max]

TOL 1012 -^:

<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"

x f x y1 :

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

X1 f 0.75 0.825 y1 y2 ≥ TOL Extrema 1 el f 0.125 0.375 y1 y2 ≤ (TOL Extrema 1 el 21 mat :

x f x y2 :

X2 f 0.25 0.5 y1 y2 ≤ (TOL Extrema 1 el :

X1 X2 21 mat 0.8014 0.2235 21 mat 0.4000 21 mat