Individual style

5fceb06e-2bbf-415d-a37d-a18f8478c593 38 4 5 decimal

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

Nikolai project ... 20220902

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 :

j 0101 range :

scale j j 0 * augment :

Z scale + 14 blue plot :

F(x) = x * Bessel(1;x) - 1,01 * Bessel(0;x).

x F x 1 x Bessel * 1.01 0 x Bessel * - eval :

Z 00 + 14 blue 10 + 14 blue 20 + 14 blue 30 + 14 blue 40 + 14 blue 50 + 14 blue 60 + 14 blue 70 + 14 blue 80 + 14 blue 90 + 14 blue 100 + 14 blue 115 mat

0 x Bessel 1 x Bessel 2 1 sys x F

recast the F(x) x f x F eval :

Plot the unary intervals to help bracket Dichotomy ... only few unary intervals to pilot spacing

0 2 0 2 - 2 2 mat x f 0 0 + 14 blue 1 0 + 14 blue 2 0 + 14 blue 3 0 + 14 blue 4 0 + 14 blue 5 0 + 14 blue 6 0 + 14 blue 7 0 + 14 blue 8 0 + 14 blue 9 0 + 14 blue 10 0 + 14 blue 11 5 mat 3 1 sys Individual style

a 1.25 : b 1.27 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 1.26063584661327

xn f 91015 -^*

iterates n 39

a 3.5 : b 4.5 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 4.08178816632211

xn f 41015 -^* -

iterates n 48

a 6.5 : b 7.5 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 7.15716948449889

sanity xn f 51015 -^*

iterates n 51

a 10 : b 10.5 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 10.2719497005655

sanity xn f 11014 -^*

iterates n 59

a 13 : b 14 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 13.3991396481971

xn f 41015 -^* -

iterates n 106

a 16 : b 17 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 16.5317616124364

xn f 41015 -^* -

iterates n 68

a 19 : b 20 : 12 mat xn b a + (2 / : ε 1014 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 19.6672349308422

roots 1.26063584661327 4.08178816632211 7.15716948449889 10.2719497005655 13.3991396481971 16.5317616124364 19.6672349308422 71 mat :

xn f 61016 -^*

iterates n 41

a 32 : b 32.25 : 12 mat xn b a + (2 / : ε 1013 -^: 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 32.2210308842134

sanity xn f 1.2 1014 -^*

iterates n 42

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 f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 31 line :

r 1.25 1.27 1014 -^D 3.5 4.5 1014 -^D 6.5 7.5 1014 -^D 10 10.5 1014 -^D 13141014 -^D 16171014 -^D 19201014 -^D 22.5 231014 -^D 25.75 261014 -^D 28.75 29.25 1014 -^D 32 32.5 1013 -^D 111 mat : 1.260635847 4.081788166 7.157169484 10.271949701 13.399139648 16.531761612 19.667234931 22.804388185 25.942613731 29.081565224 32.221030884 111 mat

r f vectorize 91015 -^* 41015 -^* - 11014 -^* 11014 -^* 41015 -^* - 41015 -^* - 61016 -^* 31015 -^* 11014 -^* 71015 -^* - 11014 -^* 111 mat

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

R 1.2606 0 . 4.0818 0 . 7.1572 0 . 10.2719 0 . 13.3991 0 . 16.5318 0 . 19.6672 0 . 22.8044 0 . 25.9426 0 . 29.0816 0 . 32.221 0.113 mat

Golden ratio

a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + :

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema 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 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

Plot the unary intervals to help bracket roots-Dichotomy ... and bracket the Golden ratio

1.2606 0 . 4.0818 0 . 7.1572 0 . 10.2719 0 . 13.3991 0 . 16.5318 0 . 19.6672 0 . 22.8044 0 . 25.9426 0 . 29.0816 0 . 32.221 0 . 11 3 mat x f 2 1 sys

y1 y2 ≥ (max in range ≡

recast the F(x) x f x F eval :

TOL 1015 -^:

y1 y2 ≤ (min in range ≡

size 10 :

G f 2.5 3 y1 y2 ≥ TOL Extrema 1 el f 5.5 6 y1 y2 ≤ TOL Extrema 1 el f 89 y1 y2 ≥ TOL Extrema 1 el f 11.5 12.5 y1 y2 ≤ TOL Extrema 1 el f 14.5 15.5 y1 y2 ≥ TOL Extrema 1 el f 17.5 18.5 y1 y2 ≤ TOL Extrema 1 el 61 mat : 2.7372 5.693 8.7677 11.8761 14.998 18.1266 61 mat

data G G f vectorize augment : 2.737 1.337 5.693 1.908 - 8.768 2.365 11.876 2.751 - 14.998 3.091 18.127 3.398 - 62 mat

