<p> Individual style </p>

0801593a-62f0-4fc5-a642-e29cb9726825 43 4 5 decimal

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

RootDichotomy ... solving roots i.e: zeros

x F x 2 π x * / sqrt * x sin * 0.008 2 π x * / sqrt * x cos * - :

x f x F x diff :

"RootDichotomy"  is a robust companion. # Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε ≥ xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

x F x f 2 1 sys

guesstimate sol x F x 010 solve : 0.0893 3.1441 6.2845 9.4256 41 mat

ε 1015 -^:

x Φ x F :

braket wisely
rootDichotomy
... etc. r x Φ 0.05 0.1 ε D x Φ 3 3.25 ε D x Φ 6 6.5 ε D x Φ 910 ε D 41 mat : 0.089323636896763 3.14413706643404 6.28445828812909 9.42562671049443 41 mat

RootSEcant ... solving intersections

guesstimate intersections sol x F x f - x 010 solve : 1.0861 3.9959 7.1062 31 mat

"RootSecant"  ... collect last iterate style Φ a b ε X Root Secant us 1 el ε : us 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : us i el us i 1 - el - abs ε ≥ us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : i i 1 + : 21 line while last iterate ... index 1 explode us us rows row 1 el 81 line :

x Φ x F x f - eval :

sols Φ 1.05 1.1 ε X Φ 3.9 4.1 ε X Φ 7 7.2 ε X 31 mat :

sols 1.08591233254523 3.99588166423646 7.10547121356862 31 mat

x f x 0.5 1 -^x - exp * : x q π sqrt x sqrt erf * : x Φ x f x q - eval : 31 sys

ε 0.5 1016 -^* :

X Φ 0.25 0.5 ε X : 0.384477669231791

x f x q X X f + 20 black 1 5 mat 3 1 sys

sanity check X f X q - 7.7 1016 -^*

X f X q - Floor 0.000000000000000

x q x 3^x 4 ≤ x 2^cases :

ε 1014 -^:

x g 32 :

unexplained
duplicates ? Xq x q x g - x 37 solve : 3.17480206422193 5.65686186786947 5.65685424818816 31 mat

x q x g 2 1 sys

x Φ x q x g - eval :

10^{-15}... division '0' sols Φ 34 ε X Φ 56 ε X 21 mat :

sols 3.1748021039364 5.65685424949238 21 mat

"RootDichotomy"  is a robust companion. # Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε ≥ xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

"RootSecant" is viable  => guess wisely tial/error Φ a b ε X Root Secant us 1 el ε : us 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : us i el us i 1 - el - abs ε ≥ us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : i i 1 + : 21 line while collect last iterate index 1 explode us us rows row 1 el 81 line :

r 0 x mwbesselj x 030 solve :

x f 0.00625 x * :

x B 0 x mwbesselj :

0 x mwbesselj x f 2 1 sys

r 2.404826 5.520078 8.653728 11.791534 14.930918 18.071064 21.211637 24.352472 27.493479 91 mat

sanity checked up to mwbesselj(0,40) 040 mwbesselj 0.007366890584237

"roots" fails ... mwbesselj(0,x) does NOT
have symbolic derivative(s) diff(▪,▪) exact 0 x mwbesselj x 1 roots :

r Clear 1

ε 1015 -^:

x Φ 0 x mwbesselj :

braket wisely
rootDichotomy
... etc. r x Φ 2.3 2.5 ε D x Φ 5.4 5.6 ε D x Φ 8.6 8.7 ε D x Φ 11.7 11.9 ε D x Φ 14.5 15.5 ε D 51 mat : 2.40482555769577 5.52007811028631 8.65372791291101 11.7915344390143 14.9309177084878 51 mat

Guesstimate for RootSecant X(Φ,a,b,ε) x x f - B x 020 solve 2.42 5.5548 8.7082 11.8657 15.0248 18.1847 61 mat

x Φ x B x f - (eval :

Intersect Φ 23 ε X Φ 56 ε X Φ 89 ε X Φ 1113 ε X Φ 1415 ε X Φ 1819 ε X 61 mat :

