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

d4f7c914-2498-4dc1-8619-814098c92213 17 4 5 false false 0 0 decimal

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

x y axes x y / - :

Golden-Ratio [min/max]... roots

Code adapted from  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 :

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 sys 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 :

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 :

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 1.2656 - 2.8161 108 -^* - 2.7656 31 mat

x y axes x y / - :

plot 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 35 mat r 1 el 0.6 black r 2 el 0.6 black 25 mat 31 sys :

plot axes 2 1 sys

plot axes 21 sys

system X X f vectorize augment (:

system 1.2656 - 6.5923 - 2.8161 108 -^* - 2 - 2.7656 39.3452 - 32 mat

fXY i 1 X rows range fX i el X i el f : for X fX augment 21 line :

fXY 1.2656 - 6.5923 - 2.8161 108 -^* - 2 - 2.7656 39.3452 - 32 mat

Export this demo section ▼. . . otherwise companion

Polyline 2 - 0.519 1.9999 - 0.5188 1.9992 - 0.5178 1.9972 - 0.5148 1.9933 - 0.509 1.9869 - 0.4996 1.9773 - 0.4854 1.964 - 0.4657 1.9462 - 0.4394 1.9235 - 0.4056 1.895 - 0.3635 1.8605 - 0.3124 1.8204 - 0.2532 1.7757 - 0.1871 1.7269 - 0.1153 1.675 - 0.039 1.6207 - 0.0405 - 1.5647 - 0.1219 - 1.508 - 0.2042 - 1.4511 - 0.2861 - 1.395 - 0.3663 - 1.3402 - 0.4438 - 1.287 - 0.518 - 1.2352 - 0.5885 - 1.1849 - 0.6548 - 1.136 - 0.7165 - 1.0886 - 0.7732 - 1.0427 - 0.8244 - 0.9981 - 0.8697 - 0.955 - 0.9087 - 0.9133 - 0.9408 - 0.873 - 0.9659 - 0.834 - 0.9839 - 0.7962 - 0.995 - 0.7595 - 0.9994 - 0.7238 - 0.9972 - 0.6889 - 0.9885 - 0.6549 - 0.9736 - 0.6216 - 0.9525 - 0.5889 - 0.9255 - 0.5567 - 0.8927 - 0.5249 - 0.8538 - 0.4935 - 0.8074 - 0.4625 - 0.7514 - 0.432 - 0.684 - 0.4019 - 0.6032 - 0.3722 - 0.507 - 0.343 - 0.3937 - 0.3143 - 0.2611 - 0.2861 - 0.1074 - 0.2583 - 0.0693 0.2311 - 0.2695 0.2043 - 0.4873 0.178 - 0.7154 0.1521 - 0.9464 0.1267 - 1.173 0.1016 - 1.3879 0.0769 - 1.5837 0.0526 - 1.753 0.0287 - 1.8886 0.005 - 1.983 0.0184 2.0312 0.0415 2.0372 0.0643 2.0071 0.087 1.9473 0.1096 1.864 0.132 1.7635 0.1544 1.6519 0.1768 1.5355 0.1992 1.4206 0.2217 1.3133 0.2442 1.2189 0.2669 1.1377 0.2899 1.0692 0.3131 1.0126 0.3367 0.9673 0.3606 0.9326 0.385 0.9079 0.41 0.8925 0.4355 0.8858 0.4617 0.887 0.4885 0.8954 0.5163 0.9093 0.5451 0.9271 0.5752 0.947 0.6067 0.9672 0.6397 0.9861 0.6746 1.0018 0.7114 1.0127 0.7504 1.0169 0.7917 1.0128 0.8354 0.999 0.8815 0.9757 0.93 0.9433 0.9806 0.9026 1.0333 0.8539 1.0881 0.7978 1.1447 0.735 1.2031 0.6659 1.2633 0.5911 1.325 0.5112 1.3881 0.4268 1.4519 0.3393 1.5154 0.2502 1.5779 0.161 1.6385 0.0734 1.6964 0.0114 - 1.7507 0.0916 - 1.8005 0.1659 - 1.845 0.2326 - 1.8833 0.2903 - 1.915 0.3379 - 1.9403 0.3761 - 1.96 0.4057 - 1.9748 0.4281 - 1.9854 0.444 - 1.9925 0.4548 - 1.9968 0.4613 - 1.9991 0.4646 - 1.9999 0.4658 - 1202 mat :

vx Polyline 1 col : vy Polyline 2 col : 12 mat x interpolatePolyline vx vy x cinterp : 21 mat

x y axes x y / - :

u 1.25 - :

track 0 u interpolatePolyline u u interpolatePolyline u 032 mat :

x interpolatePolyline axes track 3 1 sys

u u interpolatePolyline 12 mat 1.25 - 0.5686 - 12 mat

0.67 0.833 0.67 - (2 / + 0.7515

in-situ command-line 0.7515 interpolatePolyline 1.0169

x interpolatePolyline 1 - x 0.25 - 1 solve 0.1461 - 0.3191 0.67 0.833 41 mat

Golden-Ratio [min/max]... roots

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

Code adapted from  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 interpolatePolyline eval 11 line :

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}

x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 sys 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 :

X f 1 - 0.5 - y1 y2 ≤ ε Extrema f 0.125 - 0.125 y1 y2 ≥ ε Extrema f 0.25 0.5 y1 y2 ≤ ε Extrema f 0.6 1 y1 y2 ≥ ε Extrema 41 mat :

on failure , increase TOL TOL 1012 -^:

r 2 - 1.5 - TOL D 1.5 2 TOL D 21 mat : 1.648412203939 - 1.688646874251 21 mat

X X eval : 0.7535 - 0.0334 0.4444 0.7519 41 mat

x y axes x y / - :

plot x f axes 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 X 4 el X 4 el f + 20 red 45 mat r 1 el 0.5 black r 2 el 0.5 black 25 mat 41 sys :

plot

stack symbolic ▲ for plot

M X 1 el X 1 el f + 20 ded 15 mat X 2 el X 2 el f + 20 ded 15 mat X 3 el X 3 el f + 20 ded 15 mat X 4 el X 4 el f + 20 ded 15 mat 41 mat :

fXY i 1 X rows range fX i el X i el f : for X fX augment 21 line :

fXY 0.7535 - 0.9995 - 0.0334 2.0395 0.4444 0.8853 0.7519 1.0169 42 mat

system X X f vectorize augment (: 0.7535 - 0.9995 - 0.0334 2.0395 0.4444 0.8853 0.7519 1.0169 42 mat

jmG: 20231124