<p> Minimize Conjugate-Gradient </p>

56b28825-d814-4e90-a3a5-336e5ada7d2f 8 3 12 decimal

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

a 0.5 : b 0.25 - : Φ 0 : 31 line

HyperSine/HyperExponential a b Φ x f a b x Φ + (*^:

x y axes x y / - :

XY a b Φ x f a b Φ x f - a - b Φ x f Re a - b Φ x f Im 41 sys :

XY axes 2 1 sys

k f a b Φ k f a b Φ k f - a - b Φ k f Re a - b Φ k f Im 41 sys :

x f a b Φ x f a b Φ x f - a - b Φ x f Re a - b Φ x f Im 41 sys :

size 10 :

color black :

n 14 : j 3 : 21 sys

n pts k 0 : k n ≤ k k 1 + : data k 1 + el k k f j el . 10 color 15 mat : for v data 1 el : j 2 data rows range v v data j el stack : for v 41 line :

Plot n pts x f j el 21 sys :

Quick observer of function(s) values at 'n' scan n pts 1 col n pts 2 col augment :

scan 011 0.841 203 1.189 - 42 - 5 1.682 - 607 2.378 849 3.364 10011 4.757 - 128 - 13 6.727 - 140152 mat

Plot <p> Minimize Conjugate-Gradient </p>

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 :

Exact fit Hypersine

Samples expected exact 15 D XY 011 0.840896415253715 203 1.18920711500272 - 42 - 5 1.68179283050743 - 607 2.37841423000544 849 3.36358566101486 100112 mat :

Model function x β f β 1 el x * exp β 2 el x * β 3 el + sin * :

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

<= Gradient Vector Function x β φ x β f 3 φ :

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

init 0.15 π 4 / π 2 / 13 mat transpose :

iter 5 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

SSD 1.506 1027 -^*

β 0.1733 0.7854 1.5708 31 mat

corr 1.000000000000010

bar δ 0.25 BarSize :

x β f β 1 el x * exp β 1 el x * exp - data 4 1 sys

Notes about this fitting session: 1. The phase shift β3 is π/2. 2. We try β2 magnitude π/4 3. If we are correct, only the exponential growth β1 needs be found. This will arise when the Δ's will equate 0 Have we done right ? YES => exact fit !

Δ π 4 / β 2 el - 15 round π 2 / β 3 el - 15 round 21 mat : 11015 -^* - 61015 -^* 21 mat

9 β f 3.36358566101483