6c8fffcf-1f20-49a1-abcb-768b1a2ab652 32 6 5 fraction

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

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

This NON-linear solver is essentially based on the Cholesky "Gradient Matrix". In this demo, it is as robust as the Mathcad 'linfit', but superior in the sense it is not "mineer iterative", thus allowing user to further manipulate 'β". Though in this demo it shows extraordinary robustness, don't be wrong, it will not be so depending upon the model function and may require many trial in initialising "β". The module takes the sampled data set XY, the model function as algo name 'f' and the partial derivatives vector functions as algo name 'φ', as well as the the initial vector of the parameters 'β'. On difficult model, you may have to iterate manually by re-injecting the calculated 'β' in itself. The gradient vector function is built by hand. Brush over the last term, delete twice to adapt the length wrt parameters of the model fnct. To add terms, collect over the last term ________|, press coma "," and fill-in the place holder. The gradient vector function can be automated [not offered]. Cholesky LS [Least Squares] solver is one of those things in numerical maths that demonstrate "post mortem" the "ad absurdum" end proof. Why is it "so": because it is "so by the proof" !

XY 0.1 10 0.2 5 0.3 3.333 0.4 2.5 0.5 2 0.6 1.667 0.7 1.429 0.8 1.25 0.9 1.111 112 0.5 3 0.333 4 0.25 5 0.2 142 mat :

Model function

Gradient vector fnct

x β φ x β f x β f β 1 el diff x β f β 2 el diff x β f β 3 el diff x β f β 4 el diff 51 mat :

x β f β 1 el β 2 el x 1 -^* + β 3 el x 2 -^* + β 4 el x 3 -^* + :

X XY 1 col :

Y XY 2 col :

β 110141 mat :

.... iterate manually .... <= <= re-inject the last 'β'

Visit example :: Genfit Gradient Vector.sm

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-> the Cholesky Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [least squares] Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 151 line :

β XY f φ β βNlin :

β 0.0001 - 1.0002 6.4223 105 -^* - 4.5597 106 -^* 41 mat

JmGiraud: 2015_12_25

Range & populate the discrete fit

L 0.05 :

H 10 :

N 99 :

U L H N xd :

Collect initial & successive 'β'

β 0.0001 - 1.0002 6.4223 105 -^* - 4.5597 106 -^* 41 mat

β 110141 mat 0.0001 - 1.0002 6.4223 105 -^* - 4.5597 106 -^* 41 mat # # # 15 mat ≡

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

MyFit U FIT augment :

0.05 β f 20.0146

data X Y . 12 black plotG :

data MyFit 2 1 sys

β 0.0001 - 1.0002 6.4223 105 -^* - 4.5597 106 -^* 41 mat :

Export the fit

x β f β 1 el β 2 el x 1 -^* + β 3 el x 2 -^* + β 4 el x 3 -^* + :

x G 1.0002 x * 6.4223 105 -^* - + (x * 4.5597 106 -^* + 0.0001 - (x 3^* + x 3^/ :

x G confrac x convert maple 110000 / - 50015000 / x 642231000200000 / + (45564873462711000400040000000000 / x 5855274674621824334557398643740254200000 / - 9481906301177123460020761576936727739857605441 / x 292837613145564873462710 / + (/ + (/ - (/ +

x J_Frac 110000 / - 1.0002 x 6.421 105 -^* + 4.5547 106 -^* x 0.0001 - 4.567 106 -^* x 6.4268 105 -^* + / + / - / + ≡

What's the use of the J_Frac(x) ? In Smath: NO use because it does NOT have the "Mineer" solve block. In Mathcad 11: it enables to refine the original "Genfit" on 7 parameters.