Genfit Utilities

0eaeadbc-f90c-4bbf-90bb-53d5cc87fb8a 98 3 5 decimal

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Genfit Utilities

data char size clr plot 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 :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares 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 :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

About this fit . . . for more didactic about Pt100 => read more confusing from the web.

It may look simple and natural to fit this Pt100, sole recognized °C reference.

It is not so: Explain . . . so gorgeous Mathcad Minerr, it does not conclude the fit.

Under the hood, LM Minerr saturates itself of capability. The result MCD below was

obtained via a specially designed osculatrice method [not disclosed]. Differently, the simpler

Smath CG [Conjugate Gradient] that is difficult to initialize wrt many models, does not

saturate. In this fit from the model that is not reactive/reflexive, the perfect fit is reached

quickly from rough guesses.

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Pt100 XFR x β f β 1 el β 2 el 1 β 4 el β 3 el β 4 el - (/ x 259.3467 + (- β 4 el β 2 el * / exp * + β 3 el β 3 el β 4 el - (/ x 259.3467 + (- β 3 el β 2 el * / exp * - (* + :

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 :

<= ConjugateGradient Vector Fnct x β φ x β f 4 φ :

X eit90 1 col : Y eit90 2 col : XY X Y augment : n XY rows : data XY o 5 black plot : 51 line

010100141 mat

β β 1 el 010100141 mat : iter 7 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β :

β 0.0011936 - 12.5364372 228.5987334 1.6833214 41 mat

L 259.3467 - : H 961.78 : N 1221 : U L H N xd : 41 line

MCD NOT better β 0.0009996 12.4435128 228.3303460 1.7276022 41 mat :

FIT range, populate i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

Δwr i 1 XY rows range u i el X i el β f eval : for Y u - augment 21 line :

al_nleqsolve 0.00208 - 12.69498 229.25719 1.58941 41 mat

Pt100 ... mutual conformity '°C'... will produce eit90 wr ≤ ± 0.005

data FIT 2 1 sys

Δwr 0.002 0.001 - 0.005 - 0.002 0.003 0.001 0 0.002 - 0.003 - 0.001 - 0.003 0.001 - 121 mat

x y axes x y / - :

eit90 FIT axes 3 1 sys

===== Dual on-line solver =====

Genfit Smath Z 0.0011936 - : K.p 12.5364372 : α 1.6833214 : β 228.5987334 : 41 sys

Osculatice MCD
from original
Mathcad 11 Z 0.0009996 : K.p 12.4435128 : α 1.7276022 : β 228.3303460 : 41 sys

w t argument β α - (Z w - (* K.p β 1259346710000 t * + 10000 β * K.p * / - exp - (* α 1 - 259346710000 t * + 10000 α * K.p * / - exp + (* + (* + β α - / :

Solving for (t), given (w)... solve(range) => 0,t,-260,970 w t Sol w t argument 0 ≡ t 260 - 970 solve :

°C 1 :

Ω 1 :

0.00493971 t Sol °C 252.81 -

Solving for (w), given (t)... Roots(range)  => 0,w,-1,5 w t Sol w t argument 0 ≡ w 1 - 5 solve :

w 259.3467 - Sol Ω 0.0009996

°C ITS_1990 259.3467 - 248.5939 - 218.7916 - 189.3442 - 38.8344 - 0.01 29.7646 156.5985 231.928 419.527 660.323 961.78 121 mat TP Hydrogen TP Neon TP Oxygen TP Argon TP Mercury TP Water MP Gallium FP Indium FP Tin FP Zinc FP Aluminum FP Silver 121 mat 12 mat ≡

°C_wr 259.3467 - 0.0011907 0.0009996 0.0001911 248.5939 - 0.0084497 0.0110234 0.0025737 - 218.7916 - 0.091718 0.0982121 0.006494 - 189.3442 - 0.2158598 0.2149336 0.0009261 38.8344 - 0.8441421 0.8424528 0.0016893 0.01 1 0.9999903 0.0000097 29.7646 1.1181389 1.1192165 0.0010776 - 156.5985 1.6098018 1.6136646 0.0038627 - 231.928 1.8927977 1.8970493 0.0042516 - 419.527 2.5689173 2.5709767 0.0020594 - 660.323 3.3760086 3.3732862 0.0027224 961.78 4.2864205 4.2864505 0.00003 - 124 mat :

