xd(L,H,N) linfit modules

4fe9eb5c-a720-4f91-9916-4e1b877941f9 125 4 5 decimal

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

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [LS] Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

Vector gradient ... Optimiz Symbolic x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

L H N xd U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx - (dx i * + : for U 41 line :

Original XY

XY 1.6981 1.4 1.5978 5 1.4201 10 1.3042 15 1.2144 20 1.1246 25 1.0705 50 0.9956 90 0.9338 120 0.8687 150 0.7555 200 0.6384 250 0.5189 300 0.3978 350 0.2746 400 0.1498 450162 mat :

CY7 1.6981 1.4 1.5978 5 1.4201 10 1.3042 15 1.2144 20 1.1246 25 1.0705 50 0.9956 90 0.9338 120 0.8687 150 0.7555 200 0.6384 250 0.5189 300 0.3978 350 0.2746 400 0.1498 450162 mat :

1.6981 1.4 1.5978 5 1.4201 10 1.3042 15 1.2144 20 1.1246 25 1.0705 50 0.9956 90 0.9338 120 0.8687 150 0.7555 200 0.6384 250 0.5189 300 0.3978 350 0.2746 400 0.1498 450162 mat

eit90 259.3467 - 0.0011907 248.5939 - 0.00844974 218.7916 - 0.09171804 189.3442 - 0.21585975 38.8344 - 0.84414211 0.01 1.0000000 29.7646 1.11813889 156.5985 1.60980185 231.928 1.89279768 419.527 2.56891730 660.323 3.37600860 961.78 4.28642053 122 mat :

13.8033 259.3467 - 17.035 256.115 - 20.27 252.88 - 24.5561 248.5939 - 54.3584 218.7916 - 83.8058 189.3442 - 234.3156 38.8344 - 273.16 0.01 302.9146 29.7646 429.7485 156.5985 102 mat

Point triple de l'hydrogène 13,8033 -259,3467 Point d'ébullition de l'hydrogène à la pression de 33321,3 Pa 17,035 -256,115 Point d'ébullition de l'hydrogène à la pression de 101292 Pa 20,27 -252,88 Point triple du néon 24,5561 -248,5939 Point triple de l'oxygène 54,3584 -218,7916 Point triple de l'argon 83,8058 -189,3442 Point triple du mercure 234,3156 -38,8344 Point triple de l'eau 273,16 0,01 Point de fusion du galium 302,9146 29,7646 Point de congélation de l'indium 429,7485 156,5985

x f1 68.4823 40.0151 x * - : x f2 522.1143 434.5459 x * - : 21 line

eit90 °K °C 13.8033 259.3467 - 17.035 256.115 - 20.27 252.88 - 24.5561 248.5939 - 54.3584 218.7916 - 83.8058 189.3442 - 234.3156 38.8344 - 273.16 0.01 302.9146 29.7646 429.7485 156.5985 112 mat ≡

CY7 1.6981 1.4 1.5978 5 1.4201 10 1.3042 15 1.2144 20 1.1246 25 1.0705 50 0.9956 90 0.9338 120 0.8687 150 0.7555 200 0.6384 250 0.5189 300 0.3978 350 0.2746 400 0.1498 450162 mat :

x CY7 x f2 0.15 x ≤ (x 1.15 ≤ (& x f1 cases :

L 0.15 : H 1.7 : N 900 : 13 mat U L H N xd : 21 line

cy7 U U CY7 vectorize augment :

CY7 CY7 o 5 red plotG :

Explain something . . .

The long Omega data set is tabulated from only few eit90 fixed points, best

estimated from accurate supplementary fake points in the sub-cryogenic eit90 .

The collection is then tabulated from ChebyShev polynomials.

https://www.omega.com/temperature/z/pdf/z090-093.pdf

The best offer is the fit in two linear segments. The break region can be fillet

from Hermite spline, not offered from not enough information wrt this region.

The document is in reverse Engineering form Y [°K] <= X[Volt] .

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t0 1 time :

CY7 cryogenic sensor linear fit in the original domain [Ωmega data source]

cy7 CY7 2 1 sys

https://www.omega.com/temperature/z/pdf/z090-093.pdf

ChebyShev

c3 71.818025 63.799888 - 1.669931 2.314228 1.566635 0.723026 0.149503 - 0.046876 0.388555 - 0.056889 0.116823 - 0.058580 121 mat :

c2 17.304227 7.894688 - 0.453442 0.002243 0.158036 0.193093 - 0.155717 0.085185 - 0.048550 0.018312 - 0.039255 111 mat :

c4 287.756797 194.144823 - 3.837903 - 1.318325 - 0.109120 - 0.393265 - 0.146911 0.111192 - 0.028877 0.029286 - 0.015619 111 mat :

CY7 1.6981 1.4 1.5978 5 1.4201 10 1.3042 15 1.2144 20 1.1246 25 1.0705 50 0.9956 90 0.9338 120 0.8687 150 0.7555 200 0.6384 250 0.5189 300 0.3978 350 0.2746 400 0.1498 450162 mat :

c1 7.556358 5.917261 - 0.237238 0.334636 0.0568642 - 0.019929 - 0.020715 - 0.014814 - 0.008789 - 0.008554 - 101 mat :

x ω1 x 1.32412 - (1.69812 x - (- 1.69812 1.32412 - (/ :

x ω3 x 0.923174 - (1.13935 x - (- 1.13935 0.923174 - (/ :

x ω4 x 0.079767 - (0.999614 x - (- 0.999614 0.079767 - (/ :

x ω2 x 1.11732 - (1.42013 x - (- 1.42013 1.11732 - (/ :

