<p> Genfit Algo </p>

45eacb11-e314-4bc9-b3ad-cd10ac9d476b 29 6 9 decimal

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vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Applicable parts are: Optimiz "numeric" XY f φ β βNlin X XY 1 col : Y XY 2 col : n XY rows : x β f x β f : x β φ x β φ transpose : 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 + : 151 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 21 line 31 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 :

data n Decimate collects every 'n' points range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

About modeling ...

... often, two or more models will fit as well. This rational model is short, but not "normalised". 11 samples from depopulated to preserve the arc region. The flat/vertical regions were cleaned of superflous samples, just enough points. Too many samples degrade the fit from regional influence, unless you refine the fit manually. In Computer Approximation of Functions, error is given as absolute |ε| wrt their range of base approximation which is sgnificant in that case. Differently, in modeling sampled data |ε| has no meaning. Instead, we use the "relative error" ... ... ε:= 1-Y/fit(X)

data 0.5 83.67 1 74.7 2 70.26 5 67.79 10 66.76 20 65.94 50 64.88 100 64.13 200 63.65 500 62.58 1000 62.3 112 mat :

vx data 1 col :

vy data 2 col :

XY vx vy . 12 black plotG :

a 8.76 : b 66.167 : c 0.527 : d 0.008528 : x f a b x * + c x 2^* + x 1 d x * + (* / : 51 line

Depop x 0.5 100 0.75 range : i 1 x rows range pop i el x i el f eval : for x pop augment 31 line :

samples check i 1 vx rows range Y i el vx i el f eval : for

Y 83.59 74.82 70.4 67.67 66.63 65.9 64.98 64.2 63.43 62.63 62.26 111 mat

XY Depop 2 1 sys

Levenberg-Marquardt

Experiment Smath "Genfit Algorithm"

X XY 1 col :

XY 0.5 83.67 1 74.7 2 70.26 5 67.79 10 66.76 20 65.94 50 64.88 100 64.13 200 63.65 500 62.58 1000 62.3 112 mat :

Y XY 2 col :

Model function x β f β 1 el β 2 el x * + β 3 el x 2^* + x 1 β 4 el x * + (* / :

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

x β φ x β f 4 φ :

β 8.760265 66.165306 0.52011 0.008413 41 mat :

1100 0.5 0.01 41 mat

8.760493 66.178671 0.611907 0.009869 41 mat

8.760265 66.165306 0.52011 0.008413 41 mat

8.75977 66.167176 0.526315 0.008511 41 mat

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

Range & populate the discrete fit

L 0.5 : H 100 : N 199 : 13 mat

U L H N xd :

FIT i 1 U rows range FIT i el U i el β f eval : for FIT 21 line :

β 8.75977 66.167176 0.526315 0.008511 41 mat

MyFit U FIT augment :

Relative error Δ i 1 X rows range delta i el 1 Y i el X i el β f eval / - : for X delta augment 21 line :

data X Y . 12 black plotG :

data MyFit 2 1 sys Δ

Construct the J_Fraction

Define Symbolic ... Optimiz = NONE x collect x β f confrac x convert evalf maple :

Optimiz NONE x collect 1932378133125000 / 5088523791000000 / (x 115466130710000000 / + (11676737515000000 / (x 20225389811000000000 / + (/ - (/ +

61.8361 508.852379 x 115.466131 + 233.53475 x 2.022539 + / - / +