===== An inverse eit90 [-189 °C ... 972 °C] =====

Genfit Utilities

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 :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares 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 :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

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 :

eit90 eit90 412 range 12 range el :

X eit90 2 col : Y eit90 1 col : XY X Y augment : data XY o 5 black plot : 41 sys

x up 1 0.21585975 x ≤ (x 4.28642053 ≤ (& cases :

Global inverse eit90 ... ± 0.1 °C ... [-190 ... 972] x β f β 1 el x β 3 el - (β 2 el x *^(* 1 x 2 + ln + x β 3 el - (- exp + β 4 el + (* :

<= ConjugateGradient Vector Fnct x β φ x β f 4 φ :

β β 1 el 1400 0.04 2.4 - 2.5 - 14 mat transpose : iter 7 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β : 1037.95565067 0.04991287 1.28363571 - 2.20052627 - 41 mat

L 0.21585975 : H 4.28642053 : N 1151 : U L H N xd : 41 line

FIT range, populate i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

Δ i 1 XY rows range °C i el X i el β f eval : for Y Y °C - augment 21 line :

data x β f 2 1 sys

Deviation [°C]from ...
Inverse eit90 f(x,β)..
i.e: mesure Ω => °C Δ 189.34 - 0.02 38.83 - 0.09 - 0.01 0.01 29.76 0.04 156.6 0.02 231.93 0 419.53 0.03 660.32 0.04 - 961.78 0.01 92 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 :

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 :

X eit90 2 col : Y eit90 1 col : XY X Y augment : data XY o 5 black plot : 41 sys

vector of guesses ... NOT easy ! X0 1400 0.04 2.4 - 2.5 - 14 mat transpose :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 μ f rows range u k el μ k el : for μ Jac u f u Jacobian : StepMax 0 : ε 109 -^: 12 mat u X0 StepMax ε μ f μ Jac al_nleqsolve : 41 line :

... exact eit90 inverse  mid range[-218.7916  ... 29.7646] x μ Φ μ 1 el x μ 3 el - (μ 2 el x *^(* 1 x 4 + ln + x μ 3 el - (- exp + μ 4 el + (* :

<= interpolates up to ... 100 °C

Implicit equalities = 0 μ f X 3 el μ Φ Y 3 el - X 5 el μ Φ Y 5 el - X 6 el μ Φ Y 6 el - X 7 el μ Φ Y 7 el - 41 mat :

μ sol : 1400.00015 0.04277 2.49837 - 2.63968 - 41 mat

μ f 0 0.002 - 0.003 0.002 - 41 mat

0.092 200 0.092 400 - 2 2 mat x μ Φ 1.1 200 1.1 200 - 2 2 mat data 4 1 sys

x mid 1 0.09171804 x ≤ (x 1.11813889 ≤ (& cases :

X 8 el μ Φ 11 mat 156.277 11 mat

λ 4.58237449 0.68973624 - 4.15368638 - 50.94472501 - 41 mat :

... exact eit90 inverse in the subCryo  [-259.3467...-189.3442] x λ subCryo λ 1 el x λ 3 el - (λ 2 el x *^(* 1 x ln + x λ 3 el - (- exp + λ 4 el + (* :

0.21585975 λ subCryo 189.3442 -

x cryo 1 0.0011907 x ≤ (x 0.21585975 ≤ (& cases :

data x λ subCryo x cryo * 0.21585975 128 - 0.21585975 256 - 2 2 mat 3 1 sys

The picewise inverse eit90 Pt100

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 :

x up 1 0.21585975 x ≤ (x 4.28642053 ≤ (& cases :

x mid 1 0.09171804 x ≤ (x 1.11813889 ≤ (& cases :

X eit90 2 col : Y eit90 1 col : XY X Y augment : data XY o 5 black plot : 41 sys

x cryo 1 0.0011907 x ≤ (x 0.21585975 ≤ (& cases :

Picewise inverse eit90 Pt100, i.e: recover °C from measured Ω [Ohm]

x λ subCryo x cryo * x β f x up * x μ Φ x mid * data 1 0.01 + 40 blue 1 5 mat 5 1 sys data x λ subCryo x cryo * 0.21585975 128 - 0.21585975 256 - 2 2 mat 3 1 sys

