Utilities

b20b898d-649a-453b-981d-271a3e846ed5 18 4 5 decimal

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

often, Golden ratio needs be recast local definition x f N n p x hypergeom 11 line :

Utilities

user unit T_hr 1 :

# f L Δ N Δinc X 0 : i 1 N range X i 1 + el L Δ i * + : X 1 el L : 21 line for i 1 X rows range vy i el X i el f eval : for X vy augment 41 line :

in-line local definition u x π : 11 line : x f u 2^: 21 line T_hr 9.8696

x π : backspace to zap 21 line

parametric algo style ... p x n algo u x p : 11 line : x f u n^: 21 line :

p 2 ln :

Q n p x n algo vectorize augment :

n 2.5 - 2 2.4 - range :

Q 2.5 - 2.5 2.4 - 2.41 2.3 - 2.3233 2.2 - 2.2397 2.1 - 2.1591 2 - 2.0814 1.9 - 2.0065 1.8 - 1.9343 1.7 - 1.8646 1.6 - 1.7975 1.5 - 1.7329 1.4 - 1.6705 1.3 - 1.6104 1.2 - 1.5524 1.1 - 1.4966 1 - 1.4427 0.9 - 1.3908 0.8 - 1.3407 0.7 - 1.2925 0.6 - 1.246 0.5 - 1.2011 0.4 - 1.1579 0.3 - 1.1162 232 mat

x F 11 x 2^+ / :

L 2 - : Δ 0.1 : N 60 : 13 mat F x F L Δ N Δinc : 21 sys

end F 1 row F F rows row 21 sys : 2 - 0.2 12 mat 4 0.0588 12 mat 21 sys

F end 2 1 el 1 el 0.25 end 2 1 el 1 el 0.25 - 2 2 mat 2 1 sys

end 21 el 1 el 4

Q XY isolated

XY 00 0.0078 0.0041 0.0156 0.0315 0.0234 0.099 0.0313 0.2127 0.0391 0.3661 0.0469 0.5414 0.0547 0.7125 0.0625 0.8502 0.0703 0.9284 0.0781 0.9293 0.0859 0.8469 0.0938 0.6891 0.1016 0.4765 0.1094 0.2385 0.1172 0.0091 0.125 0.1798 - 0.1328 0.3034 - 0.1406 0.3482 - 0.1484 0.3137 - 0.1563 0.2127 - 0.1641 0.0679 - 0.1719 0.0921 0.1797 0.2381 0.1875 0.3452 0.1953 0.3971 0.2031 0.3882 0.2109 0.3238 0.2188 0.2184 0.2266 0.0924 0.2344 0.0319 - 0.2422 0.1348 - 0.25 0.2023 - 0.2578 0.2282 - 0.2656 0.2147 - 0.2734 0.171 - 0.2813 0.1109 - 0.2891 0.0494 - 0.2969 0 0.3047 0.0283 0.3125 0.0323 0.3203 0.0144 0.3281 0.0181 - 0.3359 0.0553 - 0.3438 0.0876 - 0.3516 0.108 - 0.3594 0.1138 - 0.3672 0.1075 - 0.375 0.0964 - 0.3828 0.0909 - 0.3906 0.1023 - 0.3984 0.1398 - 0.4063 0.2086 - 0.4141 0.3076 - 0.4219 0.4294 - 0.4297 0.5606 - 0.4375 0.6838 - 0.4453 0.7802 - 0.4531 0.8329 - 0.4609 0.8291 - 0.4688 0.7629 - 0.4766 0.6362 - 0.4844 0.4584 - 0.4922 0.245 - 0.5 0.0158 - 0.5078 0.2084 0.5156 0.4087 0.5234 0.5698 0.5313 0.6827 0.5391 0.7443 0.5469 0.7578 0.5547 0.7308 0.5625 0.6739 0.5703 0.5984 0.5781 0.5149 0.5859 0.4315 0.5938 0.3531 0.6016 0.2821 0.6094 0.2186 0.6172 0.1612 0.625 0.1089 0.6328 0.0611 0.6406 0.019 0.6484 0.0152 - 0.6563 0.0383 - 0.6641 0.0469 - 0.6719 0.0391 - 0.6797 0.0145 - 0.6875 0.0245 0.6953 0.0727 0.7031 0.1231 0.7109 0.1672 0.7188 0.1969 0.7266 0.2054 0.7344 0.1889 0.7422 0.1473 0.75 0.0844 0.7578 0.0072 0.7656 0.0743 - 0.7734 0.1494 - 0.7813 0.2081 - 0.7891 0.2425 - 0.7969 0.2482 - 0.8047 0.2249 - 0.8125 0.1768 - 0.8203 0.1116 - 0.8281 0.0394 - 0.8359 0.0284 0.8438 0.0812 0.8516 0.1103 0.8594 0.1099 0.8672 0.0781 0.875 0.0165 0.8828 0.07 - 0.8906 0.1742 - 0.8984 0.2874 - 0.9063 0.4005 - 0.9141 0.5052 - 0.9219 0.5942 - 0.9297 0.6618 - 0.9375 0.704 - 0.9453 0.7184 - 0.9531 0.7039 - 0.9609 0.6603 - 0.9688 0.5884 - 0.9766 0.49 - 0.9844 0.3677 - 0.9922 0.2256 - 1 0.0689 - 1292 mat :

