<p> plotG(vx,vy,char,size,color) </p>

2b277e9e-811d-4d82-827e-a928ad0aeb72 30 4 5 decimal

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

Both optimiz NONE ... construct/export continued fraction

Thiele on length of support points 'j' j K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 51 line :

symbolic expansion of Thiele 'Cfr' algo style 6 K X x Cfr K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el / + / + / + / + / +

https://en.wikipedia.org/wiki/Thorvald_N._Thiele

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<p> plotG(vx,vy,char,size,color) </p>

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 :

DATA km hPa °C 11227 56.5 - 12194 56.5 - 13165 56.5 - 14141 56.5 - 15121 56.5 - 205546 - 301138 - 4035 - 50 0.9 160 0.25 20 - 1004104 -^* 64 - 200 1.3 106 -^* 52.2 - 3002107 -^* 95.3 - 400 4.4 108 -^* 97.3 - 500 1.1 108 -^* 97.7 - 163 mat :

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

Courtesy: Robert Adair from my Original Excel

246789101118 mat 11 ... 60 km ≡ 2478111314151619 mat 20 ... 500 km ≡ 21 mat

data List collects index list select 246789101118 mat : 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 :

select the support points XY DATA List :

XY 1122713165151212055301140350 0.9 60 0.25 82 mat

length Thiele j XY rows : 8

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 51 line :

symbolic expansion of Thiele continued fraction algo style K X x Cfr K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el x X 6 el - K 7 el x X 7 el - K 8 el / + / + / + / + / + / + / +

recast the Cf on the system  [K,X,x] x Cf K X x Cfr :

data set vx DATA 1 col : vy DATA 2 col : 21 sys

support points X XY 1 col : Y XY 2 col : 21 sys

Thiele K's coefficients
@ support points X X Y 0 Thiele 1122713 0.0323 - 15 365.1111 - 20 0.307 30 176.4421 - 40 0.0127 - 50 296.6245 60 1.8836 82 mat

Thiele K's continued fraction K X Y 0 Thiele 2 col :

Thiele x 1160 11.125 range : i 1 x rows range v i el x i el Cf eval : for x v augment 31 line :

X XY 1 col : Y XY 2 col : 12 mat data DATA 216 range 12 range el : vx data 1 col : vy data 2 col : 12 mat support X Y + 15 green plotG : data vx vy . 10 black plotG : 51 line

round off for the visible float
expansion ... assumed legitimate
That helps a lot evaluation of the
floated coefficients. K K 6 round vectorize : 227 0.032258 - 365.111111 - 0.306962 176.44208 - 0.012708 - 296.624465 1.883647 81 mat

data Thiele support 3 1 sys

Next objective: Export a visible/executable formula

Step: 1

Define Symbolic ... Optimiz = NONE x collect x Cf confrac x convert evalf maple :

Step: 2

Expand Symbolic ... Optimiz = NONE x collect 2330303783 x * 5000000000 / 68947242310000000 / - 1518420663500000 x 3354266627500000000 / + 292922674310000000 x 48159403920000000 / - 6252855012500000 x 100920706750000000 / - (* / + (* / + (* / +

Step: 3 eval the floating coefficients

23303037835000000000 / 0.4661

Num coefficients

J_Frac coefficients

1518420663500000 / 3036.8413

3354266627500000000 / 6.7085

68947242310000000 / 68.9472

292922674310000000 / 292.9227

48159403920000000 / 24.0797

6252855012500000 / 250.1142

100920706750000000 / 20.1841

Step: 3 finish the export

x J_Frac 0.4661 x * 68.9472 - 3036.8413 x 6.7085 + 292.9227 x 24.0797 - 250.1142 x 20.1841 - / + / + / + :

J_Frac x 1160 11.125 range : i 1 x rows range v i el x i el J_Frac eval : for x v augment 31 line :

data J_Frac support 3 1 sys

This document resumes an immense amount of work/collaboration. Inspired from F.B. Hildebrand, made work in Excel, HP 48SX ... Credit to Robert Adair for the Thiele(X,Y,Δ) in Mathcad 11. Adapted/exploited by Jean. Thanks Carlos for the "Automatic" bar saturation.

β 0.4661 68.9472 3036.8413 292.9227 250.1142 6.7085 24.0797 20.1841 81 mat :

Troposphere/Mesosphere J(β,"km")≡ "hPa" β x J β 1 el x * β 2 el - β 3 el x β 6 el + β 4 el x β 7 el - β 5 el x β 8 el - / + / + / + (:

Observe: This Smath document resumes Thiele same as Mathcad [8 Pro, 2001i, 11]

Unfortunately, I could not instruct Maple to float Step 2 to arrive at the explicit Step 3

which is the final/desired J_Fraction export. The all manual work here, Mathcad 11

does it directly from Step 1 [float].

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