===== Dual command line solvers =====

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Osculatice MCD
from original
Mathcad 11 Z 0.0009996 : K.p 12.4435128 : α 1.7276022 : β 228.3303460 : 41 sys

w t argument β α - (Z w - (* K.p β 1259346710000 t * + 10000 β * K.p * / - exp - (* α 1 - 259346710000 t * + 10000 α * K.p * / - exp + (* + (* + β α - / :

Solving for (t), given (w)... solve(range) => 0,t,-260,970 w t Sol w t argument 0 ≡ t roots :

°C 1 :

Ω 1 :

0.2149336 t Sol °C 189.3442 -

Solving for (w), given (t)... solve(range)  => 0,w,-1,5 w t Sol w t argument 0 ≡ w roots :

w 189.3442 - Sol Ω 0.2149336

°C ITS_1990 259.3467 - 248.5939 - 218.7916 - 189.3442 - 38.8344 - 0.01 29.7646 156.5985 231.928 419.527 660.323 961.78 121 mat TP Hydrogen TP Neon TP Oxygen TP Argon TP Mercury TP Water MP Gallium FP Indium FP Tin FP Zinc FP Aluminum FP Silver 121 mat 12 mat ≡

°C_wr 259.3467 - 0.0011907 0.0009996 0.0001911 248.5939 - 0.0084497 0.0110234 0.0025737 - 218.7916 - 0.091718 0.0982121 0.006494 - 189.3442 - 0.2158598 0.2149336 0.0009261 38.8344 - 0.8441421 0.8424528 0.0016893 0.01 1 0.9999903 0.0000097 29.7646 1.1181389 1.1192165 0.0010776 - 156.5985 1.6098018 1.6136646 0.0038627 - 231.928 1.8927977 1.8970493 0.0042516 - 419.527 2.5689173 2.5709767 0.0020594 - 660.323 3.3760086 3.3732862 0.0027224 961.78 4.2864205 4.2864505 0.00003 - 124 mat :

ignore its90

This function is the generalized "Inverse eit90 Pt100"
i.e: from measured Ω, it recovers °C, where 'p' are 
the respective parameters wrt the applicable segments. # p 1 el x p 3 el - (p 2 el x *^(* 1 x # + ln + x p 3 el - (- exp + p 4 el + (* (≡

jmGiraud: 20190114

Pt100 XFR x β f β 1 el β 2 el 1 β 4 el β 3 el β 4 el - (/ x 259.3467 + (- β 4 el β 2 el * / exp * + β 3 el β 3 el β 4 el - (/ x 259.3467 + (- β 3 el β 2 el * / exp * - (* + ≡

Both functions are compatible with

reasonable CAS & Scientific Calculators

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Conclude the Pt100 °C user calculator

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β 1037.95565067 0.04991287 1.28363571 - 2.20052627 - 41 mat :

μ 1074.54997858 0.03763417 1.23569962 - 2.20552056 - 41 mat :

λ 4.58237449 0.68973624 - 4.15368638 - 50.94472501 - 41 mat :

Δ f(x,β) ... eit90 Δ 189.34 - 0.02 38.83 - 0.09 - 0.01 0.01 29.76 0.04 156.6 0.02 231.93 0 419.53 0.03 660.32 0.04 - 961.78 0.01 92 mat ≡

Global inverse eit90 ... ± 0.03 °C ... [-190 ... 972] x β f β 1 el x β 3 el - (β 2 el x *^(* 1 x 2 + ln + x β 3 el - (- exp + β 4 el + (* :

... exact eit90 inverse mid range[-189.3442  ... 29.7646]
... interpolates up to ... 100 °C x μ Φ μ 1 el x μ 3 el - (μ 2 el x *^(* 1 x 2 + ln + x μ 3 el - (- exp + μ 4 el + (* :

... exact eit90 inverse in the subCryo  [-259.3467...-189.3442] x λ subCryo λ 1 el x λ 3 el - (λ 2 el x *^(* 1 x ln + x λ 3 el - (- exp + λ 4 el + (* :

declare unit °C 1 :

on-line calculator
enter Ω ... read °C Ω °C Ω λ subCryo 0.0011907 Ω ≤ (Ω 0.21585975 ≤ (& Ω μ Φ 0.09171804 Ω ≤ (Ω 1.11813889 ≤ (& Ω β f cases :

