<p> Golden Ratio </p>

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Undocumented root/solve tool box

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4 appVersion 0.98.6179.21440 ≡

https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution

Part 1. An assymmetric PDF

Fréchet-II fit to 
US rainfall project K α β x f K α β / * β x / (α 1 +^* β x / (α^- exp * :

Fréchet distribution ... α > 0 ... β > 0 
Does NOT have an explicit  Inverse fnct α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

Explicit cumulative α β x CDF α β x Fr x int maple : α β x / ln * exp - exp

Part 2. Solve/inverse the assymmetric PDF

w sol α β x Fr w - root : w β * - α β x / (1 α +^* β x / (α^- exp * + β / root

recast 'root'

x λ 10 x ≤ (x 10 ≤ (& cases :

α 5 : β 1 : 12 mat w t ipdf w β * - α β t / (1 α +^* β t / (α^- exp * + β / : w t Sol_1 w t ipdf 0 ≡ t 012 solve : w t Sol_2 w t ipdf 0 ≡ w 02 solve : 41 sys

Part 3. in-situ, value the dual solver

U 0.5 :

xo 1.125 :

given ordinate find twin abscisae

X U t Sol_1 (: Y w xo Sol_2 (: 21 mat

given abscissa find ordinate

track X 1 el 0.25 - X 1 el U X 2 el U X 2 el 0.25 - 42 mat :

Smath native Simpson
very good Gaussians. α β x Fr x X 1 el X 2 el int 0.8253

Yo xo Y xo 0.25 - 22 mat :

X transpose Y 21 mat 0.7597 1.427 12 mat 1.416 21 mat

Fréchet dual-independent ▼ explorer-solver

α β x Fr x λ * track Yo 0.5 U 0.125 + U 14 red 1 5 mat X 1 el U + 40 red 1 5 mat X 2 el U + 40 red 1 5 mat xo Y + 20 black 1 5 mat 7 1 sys

ε 1015 -^:

Fréchet is a normalized PDF
well rendered SS Integrator
... acc 200 α β x Fr x ε 12 int 1.0000

α β x CDF x λ *

Golden-Ratio search [min/max]

TOL 106 -^:

x f α β x Fr :

<p> Golden Ratio </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 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

X f 0.75 1.25 y1 y2 ≥ TOL Extrema 1 el : X f 21 mat 0.9642 1.8743 21 mat