<p> Utilities </p>

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x f x 2 π * / 2 π * x / tanh * sqrt :

1. Solve & track coordintes ...

w sol t f w - t root : w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / t root

4 appVersion 0.98.6179.21440 ≡

Maple expression var root ≡ is live in Smath 5346 but undocumented ≡

For solving, we equate the Maple 'root' to 0 and include in the Smath solve block c/w the 'roots' range. Indirectly, Mathcad/Mathsoft "tracking" implements a version of Maple "root" [or similar to Maple], but active by clicking in the curve for quick coordinates.

The symmetric solver within the roots range

recast Maple for Smath solve w t Sol w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / 0 ≡ t 40 - 40 solve :

w 0.875 :

w 0.95 ≤

w t Sol 6.3582 - 6.3582 21 mat

Neg 21 - w t Sol 1 el Track : Pos 21 - w t Sol 2 el Track : show 16 w w User 14 black 15 mat : 31 sys

up dn locate Track j 1 up range v j el j dn + : u j el locate j 0 * + : 21 line for u v augment 21 line :

Curve tracking Mathcad/Mathsoft style ... view/export the solved system

x f w Neg show Pos 6 1 sys

Sol(w,t) are the coordinates that produce your demanded 'w' on the function f(x)...

w w t Sol transpose 12 mat 78 / 6.358 - 6.358 12 mat 12 mat

jmGiraud: 2015_08_23

x f 11 x 2^+ / :

2. More practical example ...

w sol t f w - t root : w 1 t 2^+ (* - 1 + 1 t 2^+ / t root

recast Maple for Smath solve w t Sol w 1 t 2^+ (* - 1 + 1 t 2^+ / 0 ≡ t 10 - 10 solve :

sanity your upper limit w 0.75 :

w t Sol 0.5774 - 0.5774 21 mat

Neg 21 - w t Sol 1 el Track : Pos 21 - w t Sol 2 el Track : show 1 w w User 14 black 15 mat : 31 sys

up dn locate Track j 1 up range v j el j dn + : u j el locate j 0 * + : 21 line for u v augment 21 line :

Curve tracking Mathcad/Mathsoft style ... view/export the solved system

x f w Neg show Pos 6 1 sys

w w t Sol transpose 12 mat 34 / 0.577 - 0.577 12 mat 12 mat

x f 1 x / :

3. Common project example ...

w sol t f w - t root : 1 w t * - t / t root

recast Maple for Smath solve w t Sol 1 w t * - t / 0 ≡ t 10 - 10 solve :

sanity your upper limit w 0.75 :

w t Sol 1.3333

sol 0 w w t Sol w w t Sol 0.25 - 32 mat : show 2 - w 0.125 + w User 14 red 15 mat : 21 sys

Adjust left 'w' position according to plot zoom.

Curve tracking Mathcad/Mathsoft style ... view/export the solved system

x f sol show 3 1 sys

w w t Sol 12 mat 34 / 1.333 12 mat

4. Statistical companion ...

Fréchet II fit to 
US rainfall project K α β x f K α β / * β x / (α 1 +^* β x / (α^- exp * :

root absent f(x) menu
believed Maple ?
recast ►  solve (,,,,) w sol K α β x f w - t root : w β * - K α * β x / (1 α +^* β x / (α^- exp * + β / t root

K 31.664 : α 6.774 : β 6.212 : 31 line

recast Maple for Smath solve w t Sol w β * - K α * β t / (1 α +^* β t / (α^- exp * + β / 0 ≡ t 015 solve :

sanity your upper limit w 6 :

w t Sol 5.2775 7.5079 21 mat

Neg 121 - w t Sol 1 el Track : Pos 121 - w t Sol 2 el Track : show 1 w w User 14 black 15 mat : 31 sys

up dn locate Track j 1 up range v j el j dn + : u j el locate j 0 * + : 21 line for u v augment 21 line :

Curve tracking Mathcad/Mathsoft style ... view/export the solved system

K α β x f w Neg show Pos 6 1 sys

w w t Sol transpose 12 mat 6 5.277 7.508 12 mat 12 mat

5. Solve/Track a smooth data set ...

'col' algo style data n ‖ data n col :

raw/smoothed Thermies xw 02116 2113.3 12121 2111.7 22099 2109.2 32120 2105.2 42113 2099.4 52098 2092.1 62067 2084.7 72071 2078.6 82070 2075.1 92056 2073.9 102083 2074.1 112090 2074.4 122072 2074.0 132064 2072.5 142081 2070.1 152074 2067.2 162054 2064.4 172056 2062.0 182058 2060.5 192061 2059.6 203 mat :

smooth xw 1 ‖ xw 3 ‖ augment :

x fit xw 1 ‖ xw 3 ‖ x cinterp :

x λ 10 x ≤ (x 19 ≤ (& cases :

convenience xw 3 ‖ Min : 2059.6

x New x fit convenience - :

