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

4b767523-95ea-4296-9ab1-0a39540c98f2 12 4 5 decimal

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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 :

Solve(f,x0,x1,TOL) ... in this utility algo bloc x0,x1,TOL are the command line at the user point of the project. The core algoritmic component is solve(4) ... solve(f(x),x,x0,x1)

Accurate Bessel roots all at once ...

x f 1.5 x mwbesselj :

Command line x0 0 : x1 18 : TOL 1014 -^: 13 mat

x f

Bsl f x0 x1 TOL Solve : 0 4.49340945790907 7.72525183693771 10.9041216594289 14.0661939128315 17.2207552719307 61 mat

sanity check Bsl f vectorize transpose 041015 -^* 31017 -^* - 41015 -^* - 51015 -^* 81015 -^* 16 mat

The ROBUST root(s) Intersection Solve scanner f q x0 x1 TOL Solve ρ x f x q - 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 * + : 12 mat xn b a + (2 / : n 0 : xn f xn q - abs TOL > xn f xn q - (a f a q - (* 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 41 line X i el xn : 21 line for X n 21 mat 31 line :

x f 6 x 3^* 7 x 2^* + 94 x * - 50 + :

x q 14010 x * + :

TOL 1013 -^:

Inclusive Parametric command line 
Bracket wisely, observing the plot ZACroots f q 6 - 5 TOL Solve :

ZACroots 4.3556 - 0.8522 - 4.0411 31 mat 16621 mat

result
iterates X n 21 mat

X ZACroots 1 el : Y X f vectorize : 12 mat

X Y 12 mat 4.35556637414312 - 0.852208956357679 - 4.04110866383413 31 mat 96.4443362585712 131.477910436424 180.411086638338 31 mat 12 mat

Install coordinates as previous example.

x f x q X 1 el Y 1 el + 20 black 1 5 mat X 2 el Y 2 el + 20 black 1 5 mat X 3 el Y 3 el + 20 black 1 5 mat 5 1 sys

x f x q - x 3 - roots x f x q - x 1 - roots x f x q - x 3 roots 31 sys 4.35556637414311 - 0.85220895635768 - 4.04110866383413 31 sys

6 ½ decimals x f x q - x 6 - 5 solve 4.3555664 - 0.8522091 - 4.0411086 31 mat

"RootSecant" intersect ... descending ▼ convergents Φ a b ε X u 1 el ε : 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 61 line :

Solve(f,q,x0,x1,TOL) ... The core component is solve(4) If the funct is not scalar wrt 'x' ... Gamma(x)is NOT. ... solve(4) & roots(3) fail Alternately RootSecant X(Φ,a,b,ε) Bracket X(Φ,■,■,ε) for last 2 convergents =

x Γ x Gamma :

N n p x hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 39 :

n 12 :

p 20 :

ε 1015 -^:

x hypergeom N n p x hypergeom :

x f 0.025 x * :

x Φ x hypergeom x f - eval :

parametric command line X Φ 6 6.5 ε X :

u X X rows el : 6.23315640883423

x hypergeom x f u u f + 20 black 1 5 mat 3 1 sys

X 11015 -^* 6.23410483488073 5.48475611649919 6.23309978239644 6.23315979534205 6.23315640883647 6.23315640883415 6.23315640883435 6.23315640883422 6.23315640883423 6.23315640883423 111 mat

<p> Golden ratio min/max </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 / 71 line :

a b + 2 / iter 21 mat

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

x f x hypergeom eval :

ε 1014 -^:

X f 45 y1 y2 ≥ ε Extrema :

X X f 12 mat 4.6491 0.2636 12 mat

jmG: 20230329