plot(data,char,size,color)

23ac2899-359e-4979-ba3e-6a9cb52d3687 74 9 12 decimal

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

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 :

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 -^* 25000 h ≤ (h 60000 ≤ (& cases :

Explain this document . . .

1. data is list-select from the discrete solved NASA(h).

2. list-select was used for Thiele continued fractions.

3. The two J_fractions collected from Mathcad 11,

α, β from the J_Frac GenfitMatrix.

4. Enter kPa . . . read altitude [m]

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Low altitude [0 . . . 20 km]

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data 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 :

XY data 410 range 12 range el :

β 909.3 3358920 - 3238.9345 8.9315 0.59215 10907209250 158.368 5.259 81 mat :

x β J22781 β 1 el x * β 2 el + β 6 el x β 3 el + β 7 el x β 4 el + β 8 el x β 5 el + / - / - / + :

XY data 513 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat 21 sys

Absolute  residuals δ22781 j 1 XY rows range Δ j el Y j el X j el β J22781 - eval : for X Y Δ 0 round vectorize augment 21 line :

δ22781 3.6 227814 5.6 199215 - 9.7 1645321 14.7 1376618 - 22.3 111157 32.4 865410 50.2 55635 - 75.5 24235 101.3 0093 mat

High altitude [20 . . . 60 km]

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α 172.915964605 789.761138489 8.81170654825 0.365358854595 256651.148563 1.63576165685 61 mat :

x α J60000 α 1 el x * α 2 el + α 5 el x α 3 el + α 6 el x α 4 el + / - / + :

XY data 16 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat 21 sys

Absolute  residuals δ60000 j 1 XY rows range Δ j el Y j el X j el α J60000 - eval : for X Y Δ 0 round vectorize augment 21 line :

δ60000 0600000 0.5 35462313074712 - 2.1 2621721 3.6 2278120 - 5.6 19921963 mat

x j1 10 x ≤ (x 3.6 ≤ (& cases : x j2 1 3.6 x ≤ (x 101.325 ≤ (& cases : support data o 5 black plot : 31 sys

x α J60000 x j1 * x β J22781 x j2 * support 3.6 22781 + 40 black 1 5 mat 4 1 sys

kPa alt[m] Δ 0600000 0.5 35462313074712 - 2.1 2621721 3.6 227814 5.6 199215 - 9.7 1645321 14.7 1376618 - 22.3 111157 32.4 865410 50.2 55635 - 75.5 24235 101.3 00143 mat

kPa altitude kPa α J60000 0 kPa ≤ (kPa 5.6 ≤ (& kPa β J22781 cases :

kPa

metres

kPa ... m 30 altitude 9160

m ... kPa 11000 NASA 22.71

al_nleqsolve λ 59814 2.56094 60887 - 0.01159 0.19977 51 mat :

modified Carlos May 31 2019 x λ f λ 1 el λ 2 el * λ 3 el x λ 5 el^* + λ 2 el x λ 4 el^+ / 1.7 - :

x c 1 9.7 x ≤ (x 101.325 ≤ (& cases :

XY data : X XY 1 col : Y XY 2 col : 12 mat 21 sys

Absolute  residuals δ j 1 XY rows range Δ j el Y j el X j el λ f - eval : for X Y Δ 1 round vectorize augment 21 line :

50034.44 2.798353 41288.11 - 0.09867 0.264406 51 mat 3.8 - 21 mat

59814 2.56094 60887 - 0.01159 0.19977 51 mat 1.7 - 21 mat

δ δ : δ δ 813 range 13 range el : 21 sys

14.7 25000 14.7 10000 - 2 2 mat x λ f x c * support 3 1 sys

δ 14.7 13766 124.1 22.3 11115 5.7 32.4 8654 6.2 - 50.2 5563 6.5 75.5 2423 4.7 101.3 0063 mat

Commercial altitude [0 . . . 14 km]

wolf and rabbit function

The wolf and rabbit function is the solution of the corresponding DE equation.

