linfit modules

f39adc06-815e-4791-b293-30f6993b0c6c 7 4 5 decimal

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linfit modules

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 :

Original XY

XY 10 - 09 - 1.282 8 - 2.201 7 - 2.64 6 - 2.755 5 - 2.768 4 - 2.779 3 - 2.755 2 - 2.653 1 - 2.509 0 2.386 1 2.3 2 2.207 3 2.07 4 1.907 5 1.76 6 1.624 7 1.431 8 1.1 9 0.605 100212 mat :

The linfitCheby Procedure

order 9 :

t0 1 time :

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin XY o 5 black plot : 41 line

4 appVersion 0.98.6179.21440 ≡

minXd maxXd 21 mat 10 - 1021 mat

β Clear 1

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 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 * β 9 el 8 x acos * cos * β 10 el 9 x acos * cos * 101 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 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 1111111111110 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 :

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 :

# # round

c 1.579327 0.416867 - 1.169473 - 0.349987 0.390714 - 0.07611 0.048831 - 0.002653 0.023863 0.010864 - 101 mat

minXd maxXd 21 mat 10 - 1021 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 - / :

maxXd x Colln x 1 - 1 int * 39.924

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

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 31 line :

Origin data 2 1 sys

x X Colln x minXd maxXd int 39.924

data

data 10 - 0.007 - 9.9 - 0.142 9.8 - 0.288 9.7 - 0.431 9.6 - 0.57 9.5 - 0.704 9.4 - 0.835 9.3 - 0.961 9.2 - 1.082 9.1 - 1.199 9 - 1.311 8.9 - 1.419 8.8 - 1.521 8.7 - 1.619 8.6 - 1.711 8.5 - 1.799 8.4 - 1.883 8.3 - 1.962 8.2 - 2.036 8.1 - 2.105 8 - 2.171 7.9 - 2.232 7.8 - 2.289 7.7 - 2.342 7.6 - 2.392 7.5 - 2.438 7.4 - 2.48 7.3 - 2.519 7.2 - 2.554 7.1 - 2.587 7 - 2.617 6.9 - 2.644 6.8 - 2.668 6.7 - 2.69 6.6 - 2.709 6.5 - 2.727 6.4 - 2.742 6.3 - 2.755 6.2 - 2.767 6.1 - 2.777 6 - 2.785 5.9 - 2.792 5.8 - 2.798 5.7 - 2.802 5.6 - 2.806 5.5 - 2.808 5.4 - 2.809 5.3 - 2.81 5.2 - 2.809 5.1 - 2.808 5 - 2.807 4.9 - 2.805 4.8 - 2.802 4.7 - 2.799 4.6 - 2.795 4.5 - 2.791 4.4 - 2.787 4.3 - 2.782 4.2 - 2.777 4.1 - 2.772 4 - 2.767 3.9 - 2.761 3.8 - 2.756 3.7 - 2.75 3.6 - 2.744 3.5 - 2.738 3.4 - 2.732 3.3 - 2.725 3.2 - 2.719 3.1 - 2.712 3 - 2.706 2.9 - 2.699 2.8 - 2.692 2.7 - 2.685 2.6 - 2.678 2.5 - 2.671 2.4 - 2.663 2.3 - 2.656 2.2 - 2.648 2.1 - 2.64 2 - 2.632 1.9 - 2.624 1.8 - 2.615 1.7 - 2.607 1.6 - 2.598 1.5 - 2.589 1.4 - 2.58 1.3 - 2.571 1.2 - 2.561 1.1 - 2.551 1 - 2.541 0.9 - 2.531 0.8 - 2.521 0.7 - 2.51 0.6 - 2.5 0.5 - 2.489 0.4 - 2.477 0.3 - 2.466 0.2 - 2.454 0.1 - 2.443 1.9 1014 -^* - 2.431 0.1 2.419 0.2 2.406 0.3 2.394 0.4 2.381 0.5 2.369 0.6 2.356 0.7 2.343 0.8 2.33 0.9 2.317 1 2.304 1.1 2.29 1.2 2.277 1.3 2.264 1.4 2.25 1.5 2.237 1.6 2.223 1.7 2.21 1.8 2.197 1.9 2.183 2 2.17 2.1 2.157 2.2 2.143 2.3 2.13 2.4 2.117 2.5 2.104 2.6 2.091 2.7 2.078 2.8 2.066 2.9 2.053 3 2.04 3.1 2.028 3.2 2.015 3.3 2.003 3.4 1.991 3.5 1.978 3.6 1.966 3.7 1.954 3.8 1.942 3.9 1.93 4 1.917 4.1 1.905 4.2 1.893 4.3 1.881 4.4 1.868 4.5 1.856 4.6 1.843 4.7 1.83 4.8 1.817 4.9 1.804 5 1.79 5.1 1.777 5.2 1.763 5.3 1.748 5.4 1.733 5.5 1.718 5.6 1.702 5.7 1.686 5.8 1.669 5.9 1.652 6 1.634 6.1 1.615 6.2 1.596 6.3 1.576 6.4 1.555 6.5 1.533 6.6 1.511 6.7 1.487 6.8 1.463 6.9 1.438 7 1.411 7.1 1.384 7.2 1.356 7.3 1.326 7.4 1.295 7.5 1.264 7.6 1.23 7.7 1.196 7.8 1.161 7.9 1.124 8 1.085 8.1 1.046 8.2 1.005 8.3 0.962 8.4 0.919 8.5 0.873 8.6 0.827 8.7 0.778 8.8 0.728 8.9 0.677 9 0.624 9.1 0.569 9.2 0.513 9.3 0.455 9.4 0.395 9.5 0.333 9.6 0.269 9.7 0.204 9.8 0.136 9.9 0.067 10 0.005 - 2012 mat :

data rows 201

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

cumulative integration

data Σ

data Σ data Σ rows row 10 39.923 12 mat

linterp, ainterp, cinterp .. hyperfast in Smath x f data 1 col data 2 col x linterp :

x c 110 - x ≤ (x 10 ≤ (& cases :

linterp on smoothed linfitCheby

Origin x f x c * 2 1 sys

x f x minXd maxXd int 39.925

1 time t0 - 26 s *

x f x X Colln :

Filling the linfitCheby project

t0 1 time :

Prepare the project

Construct the project

x f x X Colln :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

sx 10 - : N 1000 : 12 mat

xmin sx : xmax 10 : 12 mat ymin 0 : ymax 3 : 12 mat 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x f ≤ (x xmax ≤ (* :

jmGiraud: 2015/12/31 updated wrt project

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 8 #808080 13 mat :

fill Pts a b c spec pts :

Filling the fitted linfitCheby

fill data 2 1 sys

1 time t0 - 23 s *