linfit modules

3016d04e-5710-4f21-95b7-5a3f84a6c699 10 4 5 decimal

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linfit modules

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 :

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 :

Original XY

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 :

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin xy 1 col xy 2 col . 10 red plotG : 41 line

order 7 :

minXd maxXd 21 mat 1521 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 vx vy . 10 red plotG : 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 :

x β p 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 * 81 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

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

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

init transpose 1111111118 mat

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

set the 'Unary β' & 
length 'n' of the vector gradient β init : n init rows : 21 sys

vector gradient x β φ x β f n φ :

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

CWD C:\SmathFile CurrentDirectory :

Cheby CWD Cheby wfile 1

c CWD Cheby rfile :

linear coeff's c 0.393703 - 1.051076 - 0.460376 - 0.302706 0.085614 0.023682 - 0.003175 - 0.000573 81 mat

*.mcd 11 0.3938 - 1.051 - 0.4603 - 0.3027 0.08551 0.02366 - 0.003058 - 0.000538 81 mat

minXd maxXd 21 mat 1521 mat

x p x β p :

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 - / :

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

Extrapolate recovers the orginal range of 
the DE model test . Extrapolate ω 06 0.025 range : ii 1 ω rows range Ex ii el ω ii el X Colln eval : for ω Ex Re augment 31 line :

Smath 6179 creates an extra imaginary residual of low value, that must be elimitated for export.

Extrapolate Origin 2 1 sys

Applications of "linfitCheby"

1. "linfitCheby" works generally well for approximating data from solving DE's for which no model can be found easily and otherwise unsane for Polyfit. 2. As a last attempt where other fitting methods fail. For instance: simple Lagrange polynomial fit are generally limited much below order 9. Padé rational fraction are also reasonably limited in their Num/Denom order. Though they shape much more difficult data then pure Polyfit. They may produce "glitches" in unpredictable region. On that point, the user must test for discretised values of greatest fineness than the quantized values of the measuring equipment.