The fit to NASA is stricking of realism and confirms the [0 ... 14 km] can be

represented as a single NON-homogeneous layer. The upper range is included

from the wolf changing gear speed. If we can fit the speculative NASA ,

certainly we can fit the true atmosphere in the range [0 . . . ~ 60 km].

]]> plot(data,char,size,color)

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 :

n x a / n nthroot (1 n +^a * 1 - n + (* x a x / n nthroot * 1 n + (* - (* 1 - n + (n 1 + (* / a n * 1 - n 2^+ / +

data 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 101.325 0132 mat :

x a n Wolf 1 0.64 / x a / (1 n / 1 +^a 1 n / 1 + / * n n 1 - / x * a x / (1 n /^* - (* a n n 2^1 - / * + :

XY data 813 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat 21 sys

x c 10 x ≤ (x 101.325 ≤ (& cases :

a 13646.4 : n 1.231987 : 12 mat

support data o 5 black plot :

x f1 x a n Wolf 101.325 a n Wolf - :

x f1 x c * support 14.7 13766 + 150 red 1 5 mat 3 1 sys

Absolute  residuals δ j 1 XY rows (range Δ j el Y j el X j el f1 - eval : for X Y Δ 1 round vectorize augment 21 line :

δ 14.7 13766 58.7 22.3 11115 39.5 - 32.4 8654 35.5 - 50.2 5563 7.9 - 75.5 2423 0.4 - 101.325 0063 mat

Upper altitude [14 . . . 60 km]

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x a n Wolf 1 0.65 / x a / (1 n / 1 +^a 1 n / 1 + / * n n 1 - / x * a x / (1 n /^* - (* a n n 2^1 - / * + :

XY data 28 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat 21 sys

x c 10 x ≤ (x 101.325 ≤ (& cases :

a 5960.1 : n 1.052575 : 12 mat

support data o 5 black plot :

x f2 x a n Wolf 101.325 a n Wolf - :

x f2 x c * support 14.7 13766 + 150 red 1 5 mat 3 1 sys

Absolute  residuals δ j 1 XY rows (range Δ j el Y j el X j el f2 - eval : for X Y Δ 0 round vectorize augment 21 line :

δ 0.5 35462563130747105 - 2.1 26217146 - 3.6 22781214 - 5.6 19921245 - 9.7 16453108 - 14.7 13766073 mat

103 -^f2 65553

Reconciliation [0 . . . 60 km]

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x altitude x f2 0 x ≤ (x 14 ≤ (& x f1 cases :

14.7 25000 14.7 10000 - 2 2 mat x altitude x c * support 3 1 sys

Thiele J_frac 2 seg't kPa alt[m] Δ 0600000 0.5 35462313074712 - 2.1 2621721 3.6 227814 5.6 199215 - 9.7 1645321 14.7 1376618 - 22.3 111157 32.4 865410 50.2 55635 - 75.5 24235 101.3 00143 mat

Wolf-Rabbit 2 segments kPa alt[m] Δ 0.5 35462563130747105 - 2.1 26217146 - 3.6 22781214 - 5.6 19921245 - 9.7 16453108 - 14.7 137660 14.7 1376659 22.3 1111540 - 32.4 865436 - 50.2 55638 - 75.5 24230 101.325 00143 mat

___________

Observe . . .

Not only Thiele is more accurate, it runs several times faster vs Wolf.

Wolf uncludes two X^Y power function each segment.

Wolf : rates 2*~60 AOPS [Arithmetic Operations]

Thiele: total 30 AOPS.

