Data set Project

edad3ec4-74e7-472f-bb8b-aa3421200cd0 14 4 5 decimal

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

Thiele fit to a data set ... Project model

Data set Project

data 2.5 106 -^* 1.778818183229 0.0001275 38.658198254162 0.0002525 52.139062264136 0.0003775 60.256719618864 0.0005025 65.970928718218 0.0006275 70.331149007028 0.0007525 73.828283904242 0.0008775 76.730296990662 0.0010025 79.19889148293 0.0011275 81.33882355239 0.0012525 83.221655296631 0.0013775 84.898332425285 0.0015025 86.40634266391 0.0016275 87.774028734142 0.0017525 89.023309923114 0.0018775 90.171466539596 0.0020025 91.232348371986 0.0021275 92.21721602382 0.0022525 93.135340800081 0.0023775 93.994441458828 0.0025025 94.801008186617 0.0026275 95.560546827317 0.0027525 96.277765705575 0.0028775 96.956720477295 0.0030025 97.600927773333 0.0031275 98.213455329596 0.0032525 98.79699400127 0.0033775 99.353916995412 0.0035025 99.886325697416 0.0036275 100.396090487486 0.0037525 100.884882207272 0.0038775 101.354199258346 0.0040025 101.805390204363 0.0041275 102.23967276697 0.0042525 102.658149947711 0.0043775 103.061823798714 0.0045025 103.451607177651 0.0046275 103.828333813569 0.0047525 104.192767123927 0.0048775 104.545607695102 0.0050025 104.887499934698 0.0051275 105.219037803991 0.0052525 105.540769846886 0.0053775 105.853203664706 0.0055025 106.15680977511 0.0056275 106.452025124335 0.0057525 106.739256116744 0.0058775 107.018881361114 0.0060025 107.291254123411 0.0061275 107.556704446932 0.0062525 107.815541187286 0.0063775 108.068053667878 0.0065025 108.314513342556 0.0066275 108.555175173213 0.0067525 108.790278893747 0.0068775 109.020050243403 0.0070025 109.244701943269 0.0071275 109.464434706732 0.0072525 109.679438111591 0.0073775 109.889891349339 0.0075025 110.095964006988 0.0076275 110.297816717077 0.0077525 110.495601732406 0.0078775 110.689463527336 0.0080025 110.879539304713 0.0081275 111.065959445776 0.0082525 111.24884794891 0.0083775 111.428322862832 0.0085025 111.604496630064 0.0086275 111.77747640939 0.0087525 111.947364439771 0.0088775 112.114258306421 0.0090025 112.278251234782 0.0091275 112.439432277969 0.0092525 112.597886644584 0.0093775 112.753695862313 0.0095025 112.906938000714 0.0096275 113.057687822113 0.0097525 113.206017021551 0.0098775 113.351994355211 0.0100025 113.495685802984 812 mat :

Utilities...ListCollect(data,list)...plot(data,char,size,clr)

data list ListCollect ct 1 : i list a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 31 line :

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 :

list 1247121930456519 mat :

support data list ListCollect :

XY support . 14 black plot :

Exeptionally suited Thiele fit

XY data 2 1 sys

1. Plug your data set as collected 15 D. 2. Select the support points. 3. Suite the fit ... here all automated. 4. Sanity glitches from 1rst derivative. 5. Your data set may NOT be well suited. 6. Thiele outputs a scalar interpolation f(x), differently than splines NFG Derivative(s 7. Thiele is not a panacea to other methods. 8. Example 1 may have a Broyden model fnct. 9. Example 2 has a perfect rational fit.

Thiele fit this project data set ...

