<p> Construction </p>

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Smath rational fraction fit data set

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 :

Unlike Mathcad 11.2a that is based on the built-in "Genfit", that does not need initialising coefficients, this Smath model "GenfitMatrix" needs initialising coefficients ... very closely. Mathcad "GenfitMatrix" resulted from collaboration with Robert at the time we developped the "Data Analysis Extension Pack". From observing my brute force fit technique by Num/Denom, Robert toke it in hands and resumed it. Mathcad "GenfitMatrix" is in all humility a piece of gold, it beats dead NIST. Easy to use, in few seconds to get the best fit.

XY 0.881285 0.994849 1.231285 1.137237 1.581285 1.25802 1.931285 1.365855 2.281285 1.464636 2.631285 1.556524 2.981285 1.642889 3.331285 1.724676 3.681285 1.802577 4.031285 1.877117 4.381285 1.948711 4.731285 2.017692 5.081285 2.084335 5.431285 2.148866 5.781285 2.211479 6.131285 2.272338 6.481285 2.331585 6.831285 2.389344 7.181285 2.445723 7.531285 2.500817 7.881285 2.554712 8.231285 2.607484 8.581285 2.6592 8.931285 2.709922 9.281285 2.759705 9.631285 2.808601 9.981285 2.856654 10.331285 2.903908 10.681285 2.950402 11.031285 2.99617 11.381285 3.041246 11.731285 3.085661 12.081285 3.129443 12.431285 3.172618 12.781285 3.215212 13.131285 3.257246 13.481285 3.298743 13.831285 3.339722 14.181285 3.380203 14.531285 3.420204 14.881285 3.459741 412 mat :

XY XY 3 Decimate :

top_order 4 : bottom_order 4 : 21 sys

Model function

x β f β n el x n^* (n 1 top_order sum 1 β n top_order + el x n^* (n 1 bottom_order sum + / :

x β f x x β 3 el x * β 2 el + β 4 el x 2^* + (* β 1 el + (* 1 x x β 7 el x * β 6 el + β 8 el x 2^* + (* β 5 el + (* + /

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

x β φ x β f 8 φ :

<= Gradient Vector Function

X XY 1 col :

β 3.3 1.25 0.66 0.018 2.7 1 0.22 0.0001 81 mat :

Y XY 2 col :

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

Range & populate the discrete fit

L 0.125 :

H 16 :

β 3.800784 2.707459 1.011713 0.037997 3.830393 2.021455 0.333001 0.002024 81 mat

N 199 :

U L H N xd :

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

MyFit U FIT augment :

data X Y . 12 black plotG :

data MyFit 2 1 sys

3.3 1.25 0.66 0.018 2.7 1 0.22 0.0001 81 mat

Construct the J_Fraction

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

Optimiz NONE x collect 18772656710000000 / 25886610631000000 / (x 157795628710000000 / + (188974312125000000 / (x 4579701110000000 / + 139153916250000 / (x 24771125811000000000 / + 17631544831000000000 / (x 330993589110000000000 / - (/ + (/ + (/ - (/ -

18.772657 2588.661063 x 157.795629 + 75.589725 x 4.579701 + 2.226463 x 2.477113 + 1.763154 x 0.330994 - / + / + / - / -