2. Create ChebyShev Optimiz: Symbolic x T1 n 0 c1 rows 1 - (range F n 1 + el c1 n 1 + el n x acos * cos * : for F 21 line :

x T3 n 0 c3 rows 1 - (range F n 1 + el c3 n 1 + el n x acos * cos * : for F 21 line :

x T2 n 0 c2 rows 1 - (range F n 1 + el c2 n 1 + el n x acos * cos * : for F 21 line :

x T4 n 0 c4 rows 1 - (range F n 1 + el c4 n 1 + el n x acos * cos * : for F 21 line :

n1 x T1 rows : n2 x T2 rows : n3 x T3 rows : n4 x T4 rows : 41 line

n1 n2 n3 n4 41 mat 1011121141 mat

x S1 x ω1 T1 j el j 1 n1 sum 11 line :

x S2 x ω2 T2 j el j 1 n2 sum 11 line :

x S3 x ω3 T3 j el j 1 n3 sum 11 line :

x S4 x ω4 T4 j el j 1 n4 sum 11 line :

L 1.32412 : H 1.69812 : N 30 : x1 L H N xd : 41 line

L 1.11732 : H 1.42013 : N 30 : x2 L H N xd : 41 line

L 0.923174 : H 1.13935 : N 30 : x3 L H N xd : 41 line

L 0.079767 : H 0.999614 : N 100 : x4 L H N xd : 41 line

s1 j 1 x1 rows range s1 j el x1 j el S1 : for x1 s1 augment 21 line :

s2 j 1 x2 rows range s2 j el x2 j el S2 : for x2 s2 augment 21 line :

s3 j 1 x3 rows range s3 j el x3 j el S3 : for x3 s3 augment 21 line :

s4 j 1 x4 rows range s4 j el x4 j el S4 : for x4 s4 augment 21 line :

CY7 CY7 o 5 black plotG :

Sanity check

s1 s2 s3 s4 CY7 5 1 sys

1.69812 S1 1.32412 S1 21 mat ....... 1.42013 S2 1.11732 S2 21 mat ...... 1.13935 S3 0.923174 S3 21 mat ...... 0.999614 S4 0.079767 S4 21 mat 71 mat 2.08 13.33 21 mat ....... 9.97 26.35 21 mat ...... 13.8 13521 mat ...... 8848021 mat 71 mat

segment 3 neds an auxilliary rotation

1 time t0 - min 2.4

1.1246 S2 1.1246 S3 21 mat 25.1 16.3 21 mat

Convenient 3 segments Chebyshev ==========

x eit13 13.8033 1 x ≤ (x 2 ≤ (& cases :

x ω1 0.025842464 233160 x * - (* (: x Cryogenic_1 10.28 9.5333 x ω1 acos cos * + 0.4 2 x ω1 acos * cos * + Re : 21 line

x ω2 8687 - 25623 - 10000 x * + (* + 2559 / - : x Cryogenic_2 88651 x ω2 acos * cos * + 1.25 - (2 x ω2 acos * cos * + 2.5 - (3 x ω2 acos * cos * + Re : 21 line

x ω3 0.0006955 20374000 x * - (* : x Cryogenic_3 3001501 x ω3 acos * cos * + 0.75 - (2 x ω3 acos * cos * + Re : 21 line

<= Pt 100

CY7 x Cryogenic_1 x eit13 3 1 sys

~ 20 ° K

<= CY7 ...

eit 13.8033 °K

CY7 [ Ωmega ] 1.69812 Cryogenic_1 1.15

x Cryogenic_1 x Cryogenic_2 - 0 ≡ x 1.1 1.15 solve x Cryogenic_2 x Cryogenic_3 - 0 ≡ x 0.9 1 solve 21 mat 1.124 0.9531 21 mat

x Cryogenic_1 x Cryogenic_2 x Cryogenic_3 CY7 0.9531 113.2427 + 30 black 1 5 mat 5 1 sys

0.9531 Cryogenic_2 113.2427

Suggested renovation =================================

t0 1 time :

x cry_1 28.5 - 42 x 0.35 - 0.0006 x 1.07 - / + / + :

x c1 1 1.125 x ≤ (x 1.75 ≤ (& cases :

x cry_2 380 x * - 546.8 + 57 x 1.5 - 0.2 x 2 - 0.04 x 1 - 5 x 1 - / + / - / + / + :

x c2 10 x ≤ (x 1.25 ≤ (& cases :

x cry_1 x c1 * x cry_2 x c2 * 1.1246 100 1.1246 100 - 2 2 mat CY7 1.1246 25 o 25 black 1 5 mat 5 1 sys

1.1246 cry_1 1.1246 cry_2 21 mat 25.0 25.0 21 mat

1 time t0 - 0.5 s *

Comment . . .

The phenomenon behaves like discontinuous.

As published, high order Chebychev Omega aren't

much inspiring ... [handling, computation load].

Rational fraction is a good candidate for interpolating

each segment. In this proposal, we consider the break

point as an Act of God . More information might help

to conclude/renovate this type of low cryogenic sensor.

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XY 1.3042 15 1.2144 20 1.1246 25 1.0705 50 0.9956 9052 mat :

β 11.94 13.831 1.155 0.089 41 mat :

model to smooth
the break region x β f β 1 el β 2 el x β 3 el - (β 4 el / - exp * + :

XY XY o 5 black plotG : x m 1 0.975 x ≤ (x 1.5 ≤ (& cases : 21 line

XY x β f x m * 2 1 sys

Read more . . .

I'm not aware of industrial process in the sub-cryogenic of Pt 100 [13.8 °K, -259.3467 °C] .

To meet the demand starting ~ 1 °K, @ best accuracy on longer range, a dual installation is

recommanded ... CY7 switching around 20 °K to Pt 100. No more detail about installation.

This technical suggestion is subject to project requirements.

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x L - (U x - (- U L - (/