Max/Min Golden ratio

4eacc4d1-3298-4e1d-8a78-74dabc06385e 45 15 12 decimal

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

3 appVersion 0.98.6179 ≡

x f x 2 π * / 2 π * x / tanh * sqrt :

Undocumented root solutions . . .

productive scalar canvas ... slow

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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 * / :

increment wisely w 0.04 0.375 0.04 0.375 0.04 - (67 / + range :

t0 1 time :

seeded solve block  ... two simultaneous branches i 1 w rows range New i el w i el t argument 0 ≡ t 1 - 1 solve transpose : for Unest New ... collect branches solution v New 1 el : j 2 New rows range v v New j el stack : for v 31 line tan_1 w v 1 col augment : tan_2 w v 2 col augment : 51 line

5 x f * tan_1 tan_2 0.04 0 + 40 black 1 5 mat 4 1 sys

1 time t0 - 47 s *

Maple RootOf . . .

applied for solving PDF in statistics..

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

y α β x Fr ≡ x solve maple y 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 :

blue peak 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.

Max/Min Golden ratio

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 * 0.5 α β 0.5 Fr + 40 red 1 5 mat 2 1 sys

x f α β x Fr :

X f 01 y1 y2 ≥ 1012 -^Extrema 1 el :

X 0.5

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

Ν

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

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 25000 NASA 11000 NASA 0 NASA 41 mat 0.0279 2.5223 22.7074 101.4009 41 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 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 :

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 25000 NASA 25000 + 100 red 1 5 mat 11000 NASA 11000 + 100 red 1 5 mat 4 1 sys

60000 NASA 25000 NASA 11000 NASA 0 NASA 41 mat 0.0279 2.5223 22.7074 101.4009 41 mat

1 time t0 - min 1.4

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

Mach sub/super sonic

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].

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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

t0 1 time :

==========================

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

1 time t0 - 76 s *

1 time timing - min 7.5

subsonic . | . supersonic

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Overall Mach subsonic-supersonic ... a damned super-reconciliation x J 14.2683 1189.48 x 98.2703 + 66.9393 x 6.74479 + 0.726987 x 0.367336 + 0.043852 x 0.470545 - 0.293801 x 0.0549761 - 0.00515151 x 0.0199953 + / + / + / - / - / - / - :

Plot limit x c 10 x ≤ (x 15 ≤ (& cases :

x J x c * k 1 + 100 red 1 5 mat 2 1 sys

0.0125 Subsonic 0.0125 J - 0.25 Subsonic 0.25 J - 0.5 Subsonic 0.5 J - 0.75 Subsonic 0.75 J - k Subsonic k J - 51 mat 0.00 0.00 0.00 0.00 0.00 51 mat

k Supersonic k J - 5 Supersonic 5 J - 15 Supersonic 15 J - 25 Supersonic 25 J - 32 Supersonic 32 J - 51 mat 0.00 0.00 0.00 0.00 0.01 51 mat

Revisit altitude [m] vs measured presuure [kPa]

altitude data set

altitude 0.0279 60000 0.5373 35423 1.0467 30709 1.5561 28034 2.0656 26180 2.575 24868 3.0844 23719 3.5938 22745 4.1032 21901 4.6126 21155 5.122 20488 5.6314 19884 6.1409 19332 6.6503 18825 7.1597 18355 7.6691 17917 8.1785 17507 8.6879 17122 9.1973 16760 9.7067 16416 10.2162 16090 10.7256 15780 11.235 15485 11.7444 15202 12.2538 14932 12.7632 14673 13.2726 14423 13.782 14183 14.2914 13952 14.8009 13729 15.3103 13514 15.8197 13305 16.3291 13103 16.8385 12908 17.3479 12718 17.8573 12533 18.3667 12354 18.8762 12180 19.3856 12010 19.895 11845 20.4044 11684 20.9138 11527 21.4232 11374 21.9326 11224 22.442 11078 22.9515 10932 23.4609 10792 23.9703 10654 24.4797 10519 24.9891 10386 25.4985 10255 26.0079 10126 26.5173 10000 27.0267 9875 27.5362 9752 28.0456 9631 28.555 9512 29.0644 9394 29.5738 9278 30.0832 9164 30.5926 9051 31.102 8940 31.6115 8830 32.1209 8722 32.6303 8615 33.1397 8509 33.6491 8405 34.1585 8302 34.6679 8200 35.1773 8100 35.6868 8000 36.1962 7902 36.7056 7805 37.215 7709 37.7244 7614 38.2338 7520 38.7432 7427 39.2526 7335 39.762 7244 40.2715 7154 40.7809 7064 41.2903 6976 41.7997 6889 42.3091 6802 42.8185 6716 43.3279 6632 43.8373 6547 44.3468 6464 44.8562 6382 45.3656 6300 45.875 6219 46.3844 6139 46.8938 6059 47.4032 5980 47.9126 5902 48.4221 5824 48.9315 5747 49.4409 5671 49.9503 5596 50.4597 5521 50.9691 5446 51.4785 5372 51.9879 5299 52.4973 5227 53.0068 5155 53.5162 5083 54.0256 5012 54.535 4942 55.0444 4872 55.5538 4803 56.0632 4734 56.5726 4666 57.0821 4598 57.5915 4530 58.1009 4464 58.6103 4397 59.1197 4331 59.6291 4266 60.1385 4201 60.6479 4136 61.1574 4072 61.6668 4008 62.1762 3945 62.6856 3882 63.195 3820 63.7044 3758 64.2138 3696 64.7232 3635 65.2326 3574 65.7421 3514 66.2515 3454 66.7609 3394 67.2703 3334 67.7797 3276 68.2891 3217 68.7985 3159 69.3079 3101 69.8174 3043 70.3268 2986 70.8362 2929 71.3456 2872 71.855 2816 72.3644 2760 72.8738 2705 73.3832 2649 73.8927 2594 74.4021 2540 74.9115 2485 75.4209 2431 75.9303 2377 76.4397 2324 76.9491 2271 77.4585 2218 77.9679 2165 78.4774 2113 78.9868 2061 79.4962 2009 80.0056 1957 80.515 1906 81.0244 1855 81.5338 1804 82.0432 1754 82.5527 1704 83.0621 1654 83.5715 1604 84.0809 1554 84.5903 1505 85.0997 1456 85.6091 1407 86.1185 1359 86.628 1310 87.1374 1262 87.6468 1214 88.1562 1167 88.6656 1119 89.175 1072 89.6844 1025 90.1938 978 90.7032 932 91.2127 885 91.7221 839 92.2315 793 92.7409 748 93.2503 702 93.7597 657 94.2691 612 94.7785 567 95.288 522 95.7974 478 96.3068 433 96.8162 389 97.3256 345 97.835 301 98.3444 258 98.8538 214 99.3633 171 99.8727 128 100.3821 85 100.8915 43 101.4009 02002 mat :

x m_kPa altitude 1 col altitude 2 col x linterp :

x λ 10 x ≤ (x 101.325 ≤ (& cases :

NASA(h) solved [m_altitude] from measured kPa

x m_kPa x λ * 101.325 0 + 40 red 1 5 mat 2.5223 25000 + 100 red 1 5 mat 22.7074 11000 + 100 red 1 5 mat 4 1 sys