<p> linfit modules </p>

6aea5357-0d85-49b4-bfb0-2ea1495a6d64 44 4 5 decimal

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

u Φ 1 u mwbesselj u 1015 -^+ / :

x c 10 x ≤ (x 12 ≤ (& cases :

x abs Φ 1 1 sys

Voltera 2nd kind x λ f 1 λ 2^- sqrt x t - (* sin t Φ * t 0 x int :

u 9 :

in-situ explorer u 0.5 f 0.3697

Scalar plot on canvas ... 3 min

x 0.9 f x c * x 0.5 f x c * u u 0.5 f + 20 black 1 5 mat 3 1 sys

x 0121 range :

XY x x 0.5 f vectorize eval augment :

isolate XY XY :

<p> linfit modules </p>

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit 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 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 [LS] 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 :

The linfitCheby Procedure

order 10 :

t0 1 time :

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin XY . 12 black plot : 41 line

4 appVersion 0.98.6179.21440 ≡

minXd maxXd 21 mat 01221 mat

1. Convert Original XY => Cheby domain [±1] v i 1 XY rows range v i el i : for v 2 v * v Min - v Max - v Max v Min - / : 21 line : XY v XY 2 col augment : vx XY 1 col : vy XY 2 col : data XY o 7 red plot : 51 line

2. Create ChebyShev Optimiz: Symbolic x β p n 0 order range F n 1 + el β n 1 + el n x acos * cos * : for F 21 line :

0. Create the original reserve 1. The original XY is converted in ChebyShev domain. 2. The ChebyShev vector function is created. 3. Collect the model f(x,β). 4. Cheby coefficients are spooled to file. 5. Collocate the "linfitCheby" Colln(x) 6. Reverse back the fit & the data set

3. Collect  the model function x β f x β p n el n 1 order 1 + sum 11 line :

Prepare the 'Unary β' i 1 order 1 + range idx i el 1 : for

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

idx transpose 11111111111111 mat

set the 'Unary β' & length 'n' of the vector gradient

β idx : n idx rows : 21 line

<= vector gradient x β φ x β f n φ :

The "linfit" coefficients Cheby XY f φ β βlinfit :

The "linfit" coefficients Cheby XY f φ β βlinfit :

b Cheby 6 round vectorize :

Smath 6179 may create extra imaginary residual of low value, that must be elimitated for export. module has that virtue.

c b eval :

c 0.229274 0.120372 0.211426 0.186005 0.534365 - 0.309602 - 0.190889 0.086014 0.030006 - 0.012421 - 0.003152 111 mat

# # round

minXd maxXd 21 mat 01221 mat

x p x β p :

5. Collocate the "linfitCheby" x Colln c k el x p k el * (k 1 order 1 + sum 11 line :

6. reverse scale 'x domain' ω X 2 ω * minXd - maxXd - maxXd minXd - / :

5. Data set & the fit in ChebyShev domain [-1..1]

x Colln data 2 1 sys

6. linfitCheby in the original domain

Origin x X Colln 2 1 sys

◄ sanity  0.3697 9 X Colln 0.3699

Export fitted data  ► spline in-situ data ω 012 0.1 range : n 1 ω rows range Ex n el ω n el X Colln eval : for ω Ex augment 31 line :

Origin data 2 1 sys

x X Colln x minXd maxXd int 2.273

1 time t0 - 26 s *