Individual style

5fceb06e-2bbf-415d-a37d-a18f8478c593 24 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

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 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

y1 y2 ≥ (max in range ≡

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

TOL 1015 -^:

y1 y2 ≤ (min in range ≡

TOL f 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 51 mat 1.01 - 2.73722612876613 5.693008214621 8.76767453046036 11.8760735845163 51 mat