<p> Max/Min Golden ratio </p>

4eacc4d1-3298-4e1d-8a78-74dabc06385e 25 15 12 decimal

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

x f x 2 π * / 2 π * x / tanh * sqrt :

Undocumented root solutions . . .

productive scalar canvas ... slow

]]>

0.04 f 10 f 100 f 31 mat 0.0798 0.9414 0.9993 31 mat

w sol t f w - 0 ≡ t root : w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / 0 ≡ t root

timing 1 time :

rages solvability of system 0.1 - 0.1 12 mat 1 - 112 mat 10 - 1012 mat 100 - 10012 mat 41 mat 0.013 0.125 12 mat 0.04 0.375 12 mat 0.375 0.875 12 mat 0.875 0.999 12 mat 41 mat ≡

w t argument w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / : w t Sol w t argument 0 ≡ t 1 - 1 solve : x c 1 0.04 x ≤ (x 0.375 ≤ (& cases : 31 line

0.04 t Sol 0.01005 - 0.01005 21 mat

0.375 t Sol 0.88357 - 0.88357 21 mat

t0 1 time :

5 x f * x t Sol 2 el x c * x t Sol 1 el x c * 0.04 0 + 40 black 1 5 mat 4 1 sys

1 time t0 - min 7.2

Maple RootOf . . .

applied for solving PDF in statistics..

]]>

Fréchet distribution
α > 0 ... β > 0 α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

w α β x Fr ≡ x solve maple w β * α β_Z / (1 α +^* β_Z / (α^- exp * - RootOf

recast RootOf

Fr 51 x Fr 11 x Fr 21 x Fr 0.5 1 x Fr 41 sys :

plot Fr 1 - Y 1.5 Y 22 mat X 1 el Y . 14 black 15 mat X 2 el Y . 14 black 15 mat 41 sys :

α 5 : β 1 : Y 1 : 13 mat w t ipdf w β * α β t / (1 α +^* β t / (α^- exp * - : w t Sol w t ipdf 0 ≡ t 1 - 10 solve : X Y t Sol (: 41 sys

x c 1 0.001 x ≤ (x 1.874278 ≤ (& cases :

0.96419

t0 1 time :

Fréchet distribution ... α >0 ... β > 0

plot x t Sol 1 el x c * x t Sol 2 el x c * 1.874 0.9642 + 100 black 1 5 mat 3 1 sys

1 time t0 - min 3

X α β X Fr vectorize 21 mat 0.809 1.234 21 mat 1121 mat 21 mat

0.25 t Sol 0.5 t Sol 1.25 t Sol 31 mat 0.728 1.623 21 mat 0.76 1.427 21 mat 0.834 1.166 21 mat 31 mat

NOTE ... Canvas Sol(x,t) is wrt [α, β]. For oter Frechet [α, β], consider ranging the solve bloc to suit the PDF as well as finding optimum c(x). This can be done using Golden ratio.

<p> Max/Min 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"

Fréchet distribution
α > 0 ... β > 0 α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

α 1 : β 1 : 12 mat

x λ 10 x ≤ (x 10 ≤ (& cases :

α β x Fr x λ *

x f α β x Fr :

X f 01 y1 y2 ≥ 1012 -^Extrema 1 el :

X 0.5

Maple RootOf . . .

stricly a freak

]]>

x f x 2 π * / 2 π * x / tanh * sqrt :

w x f ≡ x solve maple 2 w 2^* π *_Z w 2^*_Z exp 2^*_Z w 2^* +_Z exp 2^- 1 + RootOf exp 2^1 + (*_Z w 2^*_Z exp 2^*_Z w 2^* +_Z exp 2^- 1 + RootOf exp 1 - (_Z w 2^*_Z exp 2^*_Z w 2^* +_Z exp 2^- 1 + RootOf exp 1 + (* /

w t sol 2 w 2^* π * t w 2^* t exp (2^* t w 2^* + t exp (2^- 1 + exp (2^1 + (* t w 2^* t exp (2^* t w 2^* + t exp (2^- 1 + exp 1 - (t w 2^* t exp (2^* t w 2^* + t exp (2^- 1 + exp 1 + (* / eval :

Seeded discrete solver/observer ...

