Golden ratio min/max

4b767523-95ea-4296-9ab1-0a39540c98f2 111 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/ACCURATE 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

parametric command line XX Φ 1.875 2.125 ε X :

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

XX 11015 -^* 1.91803450244985 1.50167071402424 1.9271166225024 1.92331993346546 1.9244324203465 1.92443455823882 1.92443455653425 1.92443455653433 1.92443455653431 1.92443455653431 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 : 0.1832 - 0.9831 - 0.8628 - 0.5056 - 0.1832 0.9831 0.8628 0.5056 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 1.4

f0 f1 f2 circle roots 51 sys

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

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 a c u 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 :

"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 0.05 - 121 mat 1.125 0.8 21 mat 1.5 0.5 - 21 mat 14 mat :

roots j 123414 mat : ct 1 : 12 mat 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 31 line :

r roots eval :

r 1.612343858 - 0.441471899 - 12 mat 0.034694443 - 0.999999819 12 mat 0.946948134 0.926127253 12 mat 1.645933331 0.377765061 - 12 mat 41 mat

x y axes x y / - :

f1 f2 r axes 4 1 sys

roots solve dual f(x,y) ... Examples

Solve two orthogonal parabolas ...

b ρ Clear 1

ro 0.25 - : a 1 : b 1 - : ρ 2 : 14 mat

x y p1 a x 2^(* b y 2^(* + ro - : x y p2 2 a * x 2^(* 0.5 b * y 2^(* + ρ ro * + : 21 sys

init 0.5 1 - 21 mat 0.5 - 1 - 21 mat 0.5 - 121 mat 0.5 121 mat 14 mat :

init myroots j 1 init cols range rΩ j el x y p1 x y p2 21 mat x y 21 mat init j el roots : Ωr j el rΩ j el transpose : 21 line for Ωr 21 line :

r init myroots eval :

plot p1 p2 r axes 41 sys :

plot

r 0.645497224 0.816496581 - 12 mat 0.645497224 - 0.816496581 - 12 mat 0.645497224 - 0.816496581 12 mat 0.645497224 0.816496581 12 mat 41 mat

Solve Martin functions ...

a c r 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 :

init 0.5 - 121 mat 1121 mat 2 0.5 - 21 mat 2 - 0.5 - 21 mat 14 mat :

x y axes x y / - :

init myroots j 1 init cols range rΩ j el x y f1 x y f2 21 mat x y 21 mat init j el roots : Ωr j el rΩ j el transpose : 21 line for Ωr 21 line :

r init myroots eval :

f1 f2 r axes 4 1 sys

r 0.034694443449 - 0.999999818996 12 mat 0.946948133863 0.926127253138 12 mat 1.645933330203 0.377765060923 - 12 mat 1.612343857704 - 0.441471898648 - 12 mat 41 mat

Robust Newton Quadrant method ... Alvaro 20230410

Slightly adapted from original

a c x y n 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 : 21 sys

Formula: Newton Raphson x y J x y 21 mat x y f1 x y f2 21 mat x y 21 mat Jacob (1 -^x y f1 x y f2 21 mat * - 11 line :

Iterations n 14 :

x y axes x y / - :

Algorithm Iteration x y NR xy x y stack : j 1 n range xy xy x y 21 mat x y J : augment : for xy transpose 31 line :

Quadrants 3 - 2 - NR 3 - 2 - NR rows row 31 - NR 31 - NR rows row 1 - 3 NR 1 - 3 NR rows row 3 2.5 NR 3 2.5 NR rows row 41 mat :

Quadrants 1.612343858 - 0.441471899 - 12 mat 1.64593333 0.377765061 - 12 mat 0.034694443 - 0.999999819 12 mat 0.946948134 0.926127253 12 mat 41 mat

f1 f2 3 - 2 - NR 3 1 - NR 1 - 3 NR 3 2.5 NR 0.5 3 0.5 2 - 2 2 mat 7 1 sys

All roots at once ...

a c 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 axes x y / - :

