<p> Original XY </p>

23a98419-ee02-4efc-b19b-4047fcf9dbb5 10 6 12 decimal

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

XY 10 1.16 0.136 1.32 0.269 1.48 0.39 1.64 0.494 1.8 0.572 1.96 0.623 2.12 0.641 2.28 0.625 2.44 0.576 2.6 0.493 2.76 0.378 2.92 0.238 3.08 0.069 3.24 0.115 - 3.4 0.313 - 3.56 0.516 - 3.72 0.718 - 3.88 0.913 - 4.04 1.092 - 4.2 1.251 - 4.36 1.382 - 4.52 1.481 - 4.68 1.542 - 4.84 1.564 - 5 1.543 - 262 mat :

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 :

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin XY . 10 red plot : 41 line

minXd maxXd 21 mat 1521 mat

order 7 :

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 5 black plot : 51 line

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

CWD C:\SmathFile CurrentDirectory :

x p CWD LinfitCheby wfile 1

x p CWD LinfitCheby rfile :

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

Optimiz ... SYMBOLIC x p 1 x 2 x acos * cos 3 x acos * cos 4 x acos * cos 5 x acos * cos 6 x acos * cos 7 x acos * cos 81 mat

X XY 1 col : Y XY 2 col : pts X Y . 12 black plotG : 31 line

Cholesky companion solver
........ NUMERIC ........ j 1 x p length range i 1 X length range M i j el X i el p j el : for for Φ M transpose M * : d M transpose Y * : β Φ 1 -^eval d * : 41 line

NUMERIC ... spool to file c ct 1 : j 1 β rows range u ct el β j el 6 round : ct ct 1 + : 21 line for u 31 line :

CWD C:\SmathFile CurrentDirectory :

c CWD Cheby wfile 1

c CWD Cheby rfile :

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

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

Observe: the scalar plot is auto-bounded

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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 1 1 1 1 - 2 2 mat 3 1 sys