°Cwr °C_wr 3 col °C_wr 1 col augment : support °Cwr o 5 black plot : 21 sys

read Ω from lab equipment Ω 1.5 :

Ω °C °C 127.809

up(x ... mid(x) ... cryo(x) Plot constrains

x up 1 0.21585975 x ≤ (x 4.28642053 ≤ (& cases :

x mid 1 0.09171804 x ≤ (x 1.11813889 ≤ (& cases :

x cryo 1 0.0011907 x ≤ (x 0.21585975 ≤ (& cases :

x λ subCryo x cryo * x μ Φ x mid * x β f x up * Ω Ω °C + 40 blue 1 5 mat support 1.5 50 - ----- Measured Ω ----- 12 black 1 5 mat 1 - 512 2 512 2 512 Ω Ω °C 4 2 mat Ω 50 Ω 50 - 2 2 mat 8 1 sys

Pt100 °C user calculator

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Ω °C °C 127.8

data x λ subCryo x cryo * 0.21585975 128 - 0.21585975 256 - 2 2 mat 3 1 sys Discrete incremental inverser ... disabled, export on new page

°C_wr 259.3467 - 0.0011907 0.0009996 0.0001911 248.5939 - 0.0084497 0.0110234 0.0025737 - 218.7916 - 0.091718 0.0982121 0.006494 - 189.3442 - 0.2158598 0.2149336 0.0009261 38.8344 - 0.8441421 0.8424528 0.0016893 0.01 1 0.9999903 0.0000097 29.7646 1.1181389 1.1192165 0.0010776 - 156.5985 1.6098018 1.6136646 0.0038627 - 231.928 1.8927977 1.8970493 0.0042516 - 419.527 2.5689173 2.5709767 0.0020594 - 660.323 3.3760086 3.3732862 0.0027224 961.78 4.2864205 4.2864505 0.00003 - 124 mat :

ignore its90

β 0.0009996 12.4435128 228.3303460 1.7276022 41 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 :

Pt100 XFR x f β 1 el β 2 el 1 β 4 el β 3 el β 4 el - (/ x 259.3467 + (- β 4 el β 2 el * / exp * + β 3 el β 3 el β 4 el - (/ x 259.3467 + (- β 3 el β 2 el * / exp * - (* + :

L Δ N inc X 0 : i 1 N range X i 1 + el L Δ i * + : X 1 el L : 21 line for X 31 line :

N 2000 :

be generous in the arc region

Balistic 'N' increment to start/end at support ω 0.0009996 0.0982121 0.0009996 - N / N inc :

Design a discrete function inverser  in user [L,H,N] f ω Inverse j 2 ω rows range sol j el x f ω j el - 0 ≡ x roots : sol 1 el 259.3467 - : 21 line for ω sol 2 round vectorize augment 21 line :

Sol f ω Inverse :

XY °C_wr 3 col °C_wr 1 col augment :

data XY o 5 black plot :

Sol 1001 row 0.04960585 233.04 - 12 mat

data Sol 2 1 sys

Sol 0.0009996 259.35 - 0.00104821 258.65 - 0.00109681 258.36 - 0.00114542 258.14 - 0.00119402 257.95 - 0.00124263 257.78 - 0.00129124 257.63 - 0.00133984 257.49 - 0.00138845 257.36 - 0.00143706 257.24 - 0.00148566 257.12 - 0.00153427 257.01 - 0.00158288 256.91 - 0.00163148 256.81 - 0.00168009 256.71 - 0.00172869 256.61 - 0.0017773 256.52 - 172 mat

Conformity Pt100 °C user calculator

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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 :

Deviation °C(Ω) @ eit90  points
Exceptional recovery depending
upon the accuracy of measured Ω Δwr X eit90 2 col : Y eit90 1 col : n eit90 rows : i 1 eit90 rows range u i el X i el °C eval : for Δ Y u - : Y Δ 2 round vectorize augment 61 line :

Δwr 259.3467 - 0 248.5939 - 0 218.7916 - 0 189.3442 - 0 38.8344 - 0 0.01 0 29.7646 0 156.5985 0.02 231.928 0 419.527 0.03 660.323 0.04 - 961.78 0.01 122 mat