NOTES

6ad5f2ba-be42-4223-bd74-f52c5e0b7b88 6 8 12 decimal

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

Tc 433.8 556.4 553562 619.6 51 mat :

Pc 31.6 4540 48.6 3851 mat 101.325 * :

α 6.771 6.742 6.844 6.848 7.330 51 mat :

Tr φ 36 Tr / 42 Tr ln * + 35 - Tr 6^- :

Tr Ψ 0.0364 Tr φ * Tr log10 - :

Tc Pc α T Po 10 Pc log10 0.1183 T Tc / φ * - 7 T Tc / log10 * + α 7 - (T Tc / Ψ * -^:

x c 1100 x ≤ cases :

x φ x Ψ 2 1 sys Tc 1 el Pc 1 el α 1 el x Po x c * Tc 2 el Pc 2 el α 2 el x Po x c * Tc 3 el Pc 3 el α 3 el x Po x c * Tc 4 el Pc 4 el α 4 el x Po x c * Tc 5 el Pc 5 el α 5 el x Po x c * 128 0 + 40 black 1 5 mat 6 1 sys 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 This solver is of Legendre LS [Least Square], it is due to a rediscovered unpublished work from Cholesky.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

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 20 x 1.875^x log10 x 2.0625^41 mat :

Memo 20 x 1.875^x log10 x 2.0625^51 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 10.1373 - 0.0037 - 95.5643 0.001 41 mat

1 time t0 - 0.2 s *

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 19.67 37.92 59.18 101.81 204.47 400.91 755.41 101 mat

Δ 1 0.02 5 0.25 - 10 0.24 - 20 0.33 40 2.08 60 0.82 100 1.81 - 200 4.47 - 400 0.91 - 760 4.59 102 mat

Δ 2 BarSize

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 5 blue plotG :

Complete plot: data points + the fitted model

data FIT 2 1 sys

i 1 P length range Y i el P i el log10 : for

i 1 P length range :

i Y augment