linfit modules

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linfit modules

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plotG 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 :

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 :

The linfitCheby Procedure

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin xy . 10 red plotG : 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 . 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 idx i el 1 : for

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

idx 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 idx : n idx rows : 21 line

x β φ x β f n φ :

<= vector gradient

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

CWD C:\SmathFile CurrentDirectory :

Cheby CWD Cheby wfile 1

c CWD Cheby rfile :

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

c 0.393703 - 1.051076 - 0.460376 - 0.302706 0.085614 0.023682 - 0.003175 - 0.000573 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 0.5 6 0.51 range : ii 1 ω rows range Ex ii el ω ii el X Colln eval : for ω Ex 15 round vectorize augment 31 line :

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

Two options ω Ex 15 round vectorize augment ω Ex Re augment 21 sys

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. Linearisation of cryogenic sensors: 3. 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.

1. Uknown data from collab: length 565, Cheby order 100. 2. Linearisation of cryogenic sensor: length 81, Cheby order 22. The relative error < 0.01 K° on the fitted length. Cryogenic data are from publihed specifications [Manufacturer].

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 ≡

c 0.393703 - 1.051076 - 0.460376 - 0.302706 0.085614 0.023682 - 0.003175 - 0.000573 81 mat ≡

c k el x p k el * (k 18 sum # ≡

# c 1 el x p 1 el * c 2 el x p 2 el * + c 3 el x p 3 el * + c 4 el x p 4 el * + c 5 el x p 5 el * + c 6 el x p 6 el * + c 7 el x p 7 el * + c 8 el x p 8 el * + ≡