Minimize Conjugate-Gradient

eb7ad046-be49-442d-982a-934fb46c9eb4 56 3 12 decimal

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

Smath rational fraction fit data set of unknown accuracy

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 :

init iter Minimize XY f φ β LS 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 : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop 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 :

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 :

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 :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

corr Pearson correlation χ X Mean : μ Y Mean : n XY rows : 13 mat X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el β f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el β f eval μ - (2^(i 1 n sum * sqrt / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j el β f - eval : for X Δ augment 21 line :

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

Otherwise Pearson independent of f(x,ParameterName) x y corr n x length : xy x y * vectorize : x2 x x * vectorize : y2 y y * vectorize : n xy sum (* x sum (y sum (* - n x2 sum (* x sum (2^- (n y2 sum (* y sum (2^- (* (sqrt / 51 line :

Fit.Y ii 1 n range Fit ii el X ii el β f : for Fit 21 line :

XY data

XY 0.881285 0.994849 1.231285 1.137237 1.581285 1.25802 1.931285 1.365855 2.281285 1.464636 2.631285 1.556524 2.981285 1.642889 3.331285 1.724676 3.681285 1.802577 4.031285 1.877117 4.381285 1.948711 4.731285 2.017692 5.081285 2.084335 5.431285 2.148866 5.781285 2.211479 6.131285 2.272338 6.481285 2.331585 6.831285 2.389344 7.181285 2.445723 7.531285 2.500817 7.881285 2.554712 8.231285 2.607484 8.581285 2.6592 8.931285 2.709922 9.281285 2.759705 9.631285 2.808601 9.981285 2.856654 10.331285 2.903908 10.681285 2.950402 11.031285 2.99617 11.381285 3.041246 11.731285 3.085661 12.081285 3.129443 12.431285 3.172618 12.781285 3.215212 13.131285 3.257246 13.481285 3.298743 13.831285 3.339722 14.181285 3.380203 14.531285 3.420204 14.881285 3.459741 412 mat :

Unlike Mathcad 11.2a that is based on the built-in "Genfit", that does not need initialising coefficients, this Smath model "GenfitMatrix" needs initialising coefficients ... very closely. Mathcad "GenfitMatrix" resulted from collaboration with Robert at the time we developped the "Data Analysis Extension Pack". From observing my brute force fit technique by Num/Denom, Robert toke it in hands and resumed it. Mathcad "GenfitMatrix" is in all humility a piece of gold, fully automated. Easy to use, in few seconds to get the best fit. Providing you initialise the "Smath GenfitMatrix" to convergence, you will get same result as Mathcad 11, just a bit more sweat !

0.93932 - 12.076296 9.321552 0.746425 15.207858 3.972547 0.082963 0.000431 - 81 mat

x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + 1 x x β 7 el x * β 6 el + β 8 el x 2^* + (* β 5 el + (* + / :

range/populate L 0 : H 16 : N 299 : 13 mat U L H N xd : 21 line

x β φ x β f 8 φ :

1 0.4 00 0.1 00081 mat

X XY 1 col : Y XY 2 col : 12 mat data XY . 12 black plot : n XY rows : 31 line

init β 0.5 0.5 00 0.1 00081 mat

init 0.5 0.5 00 0.1 00018 mat transpose :

iter 9 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

FIT i 1 U rows range FIT i el U i el β f eval : for U FIT augment 21 line :

SSD 0.000003

corr 1.0000004

data FIT 2 1 sys

Relative deviation: fit vs samples Δ i 1 X rows range delta i el 1 Y i el X i el β f / - eval : for X delta augment 21 line :

Δ Error

β 0.5 0.5 00 0.1 00081 mat :

Initialise helper

data XY . 12 red plot :

x β f x x β 4 el x * β 3 el + (* β 2 el + (* β 1 el + 1 x x β 7 el x * β 6 el + β 8 el x 2^* + (* β 5 el + (* + / :

This is a good example the initial model will converge this form of samples. The initial model may differ from form to form of samples.

x β f data 2 1 sys