<content> Golden ratio min/max </content>

7d99887d-755b-4318-aa70-8148a6066895 24 4 5 false false 0 0 decimal

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

3. Logistic fit . . . USCS granulometry

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 :

XY 1.125 - 3.78 1.119 - 4.054 1.075 - 5.969 0.968 - 11.919 0.824 - 27.298 0.673 - 54.513 0.523 - 81.022 0.373 - 94.521 0.223 - 98.603 0.075 - 99.725 0.073 99.916 0.223 99.941 0.373 99.959 0.525 99.972 0.676 99.983 0.823 99.991 0.949 99.997 1.038 100182 mat :

colony 1.61 :

1.125 - 3.78 1.119 - 4.054 1.075 - 5.969 0.968 - 11.919 0.824 - 27.298 0.673 - 54.513 0.523 - 81.022 0.373 - 94.521 0.223 - 98.603 0.075 - 99.725 0.073 99.916 0.223 99.941 0.373 99.959 0.525 99.972 0.676 99.983 0.823 99.991 0.949 99.997 1.038 100182 mat

x f colony β 1 el 11 β 2 el / 1 - (β 3 el x β 4 el - (* - exp * + / + :

data XY o 3 black plot :

as offered by hand β 98.39 0.9 9 0.66946 14 mat transpose :

x f x 1.125 + f :

x D 0.5 x f x diff (* :

x y axes x y / - :

x f x D data axes 4 1 sys

x f x D data axes 41 sys

u 25507531 mat :

X XY 1 col : Y XY 2 col : 21 sys

x f u - x 1.125 - 1 solve vectorize 0.8291 - 0.7033 - 0.58 - 31 mat

new X X f vectorize augment :

new 1.125 - 3.705 1.119 - 3.819 1.075 - 4.856 0.968 - 9.682 0.824 - 25.835 0.673 - 56.682 0.523 - 83.335 0.373 - 95.06 0.223 - 98.67 0.075 - 99.645 0.073 99.906 0.223 99.976 0.373 99.994 0.525 99.998 0.676 100 0.823 100 0.949 100 1.038 100182 mat

dev j 1 X rows range δ j el Y j el X j el f - : for X δ augment 21 line :

dev 1.125 - 0.075 1.119 - 0.235 1.075 - 1.113 0.968 - 2.237 0.824 - 1.463 0.673 - 2.169 - 0.523 - 2.313 - 0.373 - 0.539 - 0.223 - 0.067 - 0.075 - 0.08 0.073 0.01 0.223 0.035 - 0.373 0.035 - 0.525 0.026 - 0.676 0.017 - 0.823 0.009 - 0.949 0.003 - 1.038 0182 mat

dev 2 col sum 0.0001 -

Once a fit is "declared"
re-publish the project

XY 1.125 - 3.705 1.119 - 3.819 1.075 - 4.856 0.968 - 9.682 0.824 - 25.835 0.673 - 56.682 0.523 - 83.335 0.373 - 95.06 0.223 - 98.67 0.075 - 99.645 0.073 99.906 0.223 99.976 0.373 99.994 0.525 99.998 0.676 100 0.823 100 0.949 100 1.038 100182 mat :

colony 1.61 :

x f colony β 1 el 11 β 2 el / 1 - (β 3 el x β 4 el - (* - exp * + / + :

data XY o 3 black plot :

as offered by hand β 98.39 0.9 9 0.66946 14 mat transpose :

x y axes x y / - :

x f x 1.125 + f :

1/2 magnitude for visibility x D 0.5 x f x diff (* :

max Derivative X 0.6997 - :

Logistic inversion U X X f augment : 0.6997 - 50.7997 12 mat

plot x f x D data axes U 51 sys :

plot

u 25507531 mat :

x f u - x 1.125 - 1 solve vectorize 0.8291 - 0.7033 - 0.58 - 31 mat

Explcit/export otherwise x f 885511619933473 - 625098 x * + (* + (* 50000 / - exp + (* + 1009933473 - 625098 x * + (* + (* 50000 / - exp + (* /

x f x int maple 9839992277750000 x * + (* 50000 / - exp + ln * 1000092277750000 x * + (* 50000 / - exp ln * - 900 /

x q 9839992277750000 x * + (* 50000 / - exp + ln * 1000092277750000 x * + (* 50000 / - exp ln * - 900 / :

x λ 1 1.125 - x ≤ (x 1.038 ≤ (& cases :

Explore scalar value(s) ω 0 :

ω ω q 12 mat 0 69.5947 12 mat

x q x λ * 1.125 - 25 1.125 - 25 - 2 2 mat ω ω q 1 2 mat 3 1 sys

Golden-Ratio search [min/max]

<content> Golden ratio min/max </content>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x D eval :

ε 1012 -^:

X f 0.75 - 0.5 - y1 y2 ≥ ε Extrema :

X 0.6997 -

x f 885511619933473 - 625098 x * + (* + (* 50000 / - exp + (* + 1009933473 - 625098 x * + (* + (* 50000 / - exp + (* / :

x D2 x f x 2 diff :

x D1 x f x diff :

x f 0.5 x D1 * 0.1 x D2 * axes 4 1 sys

x f 0.5 x D1 * - x 0.625 - roots 0.5635 -

<content> hatch, scale </content>

You need that one for scaling grayscale image M lo hi Scale hi lo - (M M Min - M Max M Min - / (* lo + :

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*N vertical lines (hatching) between functions f(x) and g(x) in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

About cumulative FD [Finite Differeces]
Reputed immune to extreme noisy 'Y'
Can be splined for in-situ command line.
Rescues the not availble symbolic.

data 1.125 - 3.78 1.119 - 4.054 1.075 - 5.969 0.968 - 11.919 0.824 - 27.298 0.673 - 54.513 0.523 - 81.022 0.373 - 94.521 0.223 - 98.603 0.075 - 99.725 0.073 99.916 0.223 99.941 0.373 99.959 0.525 99.972 0.676 99.983 0.823 99.991 0.949 99.997 1.038 100182 mat :

data Σ Finite differences Cumul Integration Y 0 : 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 71 line :

Σ data Σ :

Σ Σ rows row 1.038 173.9366 12 mat

vx Σ 1 col : vy Σ 2 col : 12 mat x s vx vy x ainterp : 21 mat

Symbolic 0 69.5947 12 mat

Native FD 0 s 70.1769

x f x s - x 0.25 0.4 solve 0.2984

x f x s :

x g 0 :

Hatch f g 1.125 - 0.2984 20 hatch :

plot Σ 1.125 - 25 1.125 - 25 - 22 mat data 0.2984 150 0.2984 25 - 22 mat Hatch 51 sys :

plot