Intersect 2.37638206105862 5.6245547834828 8.45992250470978 12.1286151984414 14.4845295371415 18.7609539221604 61 mat

Exotic solve ...

x q 64 x * + x 2^+ :

datum 2 sqrt - :

x u f u 4 x * + x 2^+ :

from Golden ratio x 2 - :

u x u f datum ≡ u solve maple : 42 sqrt -

x q x u f 2 - 5 2 - 5 - 2 2 mat 3 1 sys

1. q(x) has min @ Gloden ratio -2 2. define f(x,u) wrt to 'u' to be solved 3. immadiate from Maple. 4. solve for any user datum

Jean

The Smath root(s) solve scanner

The ROBUST root(s) Solve scanner f x0 x1 TOL Solve ρ x f x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : ε TOL : 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 61 line X i el xn : 21 line for X 31 line :

x f x exp x 2^+ 5 x * - :

TOL 1014 -^:

f 02 TOL Solve 0.280490945324611 1.73399571327305 21 mat

x f

=====================================================================================

x f 6 x 3^* 7 x 2^* + 94 x * - 105 + :

TOL 1014 -^:

x f

Initialise wisely by observing the plot f 6 - 3 TOL Solve 5 - 1.5 2.33333333333333 31 mat

5 - 32 / 73 / 13 mat

===================================================================================

TOL 1014 -^:

x f 1.5 x mwbesselj :

x f

f 016 TOL Solve 0 4.49340945790907 7.72525183693773 10.9041216594289 14.0661939128315 51 mat

The Smath "root(s)intersection" solve scanner

The ROBUST root(s) Intersection Solve scanner f q x0 x1 TOL Solve ρ x f x q - 0 ≡ x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : ε TOL : n 0 : xn f xn q - abs ε > xn f xn q - (a f a q - (* 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 61 line X i el xn : 21 line for X 31 line :

x f x 2^:

x q 1.25 :

x f x q 2 1 sys

TOL 1014 -^:

f q 1.25 - 1.25 TOL Solve 1.11803398874989 - 1.1180339887499 21 mat

1.11803398874989 - f 1.25 - 1.088 1014 -^* -

=================================================================================

x f 6 x 3^* 7 x 2^* + 94 x * - 105 + :

x q 128 :

x f x q 2 1 sys

TOL 1012 -^:

f q 5 - 4 TOL Solve 4.475693 - 0.241243 - 3.55027 31 mat

====================================

ρ f q 5 - 4 TOL Solve :

sanity 10e-15 ρ 1 el f 128 - ρ 2 el f 128 - ρ 3 el f 128 - 31 mat 31013 -^* - 91013 -^* - 11012 -^* - 31 mat

=====================================================================

TOL 1014 -^:

x f x tan :

x q 2 x * :

x f x q 2 1 sys

r f q 1.75 - 1.75 TOL Solve :

This case has no other explanation than numerical instability in the negative range tan(x)

r 1 el f 2 r 1 el * - r 3 el f 2 r 3 el * - r 4 el f 2 r 4 el * - 31 mat 2.4 1015 -^* - 0 4.6 1015 -^* 31 mat

r 1.16556118520721 - 1.16556118520721 - 0 1.16556118520721 41 mat

=====================================================================

x f x 2 π * / 2 π * x / tanh * sqrt :

TOL 1015 -^:

x q 0.5 :

z f q 1.75 - 1.75 TOL Solve :

x f x q z 1 el z 1 el f . 15 black z 2 el z 2 el f . 15 black 2 5 mat 3 1 sys

z 1.571856 - 1.571856 21 mat

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 <p> Individual style </p>

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 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 101 mat : 1.260635847 7.157169484 10.271949701 13.399139648 16.531761612 19.667234931 22.804388185 25.942613731 29.081565224 32.221030884 101 mat

r f vectorize 91015 -^* 11014 -^* 11014 -^* 41015 -^* - 41015 -^* - 61016 -^* 31015 -^* 11014 -^* 71015 -^* - 11014 -^* 101 mat