NOTES ... 1. The function starts existing around 'x = 0.02, f(0.02) = 0.0564' our choice to solve is wise , down to lower possible limit. The upper limit 0.99 is practical. In fact, at 1, the solution is infinite. Pushing a bit more, 'solve' cries to murder. 2. The seeded scanner/observer resumes the inverse function in the prescribed range [0.06 .. 0.99]. 3. The seeded solver reconciliates inversing a function in native Smath ... solves in-situ, stricking of accuracy ~ 15 D !!!

Design a discrete seeded algo inverser  in the L, H range, N points f L H N Inverse dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x 0.01 25 solve : for ω sol augment 51 line :

[0.01,25] bracket solver to avoid "No real roots"

t0 1 time :

x f x 2 π * / 2 π * x / tanh * sqrt :

Sol f 0.06 0.99 201 Inverse : x spline Sol 1 col Sol 2 col x linterp : 21 sys

1 time t0 - min 0.9

low range of inversibility
high range of inversibility Sol 1 row Sol Sol rows row 21 mat 0.06 0.0226 12 mat 0.9854 20.9134 12 mat 21 mat

0.125 spline f 0.125

u 0.5 :

U x f u - 0 ≡ x 0.01 40 solve : 1.5719

U f 0.500000000000001

u spline f 0.5

u 0.875 :

U x f u - 0 ≡ x 0.01 40 solve : 6.3582

U f 0.875000000000000

u spline f 0.875

x λ 1 0.06 x ≤ (x 0.2 ≤ (& cases :

X x f x spline - 0 ≡ x 0.15 0.17 solve :

X X f 12 mat 0.1591 0.1591 12 mat

Magnify the function to see

50 x f * Sol 2 1 sys

The function and its inverse in native Smath ...

x f x spline x λ * X X f + 200 black 1 5 mat 0.06 Sol 1 row 1 2 el + 100 black 1 5 mat 4 1 sys

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

NASA(h) is kPa vs h[m] ... we solve [m] actual altitude vs measured [kPa] h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

60000 NASA 0 NASA 21 mat 0.0279 101.4009 21 mat

x nasa 11000 - x ≤ (x 60000 ≤ (& cases :

@ ISO 101.325 kPa, NASA(h) has plunged  6.33 m in the grass ! ISOerror h NASA 101.325 - 0 ≡ h 010 solve : 6.33

x NASA x nasa * 11000 64 11000 32 - 2 2 mat 25000 32 25000 32 - 2 2 mat 0 101.7 + 40 red 1 5 mat 60000 0 + 40 red 1 5 mat 5 1 sys

pressure [kPa] vs altitude [m]

L 60000 NASA : H 101.4009 : N 200 : 13 mat u L H N xd : 21 sys

altitude i 1 u rows range m_altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 4 round vectorize m_altitude 0 round vectorize augment 21 line :

t0 1 time :

NASA(h) solved [m_altitude] from measured kPa

altitude 101.325 0 + 40 red 1 5 mat 2 1 sys

1 time t0 - min 1.4

Mach sub/super sonic

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

]]>

Plot limit x c 10 x ≤ (x 5 ≤ (& cases :

t0 1 time :

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic mach x 1 + (117 mach 2^* / - (2.5^* sqrt / k - 0 ≡ mach 1 roots k x ≤ Subsonic cases : 21 line

x Subsonic x Supersonic x c * k 1 + 40 black 1 5 mat 3 1 sys

1 time t0 - 78 s *

A very accurate global model has been designed, not given here to de-confuse the intent "solve"

subsonic . | . supersonic

]]>

1 time timing - min 14