Golden ratio min/max

4b767523-95ea-4296-9ab1-0a39540c98f2 35 4 5 decimal

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

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 5 - 5 TOL Solve :

ZACroots 4.3556 - 0.8522 - 4.0411 31 mat 32721 mat

result
iterates X n 21 mat

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

X Y 12 mat 4.35556637414311 - 0.85220895635768 - 4.04110866383413 31 mat 96.4443362585693 131.477910436423 180.411086638341 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

Golden ratio 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 hypergeom eval :

ε 1014 -^:

X f 45 y1 y2 ≥ ε Extrema :

X X f 12 mat 4.6491 0.2636 12 mat

ListCollect ... Jacobian

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

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 :

sol(init,Tol) ... AlgLib Solver for a system of nonlinear equations
adapted from Viacheslav original init sol k 1 ζ f rows range u k el ζ k el : for ζ Jac u f u Jacobian : StepMax 0 : ε 109 -^: 12 mat u init StepMax ε ζ f ζ Jac al_nleqsolve : 41 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

XY 060000 0.5 35462130747 2.1 26217 3.6 22781 5.6 19921 9.7 16453 14.7 13766 22.3 11115 32.4 8654 50.2 5563 75.5 2423 101.3 0132 mat :

1. selct the support points
2. collect for al_nleqsolve
3. collect variates [x,y] list 91011121315 mat : xy XY list ListCollect : X XY 1 col : Y XY 2 col : 12 mat 31 sys

raw XY . 14 red plot : support xy + 20 blue plot : 21 line

xy 22.3 11115 32.4 8654 50.2 5563 75.5 2423 101.3 052 mat

x xy 1 col : y xy 2 col : 21 sys

al_nleqsolve COMPACTUM
... NUMERIC/SYMBOLIC x ζ Φ ζ 1 el ζ 2 el * ζ 3 el x ζ 5 el^* + ζ 2 el x ζ 4 el^+ / 0 + : init 60100361000 - 0.2 0.25 15 mat transpose : ζ f ct 1 : j 1 x rows range Q ct el x j el ζ Φ y j el - : ct ct 1 + : 21 line for Q 31 line : 31 line

t0 1 time :

ζ init sol : 59814.15162 2.56094 60887.56868 - 0.01159 0.19977 51 mat

1 time t0 - s 4

x c 1 5.6 x ≤ (x 101.325 ≤ (& cases :

user adjust wrt local kPa ζdatum 2.3 - :

datum [0] ... adjusted by user x ζ Φ ζ 1 el ζ 2 el * ζ 3 el x ζ 5 el^* + ζ 2 el x ζ 4 el^+ / ζdatum + :

Φ(x,ζ) jmG transmute fit altitude(kPa) = m

x ζ Φ x c * raw support 22.3 25000 22.3 5000 - 2 2 mat 22.3 22000 ◄ commercial 14 black 1 5 mat 7 1 sys

j 1 :

fly kPa ISA(m) 12 mat xy j row stack :

error @ flying altitude 'j' xy j 1 el ζ Φ xy j 2 el - (m * m 6.1 -

fly kPa ISA(m) 22.3 1111522 mat

Solving quaternion ... accurately given [α,β] ... solve for [p,q,r]

p ε Clear 1

α 0.25 : β 1.5 : ε 1013 -^: 13 mat

... evaluates and assign implicit p q + r - 0 ≡ p 2^q 2^+ r 2^+ 1 ≡ α p * β r * + 0 ≡ 31 mat p 1 - ≡ q 1 ≡ r 0.1 ≡ 31 mat ε FindRoot 0.64699663922063 - 0.754829412424069 0.107832773203438 31 mat

p q r 31 mat 0.64699663922063 - 0.754829412424069 0.107832773203438 31 mat

... P Q + R - 0 ≡ P 2^Q 2^+ R 2^+ 1 ≡ α P * β R * + 0 ≡ 31 mat P ε ≡ Q ε ≡ R ε ≡ 31 mat ε FindRoot 0.646996639220631 - 0.754829412424069 0.107832773203438 31 mat

P Q R 31 mat 0.646996639220631 - 0.754829412424069 0.107832773203438 31 mat

p q + r - p 2^q 2^+ r 2^+ 1 - α p * β r * + 31 mat 9.7 1016 -^* 7.9 1016 -^* - 5.1 1016 -^* - 31 mat

The constant term "modulo"
...Mathcad 11 symbolic modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt :

V β - modulo / β modulo / β α + (modulo / β α + (- modulo / α modulo / α - modulo / 32 mat :

lexicographic order
[p,q,r] two sets sols. V 0.64699663922063 - 0.64699663922063 0.754829412424069 0.754829412424069 - 0.107832773203438 0.107832773203438 - 32 mat

x y Clear 1

Accurate FindRoot(ε) ...