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

f1 f2 axes 2.1859 - 0 0.0347 - 0 1.0347 0 3.1859 0 4 2 mat 1.7321 - 0 1.7321 0 2 2 mat 5 1 sys

x 0 f2 x 3 - roots x 0 f2 x 0 roots x 0 f2 x 1 roots x 0 f2 x 3 roots 41 sys 2.1859 - 0.0347 - 1.0347 3.1859 41 sys

x 0 f1 x 2 - roots x 0 f1 x 2 roots 21 sys 1.7321 - 1.7321 21 sys

2.1859 - 0.0347 - 1.0347 3.1859 1.7321 - 1.7321 61 mat sort 00000061 mat augment 2.1859 - 0 1.7321 - 0 0.0347 - 0 1.0347 0 1.7321 0 3.1859 062 mat

Dual solver algo style ... xd(L,H,N)

U # # f Solve i 1 U rows range data 1 i el U i el y f y roots eval : for v data 1 el : j 2 data cols range v v data j el augment : for v transpose 41 line :

Incremental discretizer ...
Mesh N  within the bounds L, H L H N xd X 0 : i 1 N range X i el L H L - N / i * + : for X 31 line :

slightly inner limits L 0.5 : H 3.328427124 : N 1000 : 13 mat U L H N xd : 21 sys

t0 1 time :

Optimiz Numeric ... x y Φ x y f2 : sols U x y Φ Solve : 21 sys

CWD C:\SmathFile CurrentDirectory :

sols CWD f2Martin wfile 1

new CWD f2Martin rfile :

branch1 U new 1 col augment :

3.32842712474619

branch1 11 el 0.5 : branch1 12 el 0.5 : 12 mat

branch2 U H 0.5 + (- new 1 col reverse 1 - augment :

branch1 branch2 - 2.2324 0.5 + 200 black 1 5 mat 3 1 sys

1 time t0 - min 1.9

0.61 < u < 3.323 ... u 3.323 :

U rows 1000

x pairwise user x y Φ y solve : x user 12 mat 21 line :

pair wise u pairwise 3.32300 0.39897 0.60096 21 mat 12 mat

min
mid
max 0.61 2.2324 3.323 31 sys

L 0.61 : H 3.323 : N 100 : 13 mat u L H N xd : 21 sys

You can nest pairwise(x) extract pairwise alternate

x pairwise user x y Φ y solve eval : user 21 line :

data branch1 eval :

S data 1 row : 0.5 0.5 12 mat

data data rows row 3.3284 0.5 12 mat

S 11 el S 12 el f2 0

0.5 0.5 f2 0.000000000000000

AlgLib Solver for a system of nonlinear equations(Viacheslav) result k 1 x# Ω rows range u# k el x# k el : for x# Jac u# Ω u# Jacobian : StepMax 0 : Eps 1015 -^: u# X0 StepMax Eps x# Ω x# Jac al_nleqsolve : 51 line :

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

x y f1 x y 2 sqrt 1 ff :

x y f2 x 0.5 - y 0.5 - 22 ff : 1 - 2 x * + (2^1 - 2 y * + (2^+ (2^- 64 x y - (* 1 - 2 x * + 1 - 2 y * + (* + 16 / -

x Ω 1 - 2 x 1 el * + (2^1 - 2 x 2 el * + (2^+ (2^- 64 x 1 el x 2 el - (* 1 - 2 x 1 el * + 1 - 2 x 2 el * + (* + 16 / - 0.5 21 mat :

X0 3 0.5 21 mat :

result 3.32842712474619 0.5 21 mat

X0 2 - 0.5 21 mat :

result 2.32842712474619 - 0.5 21 mat

2.32842712474619 - 3.32842712474619 21 mat 0.5 0.5 21 mat augment 2.3284 - 0.5 3.3284 0.5 22 mat

x y axes x y / - :

f1 f2 axes 2.1859 - 0 0.0347 - 0 1.0347 0 3.1859 0 4 2 mat 1.7321 - 0 1.7321 0 2 2 mat 2.3284 - 0.5 3.3284 0.5 2 2 mat 6 1 sys

X 2.32842712474619 - 0.5 3.32842712474619 0.5 41 mat :

X 1 el 0.5 f2 X 3 el 0.5 f2 21 mat 0.000000000000000 0.000000000000000 21 mat

0.5 0.5 f2 0.000000000000000

al_nleqsolve is unique candidate solving for datum 0.5 Other datum iterative methods produce accurate solutions. 'Iter' while loop offered below.

roots multisolve algo style ...

