<p> Genfit Utilities </p>

8a186321-8c45-4ec2-8220-6071c982748d 11 3 5 decimal

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

f init GivenFind f init FindRoot :

Potential bug in FindRoot:

Snippet above is 5346 UN [Un-installed version]. The Installed version wouldn't solve [maybe in steps].

6179 solves in steps as demonstrated. A better fit is offered

The XY data set are 3 support points describing the depth of insertion of the Moulinet [a flow metering

device in large pipe size ... not very accurate but convenient]. The original equation [Miller] is empirical

and not convenient. Both sets of [K, H, L] are "perfect fit" to Miller.

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step [1] conserve 'K' K 1 ≡ H 1 ≡ L 1 ≡ 31 mat

step [2] increase 'L'
to see a fit come, K 0.12 ≡ H 1 ≡ L 10, 20 ... ≡ 31 mat

step [3] as 'L' approaches
the fit: conserve 'H' K 0.12 ≡ H 1 ≡ L 70 ≡ 31 mat

step [4]=> refine the fit
increasing 'H, L' slightly 
to your "fit preference" K 0.12 ≡ H 0.26 ≡ L 71.53 ≡ 31 mat

X 1513563531 mat :

Y 0.121229 0.119489 0.118596 31 mat :

init K 0.12 ≡ H 0.26 ≡ L 71.53 ≡ 31 mat :

XY X Y augment :

f H K L X 1 el + / + Y 1 el ≡ H K L X 2 el + / + Y 2 el ≡ H K L X 3 el + / + Y 3 el ≡ 31 mat :

Solutions are in the vector order of "init", i.e: K, H, L

f init GivenFind 0.118 0.258 71.528 31 mat

x Rational K H L x + / + :

u 0 x ≤ : (constrain the LHS trace ≡

x Rational u * X 1 el Y 1 el o 7 red X 2 el Y 2 el o 7 red X 3 el Y 3 el o 7 red 3 5 mat XY 3 1 sys

x Rational x 0635 int 75.7

sanity check wrt support points X Rational Y - vectorize 2.039 105 -^* - 1.44 105 -^* - 6.247 106 -^* - 31 mat

Alternately => here is the Smath fit

<p> Genfit Utilities </p>

vx vy char size clr plotG n vx length : plot vx 1 el vy 1 el char size clr augment : i 2 n range plot plot vx i el vy i el char size clr augment stack : for plot 41 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

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 :

XY 15 0.121229 135 0.119489 635 0.118596 32 mat :

x β f β 1 el β 2 el β 3 el x + / + :

Model function

X XY 1 col : Y XY 2 col : n XY rows : 31 line

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

Conjugate Gradient vector function x β φ x β f 3 φ :

β 0.118 0.256 68.83 31 mat :

re-inject to iterate manually

Genfit coeff's of fit β XY f φ β Minimize :

Collect initial & successive 'β' β 01031 mat 0.118 0.17 1.901 31 mat 0.118 0.175 14.379 31 mat 0.118 0.201 33.653 31 mat 0.118 0.234 55.171 31 mat 0.118 0.256 68.83 31 mat 0.118 0.261 71.975 31 mat 17 mat ≡

FIT range, populate U 07001 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

β 0.118 0.261 71.975 31 mat

data X Y . 12 black plotG :

Relative residuals Residuals i 1 XY rows range Δ i el 1 Y i el X i el β f / - eval : for X Δ augment 21 line :

data FIT 2 1 sys

SSD = "Sum Square Differences" β SSD Y i el X i el β f - (2^(i 1 n sum :

β SSD 3.4 1012 -^*