<p> Genfit modules </p>

ab916ab5-a3e0-4f0c-aa67-71fdd2decb1b 13 6 9 decimal

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

This is the fastest way to get matrix image. Collapse in 
area above the work sheet, will work anywhere down for any 
[f(x,y),L,H,N] ... typical setup for symmetric f(x,y). x y f L H N Sym U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : V i el L dx i * dx - (+ : 21 line for symmetric Gaussian r 1 : i 1 U rows range c 1 : j 1 V rows range M r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for M 71 line :

Low High step x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct eval : for U f augment 31 line :

N points in the range [L,H] L H N x fnct Npts U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for i 1 U rows range f i el U i el fnct eval : for U f augment 51 line :

t0 1 time :

x y f 255 x 2^y 2^+ (- 0.25 / exp * :

image x y f 1 - 164 Sym :

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1 time t0 - 12.6 sec *

x F 0.1 1 x 2^+ / :

x q 0.125 x sin * :

x S 0.05 x sqrt * :

Plot_1 5 - 5 4.95 - x F inc :

Plot_2 π - π π - 0.05 + (x q inc :

Plot_3 05 0.05 x S inc :

P Plot_1 Plot_2 Plot_3 31 sys :

Plot range, collect global

Plot range/mesh in canvas

3 - 3 2.95 - x F inc

Plot_1 rows Plot_2 rows Plot_3 rows 31 mat 20112610131 mat

Plot 010200 x sqrt Npts :

Plot

This double algo style turned red in 5935

Golden search demo

The working  Golden Ratio 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + a - 3 b * + 5 sqrt a b - (* + 2 / ≡ 5 sqrt ≡ 2.23606797749979 ≡

Smath is wisely design for advanced construction.  This project demonstrates the
double algo style. f(,) carries the Φ, thus it must have an empty place holder.
Φ => supports itself and does not need argument(s), just its algo name. 
This construction is the fastest in [Smath 5346], it saves some instructions.
sqrt(5) =>  2.23606797749979  drops the calculation time to "nothing".
The lack of convergence in Smath 5346  is concluded 
.... granularity in the tested region |y1-y2| # f Φ a b TOL Golden a b Φ a - 3 b * + 2.23606797749979 a b - (* + 2 / : a b - abs 1015 -^> y1 a b Φ f : y2 b a Φ f : 12 mat y1 y2 - abs TOL ≤ break y1 y2 < b x2 : x2 x1 : x1 a b Φ : y1 a b Φ f : 14 mat a x1 : x1 x2 : x2 b a Φ : y2 b a Φ f : 14 mat if if 11 line 21 line while a b + 2 / a b 12 mat 21 mat 31 line :

x f 21 x 3^+ (* - x 2^7 - x 2^+ (* + :

Explain things as it works Smath 5346. a b - abs TOL > a dummy forcing runner while # f Φ a b TOL Golden empty arg't is dummy ≡ y1 y2 < min in the range ≡ y1 y2 > max in the range ≡ 21 line 31 line

a 2 - : b 0 : TOL 105 -^: 0.2 sec 41 line

a 2 : b 4 : TOL 1015 -^: 0.4 sec 41 line

X x f Φ a b TOL Golden 1 el : 2.763932

This part below is from a separate work sheet, added for your testing, all blank in 5935

<p> Genfit modules </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 : x β f x β f : x β φ x β φ transpose : 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 + : 151 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 :

Fit Lorentz multi-peak model

Lorentz model peak fit x β f β 1 el 2 β 2 el π / * β 4 el 4 x β 3 el - (2^* β 4 el (2^+ / * (+ 2 β 5 el π / * β 7 el 4 x β 6 el - (2^* β 7 el (2^+ / * (+ 2 β 8 el π / * β 10 el 4 x β 9 el - (2^* β 10 el (2^+ / * (+ :

XY 0.1 6.905 0.3 10.567 0.5 18.504 0.7 39.878 0.8 64.004 0.9 101.6 1 126.706 1.1 101.822 1.2 64.451 1.3 40.556 1.5 19.673 1.7 12.293 1.9 9.293 3 15.124 3.2 21.711 3.4 34.35 3.6 60.908 3.7 84.199 3.8 116.214 3.9 150.8 4 167.442 4.1 150.756 4.2 116.126 4.3 84.065 4.4 60.726 4.6 34.065 4.8 21.306 5 14.572 5.5 7.542 6 5.847 6.5 10.223 6.7 22.186 6.8 42.04 6.9 101.908 7 201.789 7.1 101.681 7.2 41.582 7.3 21.492 7.5 9.026 7.7 6.026 8 3.014 8.5 1.711 9 1.175 9.5 0.883 10 0.699 452 mat :

If you have spectroscopic/chromatographic sampled data, this Smath document may be profitable to your project. The Genfit Minimize(XY,f,φ,β)wrt SSD(β)is robust enough. The peak centers are first identified, then initialise to preserve the peaks.

MCD 11 0 78.54 1 0.4 157.08 4 0.6 62.832 7 0.2 101 mat

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

ConjugateGradient vector function x β φ x β f 10 φ :

X XY 1 col : Y XY 2 col : n XY rows : data X Y . 12 black plotG : 41 line

β 0801 0.4 1504 0.5 607 0.2 101 mat :

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

β 0.32 78.148 1 0.399 154.619 4 0.582 62.622 7 0.2 101 mat

FIT range, populate U 010 0.025 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

plot data x β f 10115022 mat 40420022 mat 70725022 mat 51 sys :

plot

Discrete analytical insight

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 Integral

FIT Σ FIT 2 1 sys

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X deriv augment 101 line :

FIT Deriv