Solve an hypersurface isocurve

c 1 :

t0 1 time :

f 2 c * y * 3 x 2^* y 2^* y 4^- c 2^x 2^y 2^+ (* - 4 c 4^* + (* x cos * y sinh * 2 c * y * x 4^3 y 2^* x 2^* - c 2^x 2^y 2^+ (* - 4 c 4^* - (* x sin * y cosh * x y * x 2^y 2^+ (y 2^x 2^- 2 c 2^* + (* (* 2 c * sin * 31 mat :

x y f f 1 el f 2 el + f 3 el + :

level_0 0 : level_1 1 : level_2 2 : 13 mat

y x f0 x y f level_0 + : y x f1 x y f level_1 + : y x f2 x y f level_2 + : 31 sys

r 1 : a 1 : b 1 : 13 mat y x circle a x * (2^b y * (2^+ r - : 21 sys

symbolic x y f1 eval :

"FindRoot" needs 
the two algorithms
in explicit form. eq_1 symbolic : eq_2 a x * (2^b y * (2^+ r - : 21 sys

roots accuracy ε 1012 -^:

guess wisely ... init 0.25 - 1 - 21 mat 1 - 0.5 - 21 mat 0.25 121 mat 1 0.5 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

roots roots eval : 0.183161394215 - 0.98308285697 - 0.862776659105 - 0.50558524158 - 0.183161394215 0.98308285697 0.862776659105 0.50558524158 42 mat

◄ roots piled down ▲

f0 f1 f2 circle roots 5 1 sys

You are at liberty to select oblate/prolate ellipse ... Check initials !

very long expansion symbolic

1 time t0 - min 1.3

Solve two orthogonal parabolas

t0 1 time :

r 0.25 - : a 1 : b 1 - : ρ 2 : system x y p1 a x 2^(* b y 2^(* + r - : x y p2 2 a * x 2^(* 0.5 b * y 2^(* + ρ r * + : 31 sys

algo style

"FindRoot" needs 
the two algorithms
in explicit form. eq_1 a x 2^(* b y 2^(* + r - : eq_2 2 a * x 2^(* 0.5 b * y 2^(* + ρ r * + : 21 sys

x y axes x y / - :

guess wisely ... init 0.75 0.75 - 21 mat 0.75 - 0.75 - 21 mat 0.75 - 0.75 21 mat 0.75 0.75 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : all j el sol j el transpose stack : ct ct 1 + : 31 line for all 41 line :

roots roots eval : 0.645497 0.816497 - 12 mat 0.645497 - 0.816497 - 12 mat 0.645497 - 0.816497 12 mat 0.645497 0.816497 12 mat 41 mat

◄ roots nested ......... ▲

p1 p2 roots axes 4 1 sys

1 time t0 - min 0.2

ε 1010 -^:

k1 10.052 : k2 14.836 : k3 4.982 : 13 mat

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

init γ1 1 ≡ γ2 0.5 ≡ 21 mat :

f init ε FindRoot 19.818000000000000 10.052000000000000 21 mat

γ1 γ2 + k1 k2 + k3 + (- γ1 γ2 * k1 k2 * k1 k3 * + (- 21 mat 0.000000000000000 0.000000000000000 21 mat

jmG: 20230329

Goat function
courtesy Carlos x Goat 21 x 2^2 / - asin * 2 x 2^* x 2 / asin * + x 2^4 x 2^- (* sqrt + :

"Goat Radius" ►  Mathematica certifies 9 decimals ... Mathcad Given/Find  ►  1.15872847301812 G 21 x 2^2 / - asin * 2 x 2^* x 2 / asin * + x 2^4 x 2^- (* sqrt + π x 2^* - x 1.15 1.2 solve : 1.15872847301812

plot x Goat π x 2^* G 8 G 4 - 22 mat 31 sys :

Goat Radius intersection Goat(x) ◄ π*x^2

plot

From rough [1,2] ... 1.1587
reguess the range  [1.15,1.2] G Goat π G 2^* - 0.00000000000001

x Goat π x 2^* - x 1 roots 1.15872847301813

x Goat π x 2^* - x 1 - roots 0.852423835993046 -

x y Clear 1

x y a c f x 2^y 2^+ (2^2 c 2^* x 2^y 2^- (* - a 4^c 4^- (- :

x y f1 x y 2 sqrt 1 f :

x y f2 x 0.5 - y 0.5 - 22 f :

f1 f2 2 1 sys

"FindRoot" needs 
the two algorithms
in explicit form. eq_1 x y f1 : eq_2 x y f2 : 21 sys

roots accuracy ε 1014 -^:

guess wisely ... init 1.5 - 0.5 - 21 mat 0.1 0.9 21 mat 1.125 0.8 21 mat 1.5 0.5 - 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

r roots eval :

r 1.6123 - 0.4415 - 0.0347 - 1 0.9469 0.9261 1.6459 0.3778 - 42 mat

sanity 2nd root

f1 f2 0.0347 - 1.025 0.0347 - 0.975 2 2 mat 3 1 sys