U 32 :

X x New U - 0 ≡ x 020 solve :

X 5.0658

sol 0 U X U X 032 mat : show 2 - U 5 + U 14 red 15 mat : 21 sys

Adjust 'show'system according to plot zoom.

x New x λ * sol show 3 1 sys

U X 12 mat 32 5.066 12 mat

6. Discrete solve-scanner ... a bit freak to plot !

x f x 2 π * / 2 π * x / tanh * sqrt :

t0 1 time :

In Mathematics, an InverseFunction is the solution of this function. f(x) is the model of difficulty, documented in Mathematica 4.0. The 'roots' can't be applied wrt the derivative that does not take discrete vector of values. In lieu, the 'solve' block concludes pretty well.. a bit slow. _____________________________ Note: the function starts existing around 'x = 0.02, f(0.02) = 0.0564' our choice to solve is wise , down to lower possible limit. The upper limit 0.99 is practical. In fact, at 1, the solution is infinite. Pushing a bit more, 'solve' cries to murder. _____________________________ In the zero region, the function is very steep. Consequently, the low region of sovability is reflexive with the iterative solve bloc. That mutual granularity dictates some precauion wrt the lowest limit and the bracketed roots range in the solve bloc.

Design a discrete function inverser  in the L, H span , N points f L H N Inverse ω 0 : dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range vy j el x f ω j el - 0 ≡ x 0.001 30 solve : for ω vy augment 61 line :

N f 0.05 0.99 100 Inverse :

scale 100 - :

N N 1 col reverse scale * scale - N 2 col augment :

N 2 col Max x f * N 2 1 sys

f(x) is asymptotic to '1'

x f 1 2 1 sys

100 f 0.9993

1 time t0 - 18 s *

Confirms end range solve N 2 col Max 18.075

7. Reconciliate the project

x f x 2 π * / 2 π * x / tanh * sqrt :

'X' solves for demanded 'U' on f(x) U 0.975 :

X x f U - 0 ≡ x 0.001 40 solve : 15.8348

track 0 U X U X 032 mat : show 3 - U 0.0625 + U 14 red 15 mat : 21 sys

x f track show 3 1 sys

U X 12 mat 0.975 15.835 12 mat

The Simpson integrator is not offered ... inadequate for this steepy low region. In lieu, use the Adaptive, possibly Romberg Kernel [Smath Utilities]

x f 5953 / 11 x 2 / - (2^- (* :

8. Tracking X given Y

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1 f 3 f 2 f 1.75 f 41 mat 148.75 148.75 198.333 195.234 41 mat

x f 0 ≡ x 35 solve 4

x c 10 x ≤ (x 4 ≤ (& cases :

H 2 f 9 round eval :

algo style "iterative inverser" on range [L, H, N] .... f L H N Inverse ω 0 : dx H L - N / : sol 0 : 13 mat i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x roots : for ω sol augment 41 line :

Inv f 0 H 1000 Inverse :

x Φ Inv 1 col Inv 2 col x cinterp :

ξ 148.75 :

=============================================

track 0 ξ ξ Φ ξ ξ Φ 032 mat :

show 0.5 - ξ 20 + ξ 14 red 15 mat :

x f x c * track 2 256 2 64 - 2 2 mat show 4 1 sys

2 f Φ 1.9685

OBSERVE ... Φ(f(2)) does not return exactly 2. This is due to the inaccurate "roots" solver in the flat region. ... @ N 10000 we get 1.99 Much less detrimental ... Φ(f(0.75))

ξ Φ 1

ξ 0.75 :

ξ f Φ 0.75

10. Advanced application calibration Tracking Ω given °C

]]> <p> Utilities </p>

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 :

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

The XFR coefficients 
were obtained from MCD
via a specialy designed
..."osculatice method"...
NOT revealed [Jean Giraud] Z 0.0009996 : K.p 12.4435128 : α 1.7276022 : β 228.3303460 : 41 sys

User units °C 1 : Ω 1 : 21 line

User unit system: copy/paste argument <= in the empty place holder of result.

Pt100: Wr [Ω] vs 
temperature [t °C] t Wr Z K.p 1 α β α - (/ t 259.3467 + (- α K.p * / exp * + β β α - (/ t 259.3467 + (- β K.p * / exp * - (* + :

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

w 200 Sol Ω 1.7778549

eit90 259.3467 - 0.00119007 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 :

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

vy i 1 U rows range y i el U i el Wr : for y 21 line :

°Ceit90 eit90 1 col : Wr eit90 2 col : eit90 Wr °Ceit90 augment : eit_90 eit90 o 7 black plot : Ωr vy U augment : 51 sys

°C 500 :

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

CALIBRATE the Pt100 XTR user range ... You can use this document to calibrate the Ω Pt100 XTR from the installed Wr(t) Otherwise use BIPM. Van Dussen Calendar considered obsolete.

