<p> Genfit Utilities </p>

8a186321-8c45-4ec2-8220-6071c982748d 34 4 5 decimal

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

recast the solver f init GivenFind f init FindRoot :

The data set:

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

it can be approximated rational, simpler. Both sets of [K, H, L] are "perfect fit" to Miller.

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X 1513563531 mat :

Y 0.121229 0.119489 0.118596 31 mat :

init K 0.12 ≡ H 0.253 ≡ L 70 ≡ 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

K, H, L in lexicographic order of "init"
... captured in Rational(x) f init GivenFind 0.118 0.251 69.998 31 mat

x Rational K H L x + / + :

constrain the LHS u 0 x ≤ :

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 Y X Rational - vectorize 0.00002 0.00001 0.00001 - 31 mat

Alternately => here is the Smath refined 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 0.26 72.368 31 mat :

re-inject to iterate manually ... not needed from SSD(β)

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

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 0.118226 0.261532 72.091594 31 mat

data X Y . 12 black plotG :

data FIT XY 3 1 sys

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 :

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

β SSD 7.5 1015 -^*

FIT XY XY 3 1 sys