<p> TOC ... Minimize Conjugate-Gradient </p>

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g.e 9.8066 m s 2^/ *

Theoretical Free Fall . . . Introduction Lecture

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m 1 :

s 1 :

'g' is implicit of 'm' as it comes from functions g 9.81 : Vf 50 : 12 mat

0 ln #

Speed wrt 't' t V Vf g t * Vf / tanh * :

Time evolution wrt 'v' v T Vf 2 g * / Vf v + Vf v - / ln * :

x μ 10 x ≤ cases : x λ 10 x ≤ (x Vf ≤ (& cases : x t 10 x ≤ (x 500 ≤ (& cases : 31 sys

Terminal time Tf 2.65 Vf g / * : 13.51

X α tanh 0.99 - 0 ≡ α 23 solve : 2.65

h time Vf g / 12 g * h * - Vf 2^/ exp - sqrt atanh * :

500 time 13.51

time 13.5 :

x V x μ * Tf Vf + 250 red 1 5 mat 2 1 sys x T x λ * Vf Tf + 250 red 1 5 mat 2 1 sys x time x t * 500 500 time + 250 red 1 5 mat 2 1 sys

Instantaneous theoretical speed as function of 'h[m]'

0 ln #

h Vi Vf 12 g * h * Vf 2^/ - exp - sqrt * :

solving for 'h' wrt Vf X h Vi 0.99 Vf * - h 400600 solve : 499.1

X m 499.1

x δ 10 x ≤ (x 1.5 X * ≤ (& cases :

end 1.5 X * : h 0 end range : 12 mat

instantaneous speed  [m/s] @ user 'h' h 128 : h Vi 12 mat 128 39.8 12 mat

Instantaneous speed [m/s] along the travel 'X' ... 'X' is 'h' for the instantaneous speed = Vf the final speed

32 4 450 4 450 4 h h Vi 4 2 mat x Vi x δ * X Vf + 500 green 1 5 mat 3 1 sys

h h Vi 12 mat 128 39.8 12 mat

<p> TOC ... 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 :

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 :

4 appVersion 0.98.6179.21440 ≡

XY 0.01 1.787 5 1.519 10 1.307 20 1.002 30 0.798 40 0.653 50 0.547 60 0.467 70 0.404 80 0.355 90 0.315 100 0.282 122 mat :

<= ExpDecay model function
viscosity water cP(°C) x β f β 1 el β 2 el β 3 el x + / + exp :

0 ln #

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 3 - 50010013 mat transpose :

iter 7 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

FIT U 0100 0.1 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 31 line :

SSD 4105 -^*

corr 1.0006

data XY o 5 black plot : Title 240 Viscosity Water cP 12 blue 15 mat : 21 line

assign/isolate β β : 3.466 - 487.394 120.455 31 mat

°C μ °C β f :

cP 1 :

user search °C 25 :

°C μ cP 0.89

collect 16 - 2802802 °C °C μ 42 mat FIT data Title 41 sys :

Export user pocket calculator °C cP 3.466 - 487.394 120.455 °C + / + exp :

Relative residuals Δ i 1 X rows range resd i el 1 Y i el X i el cP / - eval : for XY resd augment 21 line :

collect

°C °C μ 12 mat cP 25 0.89 12 mat

Relative residuals Δ 0.01 1.787 0.001 - 5 1.519 0.001 10 1.307 0.002 20 1.002 0.002 30 0.798 0.001 - 40 0.653 0.002 - 50 0.547 0.003 - 60 0.467 0.004 - 70 0.404 0.001 - 80 0.355 0.001 90 0.315 0.005 100 0.282 0.011 123 mat