Thiele(X,Y,Δ) ..... plot(data,char,size,clr)

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 :

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 :

j XY rows : 9

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 x X 8 el - K 9 el / + / + / + / + / + / + / + / +

X support 1 col : Y support 2 col : 12 mat

Thiele K's coefficients @ support points [X,Y] X Y 0 Thiele

Expand Thiele Table @ support points  [X,Y] X Y 1 Thiele

Diagonal elements are Thiele 'K' coefficients ... column 1 'Y' values @ 'X' support points X Y 1 Thiele 11 mat 1.778818 00000000 38.658198 0.000003 0000000 60.25672 0.000006 82.692442 000000 73.828284 0.00001 89.030233 0.000059 00000 84.898332 0.000017 95.035229 0.000081 28.603984 0000 93.135341 0.000025 100.050117 0.000108 30.704671 0.000417 000 100.39609 0.000037 104.888277 0.000146 32.949404 0.000518 13.57942 00 106.15681 0.000053 109.018248 0.000195 35.053533 0.00064 14.570778 0.001891 0 110.879539 0.000073 112.600842 0.000255 37.032076 0.000786 15.560254 0.002209 7.878473 99 mat 11 mat

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

store/isolate K K 9 round vectorize eval :

x λ 10 x ≤ (x 0.0125 ≤ (& cases :

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 :

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

x Cf support 2 1 sys

◄ NUMERIC ▲

Thiele support derivative(s) x D x Cf x diff x λ * :

Thiele supports derivative(s).

x D

Simpson does NOT warrants
singular/infinite limits
end points of the integrand.
Wisely sanity check Adaptive. x Cf x support 1 el support 9 el int 0.0914

Spool to file Thiele

CWD C:\SmathFile CurrentDirectory :

K CWD Kthiele wfile 1

K CWD Kthiele rfile :

X CWD Xthiele wfile 1

X CWD Xthiele rfile :

Y CWD Ythiele wfile 1

Y CWD Ythiele rfile :

Export Thiele project as stand alone fnct

renew [K, X, Y] from file ... Export decimals as needed.

K 1.778818183 0.000003389 82.692441728 0.000059169 28.603984069 0.00041653 13.579419569 0.001891345 7.878473255 91 mat

X 0.0000025 0.0001275 0.0003775 0.0007525 0.0013775 0.0022525 0.0036275 0.0055025 0.0080025 91 mat

Y 1.778818183 38.658198254 60.256719619 73.828283904 84.898332425 93.1353408 100.396090487 106.156809775 110.879539305 91 mat

Export the Thiele system for else document
copied/pasted from scalar turquoise above. x f 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 x X 8 el - K 9 el / + / + / + / + / + / + / + / + :

x λ 10 x ≤ (x 0.0125 ≤ (& cases :

XY X Y augment :

distatefull (0≤x)&(x≤0.0125)

x f 0 x ≤ ( x 0.015 ≤ ( & * XY 2 1 sys

.. GR-50-AA .. Ge Ω °K 35000 0.05 2320 0.1 364.6 0.2 164 0.3 73.75 0.5 33.55 1 24.71 1.4 19.34 2 13.66 4.2 102 mat :

clear system from previous use. X Y K x Clear 1

Germanium RTD

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...GR-50-AA ...

https://www.lakeshore.com/Documents/LSTC_Germanium_l.pdf Lake Shore germanium resistance temperature sensors are recognized as “Secondary Standard Thermometers” and have been employed in the measurement of temperature from 0.05 K to 30 K for more than 40 years.

Ge Ge 210 range 12 range el : X Ge 1 col : Y Ge 2 col : 12 mat data Ge : 31 sys

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 :

plotG(vx,vy,char,size,color)

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 :

Thiele(X,Y,Δ)

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 :

data List collects index list select 12345678919 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 35000 0.05 2320 0.1 364.6 0.2 164 0.3 73.75 0.5 33.55 1 24.71 1.4 19.34 2 13.66 4.2 92 mat

x c 1 13.66 x ≤ (x 35000 ≤ (& cases :

length Thiele j XY rows : 9

Cfr(K,X,x) ... recast ... support

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 :

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

support X Y + 15 green plotG : data vx vy . 10 black plotG : 21 line

K X Y 0 Thiele 2 col :

Escape some support end points to best view the bow plot region ...

data x Cf x c * support 3 1 sys

jmG: 20230118