6c8fffcf-1f20-49a1-abcb-768b1a2ab652 47 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

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 :

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].

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 "minerr 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 [read more examples]. Cholesky solver minimises the SSE [SumSquareError] in the LS sense, LS = Least Square.

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 :

x β f β 1 el β 2 el x - β 3 el / exp * + β 4 el x - β 5 el / exp * + :

n last index of β ≡ x β φ x β f 5 φ : 21 line

x β φ β 2 el x β 3 el / - exp * β 4 el x β 5 el / - exp * + β 1 el + 1 x β 3 el / - exp x x β 3 el / - exp * β 2 el * β 3 el 2^/ x β 5 el / - exp x x β 5 el / - exp * β 4 el * β 5 el 2^/ 61 mat

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 010141 mat :

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

Applicable parts are: Optimiz "numeric" XY f φ β βNlin 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-> 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 + : 131 line :

β XY f φ β βNlin :

exp'l threshold 15 β 0.000110685 - 1.00019804 0.000064223 - 0.00000456 41 mat

range/populate L 0.05 : H 10 : N 199 : 13 mat U L H N xd : 21 line

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

data XY . 12 black plot :

0.1 β f 10

data FIT 2 1 sys

Global fitting session [0 .. 14] 1. al_nleqsolve point select equalities valid for noisy data via user piecewise linear. 2. Levenberg-Marquardt: generally valid [fails Frechet] 3. Cholesky matrix solver ... solves all, faster model function(s) parameters, linfit components, remarkable linfitCheby, polyorder, direct discrete data. ............................ rational fraction. EMERITUS => Conjugate gradient Minimize(init,iter)