∫Adaptive(f,a,b,N,ε) Carlos

fe367142-87d7-45d8-b1c6-98008cb6a1e2 2 4 5 decimal

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

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

x 0.5 1 - ^ x - exp * π sqrt x sqrt erf * 2 1 sys

Test 0.99.7818

<= Wolfram Alpha

MCD 11 default adaptive ... 0.611991 x 0.5 1 -^x - exp * x 1015 -^0.1 int Don't even attempt ≡

Infinite  limit x f x 0.5 1 -^x - exp * :

'N' iterations
precision N 2000 : p 1017 -^: 21 sys

lower limit
upper limit a 1029 -^: b 0.1 : 21 sys

t0 1 time :

f a b N p ∫Adaptive 0.611991366111712

π sqrt b sqrt erf * 0.611991366111770

1 time t0 - min 3.7

Int(f,a,b,n) ... Simpson/Romberg Kernel

Basic Romberg-Simpson ... integration acc from options f a b Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline 
Integration accuracy from user command line 'n' f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : α 0 : β 1 : h β α - n / : 14 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

RKfixed ... Simpson incompatible f(x) ... X^Y not scalar wrt 'x'

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

end 12 :

x c 10 x ≤ (x end ≤ (& cases :

0 f 403

x Y D x f :

init 0 :

sol init 0 end 1200 x Y D rkfixed :

x f x c * sol 2 1 sys

Result above max ...
Smath X^Y not scalar wrt 'x'
Rescued  via Romberg kernel x f x 012 int #

sol sol rows row 122980712 mat

f 012100 Int 29808

code or plugin
add to palette f a b N p ∫Adaptive f a b n Int 21 mat

Γ(x) [15 D] Alvaro Same otherwise

Γ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh z 0.5 + : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ 1 el Γ κ 1 + el z κ + / κ 114 sum + (* zp 2^* zg - exp * eval 21 line :

Valid for Re(ζ)>0

Hypergeom ...

Smath Gamma(x) incompatible solve x Γ x Gamma :

x N n p hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 39 :

n 12 :

p 20 :

x hypergeom x N n p hypergeom :

x hypergeom 0.125 - 0 ≡ x 23 solve 2.8142

Continuous PDF Hypergeometric

0 x ≤ x hypergeom outside lower bound if 0.125 2 1 sys x Γ