fdcef06f-4f8f-4a90-b5f3-4302e01ec390 27 4 5 false false 0 0 decimal

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

x Γ 2 π * x / sqrt 1 e / x 112 x * 110 x * / - / + (* (x^* :

2 π * x / sqrt 1 e / x 112 x * 110 x * / - / + (* (x^*

x g 2 :

x Γ x g 2 1 sys

COMMENTS:
1. Plot first the system to enable the bracketing.
2. Sometimes, "solve" is hard to refine ....
=> get higher accuracy from "RootDichotomy"

"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 n 21 mat 51 line :

x Φ x Γ x g - eval :

ε 1015 -^:

R x Φ 24 ε D : 3.000003 4821 mat

Sanity check R 1 el Φ 2.2 1016 -^* -

"RootSecant" is viable  => refine from "sanity check" Φ a b ε X Root Secant u 1 el 1015 -^: u 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : u i el u i 1 - el - abs ε ≥ u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : i i 1 + : 21 line while u 71 line :

x Φ x Γ x g - eval :

a 0.44 : b 0.46 : ε 109 -^: 13 mat

solΓ Φ a b ε X :

solΓ 11015 -^* 0.439286 0.439286 2.507237 1027 -^* i * - 31 mat

sanity check solΓ 2 el Φ 0

a 2.99 : b 3 : ε 1012 -^: 13 mat

solΓ Φ a b ε X :

solΓ 11015 -^* 3.000003 3.000003 5.523334 1030 -^* i * - 31 mat

sanity check solΓ 2 el Φ 3.321 108 -^*

π Gamma 2.288

x Γ x g - 0 ≡ x 2.875 3.125 solve 3.000002655640000

x Γ 2 π * x / sqrt 1 e / x 112 x * 110 x * / - / + (* (x^* eval :

x q 0.25 0.5 x * + :

x f x Γ :

Newton-Raphson looks like a curiosityrather disastrous in this example. f q init ε Zero Newton-Raphson i 1 : x 1 el init : x 2 el x i el x i el f x i el q 1012 -^+ / - : x i 1 + el x i el - abs ε > i i 1 + : x i 1 + el x i el x i el f x i el q 1012 -^+ / - : 21 line while x 61 line :

init 2.78 :

zero f q init 109 -^Zero :

zero 2.78 1.775 0.962 0.439 - 229.777 - 44.656 i * - 229.777 - 44.656 i * - 61 mat

sol x f x q - 0 ≡ x 1 1.6 solve x f x q - 0 ≡ x 2.5 3 solve 21 mat : 1.296 2.77 21 mat

x f x q sol 1 el sol 1 el f + 20 black 1 5 mat sol 2 el sol 2 el f + 20 black 1 5 mat 4 1 sys

Sanity check sol f sol q - vectorize 8.8 1010 -^* 9.9 1012 -^* - 21 mat

u 2 v * + 1 ≤ (5 u * 2 v * - 5 ≥ (* u 2 v * - 9 ≤ (*

'N' random in critical region N 1000 :

t0 1 time :

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

size/define critical region.Plot first before filling xmin 1 - : xmax 5 : ymin 5 - : ymax 0 : 22 mat space_x xmax xmin - (: space_y ymax ymin - (: 21 line 21 sys

A function that tests an arbitrary point for its membership in the critical region. x y Test x 2 y * + 1 ≤ (5 x * 2 y * - 5 ≥ (* x 2 y * - 9 ≤ (* :

a b c spec . 6 #808080 13 mat :

A routine that finds an array of N random pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 6 lime 13 mat : fill Pts a b c spec pts : 21 line

Filling the critial domain + CenterGravity

1 x - 2 / 5 5 x * - ( 2 / - 9 x - ( 2 / - 3 1 sys fill 1.6667 2.333 - . 15 black 1 5 mat 3 1 sys

1 time t0 - 4.5 s *