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

016c2377-6cac-4d96-b939-d668a7023dc3 3 4 5 decimal

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Golden-Ratio search [min/max]

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 1012 -^:

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 1.2655644606693 - 2.81601830000001 108 -^* - 2.76556437660199 31 mat

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 1.5 - 1 - y1 y2 ≤ ε Extrema 1 el f 0.5 - 0.5 y1 y2 ≥ ε Extrema 1 el f 2.5 3 y1 y2 ≤ ε Extrema 1 el 31 mat : 1.265564 - 2.816018 108 -^* - 2.765564 31 mat

iterates f 1.5 - 1 - y1 y2 ≤ ε Extrema 2 el f 0.5 - 0.5 y1 y2 ≥ ε Extrema 2 el f 2.5 3 y1 y2 ≤ ε Extrema 2 el 31 mat : 56585631 mat

View steps of an iterative solution ...

x Clear 1

x f x 0.5 1 -^x - exp * :

x q π sqrt x sqrt erf * :

"RootSecant" is viable  => guess wisely tial/error Φ a b ε X Root Secant us 1 el ε : us 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : us i el us i 1 - el - abs ε ≥ us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : i i 1 + : 21 line while us 71 line :

x f x q 2 1 sys

x Φ x f x q - eval :

a 0.35 : b 0.425 : ε 1015 -^: 13 mat

sol Φ a b ε X :

manage the brackets [a,b]
for the last two sols = sol 11015 -^* 0.386800321029649 0.386800320927094 0.384468677669743 0.384477703937892 0.38447766923231 0.38447766923179 0.38447766923179 81 mat

sub sol 2 sol rows range 1 el :

convergents sub sub f vectorize augment :

convergents 0.386800321029649 1.09212246851665 0.386800320927094 1.09212246877344 0.384468677669743 1.0979862280579 0.384477703937892 1.09796342887793 0.38447766923231 1.09796351653823 0.38447766923179 1.09796351653955 0.38447766923179 1.09796351653955 72 mat

sanity junction sub 7 el Φ 41015 -^*