<p> Genfit Utilities </p>

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vx vy char size clr plotG n vx length : plot vx 1 el vy 1 el char size clr augment : i 2 n range plot plot vx i el vy i el char size clr augment stack : for plot 41 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 :

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 :

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].

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 :

Data 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 :

Model f(x,β) => Courtesy Jean Giraud. ------------------------------------ ...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, OVNI model will digest. Observe this model 'pins' the fit to datum 0 ! Reference: Kirby, R., NIST (1979). Scanning electron microscope line width standards.

μ 3 :

Model function 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 -^* :

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 :

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

<= Gradient Vector Function x β φ x β f 4 φ :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

β 2916.7187 1.5688 54522.6267 89.4639 - 41 mat :

initials 3000 1.5 5500090 - 41 mat

........

re-inject to iterate

Genfit coeff's of fit β XY f φ β Minimize :

β 2916.71 1.569 54522.586 89.464 - 41 mat

range/populate L 0 : H 400 : N 400 : 13 mat U L H N xd : 21 line

Collect initial & successive 'β'

the recovered function x β f x * ≡

β 3000 1.5 5500090 - 41 mat 2672.58 1.6 54440.1 89.72 - 41 mat 2836.32 1.57 54333.71 89.33 - 41 mat # 2916.71 1.569 54522.586 89.464 - 41 mat 15 mat ≡

Kirby i 1 U rows range vy i el U i el β f U i el * eval : for U vy augment 21 line :

Kirbydata Data 1 col Data 2 col . 12 black plotG : reduced X Y . 12 black plotG : Title 16025 Transmute OVNI fit to Kirby2 12 black 15 mat : 31 line

Y Data 2 col :

Absolute  residuals δ j 1 XY rows range Δ j el Y j el X j el β f X j el * - eval : for X 100 Δ * augment 21 line :

bar δ 1.5 BarSize (:

δ Error

bar Kirby Kirbydata Title 4 1 sys

Reduced data set to be fitted

reduced

Auto-iterate the fit ==============================

Often and ideally, a single set of initials is suitable for re-injection of β's. In this case, you can subject the fit to Auto-iterate, helped by watching the error bar plot as well as the SSD. This fitting session is an ideal one ... Explain: some modelling/fitting may not stabilise, in the sense that they may push/accumulate [so to speak] in either region.. in either left/right region. The proposed "Error bar" will help you set the preferred fit wrt to β's ... Kirby error bar DOES STABILISE. Some experimental data will not have model function(s). In this case, an alternate fit is the "Padé rational fraction" ... a piece of cake in Mathcad GenfitMatrix, lot more difficult in Smath [some were done].

β β 1 el 3000 1.5 5500090 - 41 mat : iter 5 : 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 :

Y Data 2 col :

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

β 2916.7112 1.5688 54522.5935 89.4639 - 41 mat

bar δ 2.5 BarSize (:

bar

SSD = "Sum Square Differences" SSD Y i el X i el β f X i el * - (2^(i 1 n sum : 0.059