Create/Xport Thiele Continued Fraction

625871cb-8ae2-4fe7-9ef6-16dc6e7074da 19 4 5 decimal

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

Part.1 Create/Export a Thiele Continued Fraction

user length j 7 :

Thiele on length 'j' of support points 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 - (1 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 :

Expandable Thiele wrt X's, K's
Export, enable, set 'j' on length of support points 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 / + / + / + / + / + / +

Both Optimize NONE

Part.2 Open style 'K, Table' ... fill-in Part.1 wrt Project

t0 1 time :

X 0 0.0917 0.1842 0.2779 0.3732 0.4703 0.5699 0.6721 0.7776 0.8867 1111 mat :

Y 0 0.1543 0.2943 0.4188 0.5315 0.6327 0.7236 0.8049 0.8776 0.9424 1111 mat :

Thiele table of inverse differences ... Carlos i 1 Y rows range Table i 1 el Y i el : for j 2 Y rows range i j Y rows range Table i j el X j 1 - el X i el - Table j 1 - j 1 - el Table i j 1 - el - / : cont j : 21 line break try for for i 1 cont range K i el Table i i el : for 31 line

K K 6 round vectorize :

... NUMERIC ...

K 0 0.594297 2.927667 0.391282 - 0.469189 - 1.153435 - 4.339277 - 0.061307 2.535741 0.867401 0.230456 111 mat

Expand to View/Export as required Table 11 mat

4 appVersion 0.98.6179.21440 ≡

Part.3 Recast the Project

isolate 'K'

Thiele on length of support points 'j' K X x Cfr j K rows 1 - : N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (1 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 61 line :

K 0 0.594297 2.927667 0.391282 - 0.469189 - 1.153435 - 4.339277 - 0.061307 2.535741 0.867401 0.230456 111 mat :

x λ 10 x ≤ (x 1 ≤ (& cases :

Recast inclusive of
parameters K's , X's x Cf K X x Cfr :

NUMERIC

SYMBOLIC

x 0.5 :

x Cf x diff 1.4026

g 0.17 :

x g :

x Cf x diff 800 -

x Cf x λ *

1 time t0 - 3 s *

Glitches are almost not visible in the function. They may silently offend more maths. Glitches reveal not true enough data set. Interpolate for the offending from good near values ... fails ... Thiele fits/interpolates more than any others. The competitive is the Padé style rational fraction, subject to glitches as well.

x Cf x diff x λ * g 2 g 0.25 - 2 2 mat 2 1 sys

1 time t0 - 18 s *

Part.4 QED Thiele free of glitches

QED 4 points circle absent of glitches TRANSIT via FILE

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 :

K Clear 1

x Clear 1

data 1 - 0 0.86602540378444 - 0.5 0.5 - 0.86602540378444 0142 mat :

length Thiele j data rows : 4

X data 1 col :

Y data 2 col :

Continued on length 'j'  of support points 'X' K X x Th j data rows : 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 61 line :

Optimiz NONE ... Export the RHS member K X x Th K 1 el x 1 - (- K 2 el x 2165063509461125000000000000 / - (- K 3 el x 12 / - (- K 4 el / + / + / +

CWD C:\SmathFile CurrentDirectory :

K X x Th CWD Th wfile 1

K X x Q CWD Th rfile :

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 :

K X Y 0 Thiele 2 col (: 0 0.26794919243112 1.18301270189221 1.34969465094621 1014^* 41 mat

data data . 14 black plot :

x c 11 - x ≤ (x 0 ≤ (& cases :

K 0 0.26794919243112 1.18301270189221 1.34969465094621 1014^* 41 mat :

x f K X x Q :

0.5 - f 0.866

x f x diff x c *

accuracy of 4 points fit is not the question. Rather derivative glitches ... none to be seen.

0.125 x f x diff x c * ( * x f x c * data 1 x 2 ^ - sqrt ± 7 1 sys

Sources ... Contributions: 1. Jean adapted from F.B. Hildebrand book. 2. Jean installed/exploited in Excel. 3. Robert A. Mathcad 11 Thiele(X,Y,Δ). 4. Carlos Smath open style Part.2 5. Theodore Bones [age 85 +] Mathcad 11. 6. C.T. Fikes [IBM] 'J_Frac' reduction. 7. X,Y Part.2 Valentino Thiele-McCabe project.

1 time t0 - 20 s *

Part.5 Application type 'T' T/C [ThermoCouple]

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 :

Type_T 6.256 - 42.9883 6.171 - 0.031436074 - 5.873 - 8.430462201 - 5.315 - 0.224722515 - 2.741 - 13.914251736 - 1.435 - 2.35326963 - 1.068 1.901857112 - 5.341 1.371713776 - 11.013 6.086659317 - 16.095 17.443238892 18.218 0.478252126 112 mat :

mV Type_T 1 col : K Type_T 2 col : 21 sys

x T K 1 el x mV 1 el - K 2 el x mV 2 el - K 3 el x mV 3 el - K 4 el x mV 4 el - K 5 el x mV 5 el - K 6 el x mV 6 el - K 7 el x mV 7 el - K 8 el x mV 8 el - K 9 el x mV 9 el - K 10 el x mV 10 el - K 11 el / + / + / + / + / + / + / + / + / + / + :

ET 6.256 - 268.935 - 6.229 - 259.34 - 6.198 - 252.87 - 6.171 - 248.595 - 6.154 - 246.048 - 5.873 - 218.789 - 5.753 - 210.002 - 5.536 - 195.802 - 5.315 - 182.962 - 2.741 - 78.476 - 1.435 - 38.862 - 00 1.068 26.87 4.277 100 5.341 122.37 7.036 156.634 11.013 231.938 13.219 271.442 16.095 321.108 16.473 327.502 18.218 356.66 212 mat :

XY ET 1 col ET 2 col . 10 black plotG :

Reverse Thiele fit T t 6.25 - 18 6.125 - range : i 1 t rows range u i el t i el : v i el t i el t i el T * eval : 21 line for u v augment 31 line :

Absolute residuals over entire fixed points Resdl mV ET 1 col : °C ET 2 col : i 1 ET rows range Δ i el mV i el mV i el T * °C i el - 2 round : for mV °C Δ augment 41 line :

Type 'T' from Thiele reduced/recovered ... watch for glitch in your application

0 T XY 3 1 sys

Resdl 6.256 - 268.935 - 0 6.229 - 259.34 - 0.73 - 6.198 - 252.87 - 0.33 - 6.171 - 248.595 - 0 6.154 - 246.048 - 0.04 - 5.873 - 218.789 - 0 5.753 - 210.002 - 0 5.536 - 195.802 - 0.03 - 5.315 - 182.962 - 0 2.741 - 78.476 - 0 1.435 - 38.862 - 0000 1.068 26.87 0 4.277 100 0.06 - 5.341 122.37 0 7.036 156.634 0.35 11.013 231.938 0 13.219 271.442 0.02 16.095 321.108 0 16.473 327.502 0 18.218 356.66 0213 mat

NOTE ... The long continued fraction may be reduced to the J_Fract style. The procedure is via Smath/Maple companion. The steps are manual. Often successful, some evaluation may fail from Maple too long expansion of coeffients. Example not offered in this document. Considering this example, the counter part of Thiele is the normalized Padé rational fraction ... not offered. Thiele-McCabe is implicit of Thiele fit, 'possibly' from otherwise model function fit.

Jean Giraud: 2015/12/14 refreshed: 2020/10/12