Minimize Conjugate-Gradient

eb7ad046-be49-442d-982a-934fb46c9eb4 92 4 5 decimal

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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 :

0.881285

0.994849

Normalized Rational Fractions [Padé type]
have leading denom '1'. This insures numerical
stability . Most if not all numerical functions
in CAS are of this type, many resident in Win. x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + 1 β 6 el x * + β 7 el x 2^* + β 8 el x 3^* + / :

x β φ x β f 8 φ :

About rational fit/interpolation: 1. Though the model is a function, it is NOT a "model fnct" to data set. 2. Use the helper in separate document. 3. In this example, the variate 'x' is of moderate magnitude wrt highest order x^4 [num] ... thus, you can attempt rounding 'β', while observing residuals. 4. set 'iter' to best fit. 5. NormRatFrac extrapolate fairly well. 6. This example is stricking of realism. 7. Wiggling ± 0.000003 of XY data 6 D.

X XY 1 col : Y XY 2 col : 12 mat data XY . 12 black plot : x c 10 x ≤ (x 16 ≤ (& cases : 31 line

init 0.625 0.5 000 0.1 0018 mat transpose :

iter 16 :

algo solving for β β init iter Minimize :

assign/isolate .. β β 9 round vectorize :

SSD 1.3 1010 -^*

corr 1.000000000

data x β f x c * 2 1 sys

Absolute residuals... fit vs samples Δ i 1 X rows range delta i el Y i el X i el β f - eval : for X delta 6 round vectorize augment 21 line :

β 3.192872516 - 29.804815616 25.73425962 2.534286109 0.018518514 39.260656763 11.818721113 0.378825727 81 mat

Δ Error

Δ 2 col Max Δ 2 col Min 21 mat 0.000003 0.000004 - 21 mat

β 0.625 0.5 000 0.1 0081 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 to fit the data set very well. Not attractive but Ad Absurdum.

x β f data 2 1 sys