<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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

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 :

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 :

Minimize model "ConjugateGradient" wrt parameters [β's]

This Smath document will minimize the model for the best set of the parameters. The success will depend upon the wise choice of the initial β's. This algo is not as robust as the Levenberg-Marquardt, the final fit is iterative but will finally fit<=>LM. At each run, you can observe the goodness of fit on the graph. The core algorithm Minimize(XY,f,φ,β) operates indirectly on the SSD(β). In other words: Minimize(XY,f,φ,β) is implicit of the objective criterium SSD(β). Many models are pretty intuitive, others are very reactive [reflexive] wrt the parameters. If not wisely initialised, may crash Smath. Good idea to start the fitting session on a scratch pad work sheet. Hover φ(x,β), for ConjugateGradient. This open code model "Minimize" is identical in fit to Mathcad/Mathsoft, OriginLab.

XY 1 5.308 2 7.24 3 9.638 4 12.866 5 17.069 6 23.192 7 31.443 8 38.558 9 50.156 10 62.948 11 75.995 12 91.972 122 mat :

Model function x β f β 1 el 1 β 2 el β 3 el x * exp * + / :

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

<= ConjugateGradient Vector Fnct x β φ x β f 3 φ :

X XY 1 col : Y XY 2 col : n XY rows : data X Y . 12 black plotG : 41 line

20050 0.5 - 31 mat

β β 1 el 20050 0.5 - 31 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 :

From Carlos LM β 196.186261720221 49.09163944894 0.313569729961 - 31 mat :

Relative residuals Residuals i 1 XY rows range Δ i el 1 Y i el X i el β f / - eval : for X Δ augment 21 line :

β 196.164 49.087 0.314 - 31 mat

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 2.59

FIT range, populate U 020 0.01 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

RMS difference SSD n / sqrt 0.5

data FIT 2 1 sys

Relative residuals

Residuals 0.1 BarSize 1 1 sys