NOTES

9851b32e-a59b-40a9-a775-388786670ff5 33 15 12 decimal

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

NOTES

NOTE of general interest --------------------------------------------------- The function φ(x) is a phantom function to generate the n_rows, k_cols to feed the numerical solver. This solver is of Legendre LS [Least Square], it is due to a rediscovered unpublished work from Cholesky. It can be used for fitting a polynomial of degree 'n' to a noisy data set of at least n+1 data points. Pure polynomial fit is a myth whereas in nature there are few "polynomial phenomenon". Rational fraction is a much different story and better fitter... About the CholeskyMatrixSolver --------------------------------------------- You can tweak the vector of initial guesses φ(x), but you don't have much leaway. It is not a "Minerr" solver like in Mathcad. Hower this example fits the model pretty well with little sweat ! The model is not a polynomial, the 3rd & 4th terms aren't monomials x^2, x^3

Unlike the other "Genfit models" this one is not based on the vector of deviation. Rather on the calculated values of one component. It does not need a vector of "β" guesses

plotG

T P char size color plotG n T length : plot T 1 el P 1 el char size color augment : i 2 n range plot plot T i el P i el char size color augment stack : for plot 41 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 :

Example of Non-linear Fit [Adapted from Smath source]. We will fit a model fnct using the CholeskyMatrixSolver

t0 1 time :

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat :

P 1510204060100200400760101 mat :

Install 'Y' non-linear i 1 P length range Y i el P i el log10 : for

Y 0 0.699 1 1.301 1.6021 1.7782 2 2.301 2.6021 2.8808 101 mat

i 1 P length range :

i Y augment

In this non-linear fit, the all thing is about installing 'Y' the non-linear component.

Just to show what equation we are fitting i.e: the model function to fit data points

P log10 b 1 el b 2 el T / + b 3 el T log10 * + b 4 el T 2^* + ≡

[1]

x φ 100 x 1.875^x log10 x 2^41 mat :

Memo 100 x x log10 x 2^51 mat

Prepare Cholesky n T length : k x φ length : 21 line

n k 21 mat 10421 mat

Cholesky  solver Num. j 1 k range i 1 n range X i j el T i el φ j el : for for Φ X transpose X * : d X transpose Y * : b Φ 1 -^eval d * : 41 line

Let recast the model function in [1]
to eliminate the log10 x fit 10 b x φ *^eval :

i 1 n range Pc i el T i el fit eval : for

Deviations @ support points Δ i 1 T length range res i el P i el T i el fit - eval : for P res augment 21 line :

b 2.0594 - 0.0055 - 97.3429 0.0023 41 mat

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat

P 1510204060100200400760101 mat

Pc 0.98 5.25 10.24 19.66 37.91 59.16 101.77 204.42 400.84 755.02 101 mat

Δ 1 0.02 5 0.25 - 10 0.24 - 20 0.34 40 2.09 60 0.84 100 1.77 - 200 4.42 - 400 0.84 - 760 4.98 102 mat

Δ 2 BarSize

1 time t0 - 0.7 s *

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 :

U 225375300 xd :

FIT i 1 U rows range vy i el U i el fit eval : for U vy augment 21 line :

data T P o 7 blue plotG :

Complete plot: data points + the fitted model

data FIT 2 1 sys

SSD P i el T i el fit - (2^(i 1 n sum : 53.455

This project is NON-linear in two counts . . .

1. When the model function is a vector of monomials, by convention LINEAR.

2. Here, two terms of the model function are not monomials thus: NON-LINEAR.

3. Cholesky solver digests either style vector function providing it does not encounter

matrix is non-invertible . . . in that case: freak something in there.

4. log10(x) is specific for this parabolic physical property.

5. x^1.875 is hand offered from visual inspection between the extereme deviations.

6. Once you have a fit, whether one is better than another is a matter of preference,

whereas at the end the preference is dictated by the most common point of operation.

7. SSD is just an observer . . . Pearson correlation is generally prefered.

8. Germanium RTD is non-linear in 8 parameters, exceptionally robust Genfit.

The model terms result from trial/error and perseverance. In that example, Thiele is

a better candidate, Explain: if the data set is not sufficiently true, Thiele will produce

glitche(s), generally 1 glitch. At this point, if Thiele is compatible, you may attempt to

play with the data sets to eliminate the glitch. Remember: data sets are just data, they

are never true unless created from true functions.

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... 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 24.73 1.4 19.32 2 13.66 4.2 102 mat :

Germanium RTD

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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.

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 :

Ge Ge 210 range 12 range el :

Minimize Conjugate-Gradient

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 :

init iter Minimize XY f φ β LS 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 : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop 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 :

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 :

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 :

corr Pearson correlation χ X Mean : μ Y Mean : n XY rows : 13 mat X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el β f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el β f eval μ - (2^(i 1 n sum * sqrt / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j el β f - eval : for X Δ augment 21 line :

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 XY rows sum :

x β f β 1 el β 2 el x 1.125^/ + β 3 el x 2.25^/ + β 4 el x 3.375^/ + β 5 el x 1^/ + β 6 el x 0.5^/ + β 7 el x 0.25^/ + β 8 el x 0.125^/ + :

<= ConjugateGradient Vector Fnct x α φ x β f 8 φ :

X Ge 1 col : Y Ge 2 col : 12 mat XY X Y augment : data Ge o 6 black plot : n Ge rows : x c 1 13.66 x ≤ (x 35000 ≤ (& cases : 51 sys

init 1111111118 mat transpose :

iter 5 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

assign to isolate α β :

SSD 0.000004

α α 4 round vectorize : 3.844 - 2401.821 5401.639 - 52172.807 1901.257 - 172.7 82.887 - 33.532 81 mat

corr 1.000002

Conformity from absolute residuals δ i 1 Ge rows range Δev i el Y i el X i el α f - 3 round eval : for Ω °K Δ°K 13 mat X Y Δev augment stack 21 line :

Quick linfit ... The 4th support point is unlikely exact 164 ... Smath 6179

data x α f x c * 2 1 sys

δ Ω °K Δ°K 35000 0.05 02320 0.1 0 364.6 0.2 0164 0.3 0.001 73.75 0.5 0.001 - 33.55 1 0.001 24.73 1.4 0.001 - 19.32 20 13.66 4.2 0103 mat