XY rows 129

X XY 1 col :

Y XY 2 col :

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

continuous up to 2nd derivative x f X Y x cinterp :

N XY rows : 129

4 appVersion 0.98.6179.21440 ≡

ω 2 π N / * : 0.048707

Δx X N el X 1 el - (N 1 - / : 0.007812

a b c spec . 11 black 13 mat : pts XY a b c spec pts : 21 sys

u λ u X 1 el - (ω Δx / * :

0.05 λ 0.311724

n 0 N range A n 1 + el 2 N / Y k 1 + el n k * ω * cos * k 0 N 1 - sum * eval : B n 1 + el 2 N / Y k 1 + el n k * ω * sin * k 0 N 1 - sum * eval : 21 line for

½ N to avoid aliasing N 2 / Floor

A 1 el 2 / 0.006326 -

Y Mean 0.006326 -

x μ 10 x ≤ (x 1 ≤ (& cases :

local definition u B39 A 1 el 2 / A n 1 + el u λ n * cos * B n 1 + el u λ n * sin * + (n 1 N 2 / Floor sum + 11 line :

local in-line definition

Fourier Budapest "B39" polynomial adjustable smoothing

pts x B39 x μ * 2 1 sys

Unform ...  -1 < rnd < 1 n λ rnd μ n 11 line : λ μ Random * λ 2 / - 21 line :

'N' decimals ...Uniform ... -1 < rnd < 1 n λ N Nrnd i 1 n range r i el n λ rnd i el N round : for r 21 line :

noise 1153 Nrnd :

noise 1.057 - 0.669 - 1.965 - 1.462 1.425 0.259 1.938 1.851 2.039 0.903 - 1.085 - 111 mat

13 Random 0.681439 0.990598 0.506937 13 mat

15 Random transpose 0.084070709573138 0.438619443419678 0.334288367225923 0.614369168232367 0.307396553599926 51 mat

plotG(vx,vy,char,size,color)

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

X 1.5 35745 0.5 - 71 mat :

Y 010424671 mat :

t0 1 time :

Cntrl X Y . 14 black plotG :

r 1 :

X X r RepEnd : Y Y r RepEnd : 21 line

<= isolate the repeated polygon XY X Y augment eval :

DeBoor 'u' vector

u 1 X rows n 2 + (+ range :

v j N0 μ j 11 line : u μ el v ≤ (v u μ 1 + el < (& 10 if 21 line :

4 appVersion 0.98.6179.21440 ≡

1. correct
2. wrong u μ el v ≤ (v u μ 1 + el < (& u μ el v ≤ (v u μ 1 + el ≤ (& 21 sys

v j N1 μ j 11 line : v u μ el - u μ 1 + el u μ el - / v μ N0 * u μ 2 + el v - u μ 2 + el u μ 1 + el - / v μ 1 + N0 * + 21 line :

v j N2 μ j 11 line : v u μ el - u μ 2 + el u μ el - / v μ N1 * u μ 3 + el v - u μ 3 + el u μ 1 + el - / v μ 1 + N1 * + 21 line :

respective DeBoor components v j N μ j 11 line : v μ N1 21 line μ j 11 line : v μ N2 21 line 21 mat :

n 2 :

Defined as local definition v Sx v k N n el X k el * (k 1 X rows sum 11 line :

Defined as local definition v Sy v k N n el Y k el * (k 1 X rows sum 11 line :

<= n=2 populate for more points at will vx 310 3.05 range :

i 1 vx rows range SX i el vx i el Sx eval : for

i 1 vx rows range SY i el vx i el Sy eval : for

DeBoor SX SY augment :

1 time t0 - 25.9 s *

De Boor parametric order 2 wrt Smath ... kind of Convex Hull

XY DeBoor Cntrl 3 1 sys

De Boor components

vx SX augment vx SY augment 2 1 sys

recast Smath parametric Bessel plugin ... x p f u x 11 line : u 1 u Bessel * p 0 u Bessel * - 21 line :

p 5 :

u 4 :

u p f 1.7216

x p f u u p f + 20 red 1 5 mat 2 1 sys $TOL 1014 -^:$

re-bracket rootDichotomy wrt 'p'

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 line n 0 : xn p f abs ε > xn p f a p f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 31 line :

r 13 TOL D 46 TOL D 6 8.5 TOL D 1012 TOL D 1315 TOL D 51 mat : 1.9898 4.7131 7.6177 10.6223 13.6786 51 mat

r p f vectorize 71015 -^* 91015 -^* 61015 -^* 21015 -^* - 81015 -^* - 51 mat