NOTES

cfbdd929-4ca6-40dc-b81e-da27631976b3 17 4 5 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 This solver is of Legendre LS [Least Square], it is due to a rediscovered unpublished work from Cholesky.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 1: NON-linear Cholesky

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

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^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 *^:

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 2.8867 - 0.3416 - 149.9436 0.0002 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 5.26 10.25 19.66 37.86 59.07 101.67 204.62 401.88 753.68 101 mat

Δ 1 0.02 5 0.26 - 10 0.25 - 20 0.34 40 2.14 60 0.93 100 1.67 - 200 4.62 - 400 1.88 - 760 6.32 102 mat

Δ 2 BarSize

1 time t0 - 0.5 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 : 73.297

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

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

2. Here, one term of the model function is 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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Example 2: Genfit init-iter

... 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

]]>

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 4.127704 106 -^*

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

Example 3: Genfit init-iter ... Reduced Technique

ListCollect(data,list)

data list ListCollect collects index list select list : 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 :

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 :

Reduced Kirby2 [Y/X] XY 9.65 0.00085 24 0.00538 62.63 0.0515 104.46 0.1376 134.79 0.1918 157.76 0.2222 176.25 0.2401 191.48 0.251 207.58 0.2591 229.92 0.2657 254.22 0.2684 278.83 0.2675 302.02 0.2645 324.42 0.2602 347.02 0.2548 371.3 0.2483 162 mat :

1. selct the support points
2. collect for al_nleqsolve list 159121515 mat : xy XY list ListCollect : 21 sys

raw XY o 5 black plot : support xy . 14 red plot : 21 line

pre-graph of the support points

raw support 2 1 sys

μ 3 :

optimum for this fit

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]

NIST Kirby2 data Kirby 9.65 0.0082 24 0.129 62.63 3.223 104.46 14.377 134.79 25.853 157.76 35.06 176.25 42.31 191.48 48.06 207.58 53.78 229.92 61.1 254.22 68.23 278.83 74.6 302.02 79.88 324.42 84.4 347.02 88.43 371.3 92.2 162 mat :

x β f β 1 el x β 2 el^* β 3 el x β 4 el - (1 μ -^* - exp * x β 4 el - (μ -^* β 5 el + :

vector gradient function x β φ x β f 5 φ :

init 2000 1.6 5000090 - 015 mat transpose :

iter 5 :

X Kirby 1 col : Y Kirby 2 col : 12 mat n Kirby rows : 21 line

algo solving for β β init iter Minimize :

assign/isolate .. β β :

recast/recover Kirby2 x β f x x β f * :

Minimize CG β 2371.3622 1.5984 52309.9886 86.1521 - 0.001 51 mat

Kirby2 x β f x * ≡

SSD corr 21 mat 0.0304 1.0001 21 mat

Kirbyfit x 04001 range : i 1 x rows range vy i el x i el β f eval : for x vy augment 31 line :

Kirbydata Kirby . 12 black plot : Title 16025 Transmute OVNI fit to Kirby2 12 black 15 mat : 21 line

j 1 XY rows range ΔK j el Y j el X j el β f - eval : for δ X 100 ΔK * augment : bar δ 1.5 BarSize (: SSD Y i el X i el β f - (2^(i 1 XY rows sum : 41 line

bar Kirbyfit x β f Kirbydata Title 5 1 sys

Model f(x,β) => ------------------------------------ ...This fit results from 'data reduction Y/X' from which the model is developped and reconstructed Kir(x)... Kirby2(x)*x. The reduced Y/X looks weird to fit, a delight for OVNI model. Observe this model 'pins' the fit to datum 0 ! Reference: Kirby, R., NIST (1979). Scanning electron microscope line width standards.

jmGiraud: 2018-08-09

Search β's from robust al_nleqsolve

Reduced Kirby2 [Y/X] XY 9.65 0.00085 24 0.00538 62.63 0.0515 104.46 0.1376 134.79 0.1918 157.76 0.2222 176.25 0.2401 191.48 0.251 207.58 0.2591 229.92 0.2657 254.22 0.2684 278.83 0.2675 302.02 0.2645 324.42 0.2602 347.02 0.2548 371.3 0.2483 162 mat :

1. selct the support points
2. collect for al_nleqsolve list 159121515 mat : xy XY list ListCollect : 21 sys

raw XY o 5 black plot : support xy . 14 red plot : 21 line

pre-graph of the support points

raw support 2 1 sys

μ 3 :

optimum for this fit

Declare the model function x ζ Φ ζ 1 el x ζ 2 el^* ζ 3 el x ζ 4 el - (1 μ -^* - exp * x ζ 4 el - (μ -^* ζ 5 el + ≡

The support points x xy 1 col : y xy 2 col : 12 mat

vector of guesses X0 3000 1.5 50000100 - 015 mat transpose :

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

The support points xy 9.65 0.0008 134.79 0.1918 207.58 0.2591 278.83 0.2675 347.02 0.2548 52 mat

Model function x ζ Φ ζ 1 el x ζ 2 el^* ζ 3 el x ζ 4 el - (1 μ -^* - exp * x ζ 4 el - (μ -^* ζ 5 el + :

The system of equalities
is constructed manually.
Expand/reduce to match
length of support points.
Order of equalities is
immaterial  al_nleqsolve.
Solver fits to support 'xy'
if noisy, refit CG. ζ f x 1 el ζ Φ y 1 el - x 2 el ζ Φ y 2 el - x 3 el ζ Φ y 3 el - x 4 el ζ Φ y 4 el - x 5 el ζ Φ y 5 el - 51 mat :

ζ sol : 2065.0903 1.6168 50207.6598 82.5509 - 0.0001 - 51 mat

2065.0903 1.6168 50207.6598 82.5509 - 0.0001 - 51 mat

Accepted as fitted suite CG advised given above ...

raw x ζ Φ 2 1 sys

Mutual conformity
@ support points [SP]
within accuracy of SP ζ f 0.0007 - 0.0009 0.0005 0.0001 0.0009 - 51 mat