Ω w °C Sol (: Ω 2.8467044

track 0 °C Ω °C Ω 50 - 32 mat :

show 0.25 - °C 30 + °C 14 blue 15 mat :

track Ωr eit_90 show 4 1 sys

°C Ω 12 mat 500 2.8467 12 mat

°C_wr 259.3467 - 0.00119007 0.0009996 0.0001911 248.5939 - 0.00844974 0.0110234 0.0025737 - 218.7916 - 0.09171804 0.0982121 0.006494 - 189.3442 - 0.21585975 0.2149336 0.0009261 38.8344 - 0.84414211 0.8424528 0.0016893 0.01 1 0.9999903 0.0000097 29.7646 1.11813889 1.1192165 0.0010776 - 156.5985 1.60980185 1.6136646 0.0038627 - 231.928 1.89279768 1.8970493 0.0042516 - 419.527 2.56891730 2.5709767 0.0020594 - 660.323 3.37600860 3.3732862 0.0027224 961.78 4.28642053 4.2864505 0.00003 - 124 mat :

Explain "sanity"

Here, 'w' are the 'Wr' calculated from formula Wr(t). In 'sanity', we re-inject those values in the solver Sol(w,t), collect those values and express the Δt vs eit90. Conformity formula Wr(t) is stricking of realism vs BIPM eit90 ! STATED OTHERWISE ... by back-feeding the measured 'w' in sanity(w) we retrieve exactly the fixed eit90 support points collected Δt

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,°C] given [w ,Ω] w t Sol w t argument 0 ≡ t 260 - 970 solve :

w sanity w t Sol :

°C 1 :

1.8970493 sanity °C 231.928

Sanity Δt vs eit90
fixed points. Δt w °C_wr 3 col : T 1 el 259.3467 - : i 212 range T i el w i el t Sol : for °C_wr 1 col T - 41 line :

Δt 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 121 mat

°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 ≡

11. Solve/track . . . ant colony logistic

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Model from another
Smath document ... x μ colony f colony μ 1 el 11 μ 2 el / 1 - (μ 3 el x μ 4 el - (* - exp * + / + :

x λ 10 x ≤ (x 10 ≤ (& cases :

x f x μ colony f :

Verhulst fit to 'ant' data μ 11000 0.76 1.35 5.037 41 mat : colony 0 : 12 mat

Y 5000 :

X x f Y - 0 ≡ x 010 solve : 4.0481

track 0 Y X Y X 1000 - 32 mat :

show 0.75 - Y 1000 + Y 14 red 15 mat :

plot x μ colony f x λ * track show 31 sys :

plot

Y X 12 mat 5000 4.048 12 mat

Maple solves Verhulst model
thus you can export ... x σ ant Φ ant σ 1 el 11 σ 2 el / 1 - (σ 3 el x σ 4 el - (* - exp * + / + :

y inverse y x σ ant Φ ≡ x solve maple :

Verhulst fit to 'ant' 
data American Scientist, 
Jan-Feb 1995 σ 11000 0.76 1.35 5.037 41 mat : ant 500 : 12 mat

x Inverse σ 3 el σ 4 el * σ 2 el x ant - σ 1 el - (* x - x σ 2 el * + ant + ant σ 2 el * - / ln - σ 3 el / :

x inverse ant 5 ant 5 - 2 2 mat 2 1 sys x Inverse ant 2 ant 2 - 2 2 mat 2 1 sys

x λ 1 ant x ≤ (x 11000 ant + ≤ (& cases :

x Inverse σ ant Φ

The function of its inverse is linear 'x' x sanity x Inverse x λ * σ ant Φ :

x sanity x 2 1 sys

up 2 :

size 12 :

OBSERVE: data/text is produced in reverse order than created. It is a quickplot stripped of grids and axes.

...................  eit90 fixed points [°C ] eit90 00 FP Silver [961.78] size red 00 up + FP Aluminum [660.323] size blue 002 up * + FP Zinc [419.527] size blue 003 up * + FP Tin [231.928] size blue 004 up * + FP Indium [156.5985] size blue 005 up * + MP Gallium [29.746] size blue 006 up * + TP Water [0.01] size red 007 up * + TP Mercury [-38.8344] size blue 008 up * + TP Argon [-189.3442] size blue 009 up * + TP Oxygen [-218.7916] size blue 0010 up * + TP Neon [-248.5939] size blue 0011 up * + TP Hydrogen [-259.3467] size red 125 mat :

...... eit90 fixed points [°C ]

eit90