Plot the unary intervals to help bracket roots-Dichotomy ... and bracket the Golden ratio

1.2606 0 . 4.0818 0 . 7.1572 0 . 10.2719 0 . 13.3991 0 . 16.5318 0 . 19.6672 0 . 22.8044 0 . 25.9426 0 . 29.0816 0 . 32.221 0 . 11 3 mat x f 2.7372 1.3373 + size black 5.693 1.908 - + size black 8.7677 2.3647 + size black 11.8761 2.751 - + size black 14.998 3.0909 + size black 18.1266 3.3977 - + size black 6 5 mat 3 1 sys

data x TOL 10 0.1 range : i 1 x rows range Y i el x i el f eval : for x Y augment 31 line :

The function f(x) is a freak function i.e: not scalar wrt 'x'. NOT compatible for integration/derivative. 1. Discretize for Finite Differences [FD] 2. Alternately, recast f(x) from spline.

data isolated

data 11015 -^* 1.01 - 0.1 1.0025 - 0.2 0.98 - 0.3 0.9429 - 0.4 0.8916 - 0.5 0.8267 - 0.6 0.7491 - 0.7 0.6597 - 0.8 0.5597 - 0.9 0.4502 - 1 0.3328 - 1.1 0.2088 - 1.2 0.0799 - 1.3 0.0523 1.4 0.1862 1.5 0.32 1.6 0.4519 1.7 0.5802 1.8 0.7033 1.9 0.8196 2 0.9273 2.1 1.0251 2.2 1.1117 2.3 1.1856 2.4 1.2459 2.5 1.2916 2.6 1.3219 2.7 1.3362 2.8 1.3341 2.9 1.3153 3 1.2798 3.1 1.2278 3.2 1.1597 3.3 1.0759 3.4 0.9773 3.5 0.8648 3.6 0.7394 3.7 0.6024 3.8 0.4553 3.9 0.2996 4 0.1369 4.1 0.0309 - 4.2 0.202 - 4.3 0.3745 - 4.4 0.5465 - 4.5 0.716 - 4.6 0.881 - 4.7 1.0397 - 4.8 1.19 - 4.9 1.3302 - 5 1.4585 - 5.1 1.5734 - 5.2 1.6734 - 5.3 1.757 - 5.4 1.8232 - 5.5 1.871 - 5.6 1.8995 - 5.7 1.9082 - 5.8 1.8966 - 5.9 1.8646 - 6 1.8123 - 6.1 1.7398 - 6.2 1.6478 - 6.3 1.537 - 6.4 1.4082 - 6.5 1.2627 - 6.6 1.1017 - 6.7 0.9267 - 6.8 0.7395 - 6.9 0.5419 - 7 0.3359 - 7.1 0.1235 - 7.2 0.0931 7.3 0.3117 7.4 0.5298 7.5 0.7454 7.6 0.9559 7.7 1.1592 7.8 1.353 7.9 1.5352 8 1.7037 8.1 1.8566 8.2 1.9922 8.3 2.1087 8.4 2.2048 8.5 2.2792 8.6 2.3309 8.7 2.3592 8.8 2.3635 8.9 2.3434 9 2.299 9.1 2.2305 9.2 2.1383 9.3 2.0231 9.4 1.8859 9.5 1.7279 9.6 1.5505 9.7 1.3554 9.8 1.1444 9.9 0.9196 10 0.6831 1012 mat :

vx data 1 col : vy data 2 col : 12 mat x S vx vy x ainterp : 21 sys

x S x TOL 10 int 2.4487

x λ 10 x ≤ (x 10 ≤ (& cases :

x f x S :

h 0.01 :

x D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

x D1 x λ *

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 + el Y i el - X i 1 + el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X 1 deriv * augment 101 line :

D data Deriv :

D 1 1 sys

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 :

idx 65 :

Σidx Σ idx row : 6.4 1.5639 - 12 mat

track 0484 Σidx 11 el Σidx 12 el 32 mat :

Σ track 2 1 sys

Σidx 6.4 1.564 - 12 mat

Σ Σ rows row 10 2.4467 12 mat