b7b0657c-f0a2-494c-9156-c630648b38c4 7 6 12 decimal
&[DATE] &[TIME] - &[FILENAME]
&[PAGENUM] / &[COUNT]
 <p> linfit modules </p>

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 + : 13 1 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 : 2 1 line ct ct 1 + : 4 1 line for V 3 1 line : <p> Original XY </p>

The linfitCheby Procedure

0. reserve for backscale in original domain

xy XY : minXd xy 1 1 el : maxXd xy xy rows 1 el : Origin XY o 5 black plot : 4 1 line 10 - 10 2 1 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 - / : 2 1 line : XY v XY 2 col augment : vx XY 1 col : vy XY 2 col : data XY o 5 black plot : 5 1 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 2 1 line : β 1 el β 2 el x * β 3 el 2 x acos * cos * β 4 el 3 x acos * cos * β 5 el 4 x acos * cos * β 6 el 5 x acos * cos * β 7 el 6 x acos * cos * β 8 el 7 x acos * cos * β 9 el 8 x acos * cos * β 10 el 9 x acos * cos * 10 1 mat

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 1 1 line :

Prepare the 'Unary β'

i 1 order 1 + range idx i el 1 : for 1 1 1 1 1 1 1 1 1 1 1 10 mat

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

<= vector gradient

The "linfit" coefficients

Cheby XY f φ β βlinfit : 1 1.579327 0.416867 - 1.169473 - 0.349987 0.390714 - 0.07611 0.048831 - 0.002653 0.023863 0.010864 - 10 1 mat

Smath 6179 may create extra imaginary residuals of low value, that must be elimitated for export. The round(,,) module has that virtue .

 iVBORw0KGgoAAAANSUhEUgAAABkAAAAZCAYAAADE6YVjAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAD0SURBVEhLvdZRCoJAFIXhiFmetJBwteFKwkdp6gyduF7PHWcKe/gJx/F+qCCdlnPKR9eNjNdRrtfqQobLkNP7V52PakYIsB6oCfFAL7SLRABrgapIeix5uk1yOCvnX/vU9SxEcCGLIAJMzUESwSPwAzykzkePboPYdxBB0TpS0AqxAPMD5/u8OrYA89AHUQDzUA1gFioIPhVqo83fQTkW+2z8BP3vTmqQB356J8xCHuDAaF0BaIMgbKwNiiAFIIkgP8ACEaTmoBBBNYARUtezKoLsO1JFj8i2i6AIagFQE4I81AqgZgQR6gFQF4IO/7fyXSk/AZHy3IrSyGP2AAAAAElFTkSuQmCC 10 - 10 2 1 mat

5. Collocate the "linfitCheby"

x Colln c k el x p k el * ( k 1 order 1 + sum 1 1 line :

6. reverse scale 'x domain'

ω X 2 ω * minXd - maxXd - maxXd minXd - / : 39.924

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

 x Colln data 2 1 sys

6. linfitCheby in the original domain

 x X Colln Origin 2 1 sys

Export fitted data for tracability

data ω 10 - 10 9.9 - range : n 1 ω rows range Ex n el ω n el X Colln eval : for ω Ex augment 3 1 line : Origin data 2 1 sys 39.924 <p> data </p>

cumulative integration

 data Σ 10 39.923 1 2 mat

linterp, ainterp, cinterp .. hyperfast in Smath

x f data 1 col data 2 col x linterp :

linterp on smoothed linfitCheby

 Origin x f x c * 2 1 sys 39.925 25 s *

Export this formula to scientific calculator

 x q 3.992