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Rescue pilot [0 . . . 12 km]

for minimal pocket calculator

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data 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 101.325 0132 mat :

x KHL K H L x + / + :

x c 1 14.7 x ≤ (x 101.325 ≤ (& cases :

K 13410.7 - : H 2338457 : L 73.0471 : 13 mat

XY data 813 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat 21 sys

support XY o 5 black plot :

22.3 20000 22.3 10000 - 2 2 mat x KHL x c * support 3 1 sys

Absolute  residuals δ j 1 XY rows (range Δ j el Y j el X j el KHL - eval : for X Y Δ 1 round vectorize augment 21 line :

δ 14.7 13766 526.7 22.3 111150 32.4 8654 111.9 - 50.2 55630 75.5 2423 91.5 101.325 0063 mat

jmGiraud: June 01 2019

Corrected [0 . . . 14 km]

for minimal pocket calculator

0 error @ sea level

]]> plotG, discrete poly

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

Low High step x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct : for U f augment 31 line :

Cholesky

data 14.7 13766 526.7 22.3 111150 32.4 8654 111.9 - 50.2 55630 75.5 2423 91.5 101.325 0063 mat :

x φ .25 x x 2^x 3^x 4^51 mat :

X data 1 col : Y data 3 col : pts X Y . 14 black plotG : 31 line

Cholesky companion solver j 1 x φ length range i 1 X length range M i j el X i el φ j el : for for Φ M transpose M * : d M transpose Y * : b Φ 1 -^eval d * : 41 line

b 11145.317138081 236.549551777 - 6.585866153 0.073542888 - 0.000285143 51 mat

b 11145.3 236.5496 - 6.58587 0.073543 - 0.000285143 51 mat :

x fit b x φ * :

<= recast the scalar model

Poly 14.7 101.235 15 x fit inc :

pts Poly 2 1 sys

'0' uncorrected

'1' corrected

data 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 101.325 0132 mat :

x KHL K H L x + / + 0 x fit 15.9 + (* + :

x c 1 9.7 x ≤ (x 101.325 ≤ (& cases :

K 13410.7 - : H 2338457 : L 73.0471 : 13 mat

XY data 713 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat 21 sys

support XY o 5 black plot :

14.7 20000 14.7 10000 - 2 2 mat x KHL x c * support 3 1 sys

Absolute  residuals δ j 1 XY rows (range Δ j el Y j el X j el KHL - eval : for X Y Δ 1 round vectorize augment 21 line :

δ 9.7 16453 1603.4 14.7 13766 526.7 22.3 111150 32.4 8654 111.9 - 50.2 55630 75.5 2423 91.5 101.325 0073 mat

... Uncorrectd 9.7 16453 1603.4 14.7 13766 526.7 22.3 111150 32.4 8654 111.9 - 50.2 55630 75.5 2423 91.5 101.325 0073 mat

... Corrected 9.7 16453 540.6 14.7 13766 1.1 - 22.3 11115 57.2 - 32.4 8654 23.6 50.2 5563 31.3 - 75.5 2423 6.8 - 101.325 0073 mat

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 :

data 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 101.325 0132 mat :

β 2243.6152 - 39.8554 - 893456.15 40.90062 41 mat :

Exact4 [0 . . . 10 km]

0 error @ sea level

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x β Exact4 β 1 el β 2 el x * + β 3 el β 4 el x + / + :

XY data 913 range 12 range el : X XY 1 col : Y XY 2 col : 12 mat data data o 5 black plot : 31 sys

x j 10 x ≤ (x 101.325 ≤ (& cases :

Absolute  residuals Δ4 j 1 XY rows range Δ j el Y j el X j el β Exact4 - eval : for res X Y Δ 3 round vectorize augment : kPa alt[m] Δm 13 mat res stack 31 line :

Δ4 kPa alt[m] Δm 22.3 11115 110.6 32.4 86540 50.2 55630 75.5 24230 101.3 0063 mat

Δm j 1 XY rows range Δ j el Y j el X j el β Exact4 - : for Δ 21 line :

22.3 20000 22.3 10000 - 2 2 mat x β Exact4 x j * data 3 1 sys

safe from commercial altitude ~ 11 km

Δm 110.565 0 0.001 0.001 - 051 mat