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 :

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 :

datum algoV j 1 datum rows range rx j el x datum j el f2 x 3 - roots : ry j el datum j el : 21 line for rx ry augment 21 line :

The all trick is to initialise the 'roots solver' regions.

datum algoS j 1 datum rows range rx j el x datum j el f2 x 1 - roots : ry j el datum j el : 21 line for rx ry augment 21 line :

datumV datum algoV eval : datumS datum algoS eval : 21 sys

datum 0.375 - 0.25 - 0 0.25 0.375 0.5 1 1.375 81 mat :

datumV 1.774 - 0.375 - 1.9664 - 0.25 - 2.1859 - 0 2.2947 - 0.25 2.3201 - 0.375 2.3284 - 0.5 2.1859 - 1 1.774 - 1.375 82 mat

datumS 0.6392 - 0.375 - 0.3898 - 0.25 - 0.0347 - 0 0.246 0.25 0.3745 0.375 0.5 0.5 0.0347 - 1 0.6392 - 1.375 82 mat

f1 f2 datumV datumS 4 1 sys

Mdatum datumV datumS stack :

Mdatum . 12 red plot

datum 1.375 :

t0 1 time :

al_nleqsolve system core component

AlgLib Solver for a system of nonlinear equations(Viacheslav) result k 1 x# Ω rows range u# k el x# k el : for x# Jac u# Ω u# Jacobian : StepMax 0 : Eps 1015 -^: u# X0 StepMax Eps x# Ω x# Jac al_nleqsolve : 51 line :

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

x y f1 x y 2 sqrt 1 ff :

x y f2 x 0.5 - y 0.5 - 22 ff : 1 - 2 x * + (2^1 - 2 y * + (2^+ (2^- 64 x y - (* 1 - 2 x * + 1 - 2 y * + (* + 16 / -

x Ω 1 - 2 x 1 el * + (2^1 - 2 x 2 el * + (2^+ (2^- 64 x 1 el x 2 el - (* 1 - 2 x 1 el * + 1 - 2 x 2 el * + (* + 16 / - datum 21 mat :

al_nleqsolve target recursive Iter X0 2 - datum 21 mat : j 0 : result 2 el datum - abs 1015 -^> X0 result 1 el datum 21 mat : j j 1 + : 21 line while result j 21 mat 41 line :

Iter 1.77395397397543 - 1.375 21 mat 8421 mat

Timing varies wrt 'j iter' 9 iter only 0.2 s

1 time t0 - 17.1 s *

sol Iter 11 el (1 el Iter 11 el (2 el augment eval :

f1 f2 sol 3 1 sys

sol 1.774 - 1.375 12 mat

datum 1.375

faster + accurate 'roots'

Watch where you thread init

X x datum f2 x 1 - roots :

X 0.639246824395971 -

Solving 'divide by zero' ...

x Vf v f 12 x * g * v 2^/ - exp - sqrt atanh v 2^* g Vf * / :

g 9.80665 :

Vf 10 : v 0.625 : 12 mat

x 01 0.001 range :

V k 1 : j 1 x rows range x j el Vf v f IsDefined 1 ≡ Y k el x k el Vf v f : k k 1 + : 21 line continue if break try for 21 line Y 21 line :

x x 1 V rows range el :

V rows 702

data x V augment :

XY data data rows row : 0.701 0.0701 12 mat

data XY 1 1 el XY 1 2 el + 100 red 1 5 mat 2 1 sys

jmG: 20230410

x Vf v f