Γ(x) [15 D] Alvaro Same otherwise

6951be4a-a62f-4e2e-9c30-84f8657d87a3 36 4 5 decimal

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

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

Difficult. . . solving Hypergeometric PDF

]]>

NOT compatible
crashes Smath. # x Gamma :

4 appVersion 0.98.6179.21440 ≡

t0 1 time :

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 : max 4.649081861435 : 41 sys

N population size ≡ n number of success in population ≡ p number of draws per trial ≡ x evolution of observed successes ≡ 41 sys

x hypergeom x N n p hypergeom :

x hypergeom 0.125 - 0 ≡ x 010 solve 2.8142 6.5373 21 mat

x λ 10 x ≤ (x max ≤ (& cases :

recast for Golden search ... suite. x f x hypergeom :

TOL 1012 -^:

Golden search

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 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 71 line :

Continuous PDF Hypergeometric

x hypergeom x λ * x hypergeom 0.125 3 1 sys

X f 45 y1 y2 ≥ TOL Extrema 1 el : 4.649081861435

important insights 0 hypergeom max hypergeom 21 mat 0.0009 0.2636 21 mat

Cumulative is asymptotic '1'
  ► Hypergeometric Normalized x f x 012 int 1

mesh the recursive inversers N 50 :

The two branches inversers

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 0 : H max hypergeom : N N : 13 mat

u L H N xd :

idx 2 u rows 1 - (range :

Recursive seeded solve ... greedy but productive.
Once 'sol' in hand  ► spline interpolate. sol i 2 u rows 1 - (range hyper i 1 - el x f u i el - 0 ≡ x 0 4.7 solve : for hyper 4 round vectorize 21 line :

sol u idx el sol augment :

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L max hypergeom : H 0 : N N : 13 mat

u L H N xd :

idx 2 u rows 1 - (range :

Recursive seeded Smath solve ... much greedy but productive.
Once 'sol' in hand  ► spline interpolate. Sol i 2 u rows 1 - (range hyper i 1 - el x f u i el - 0 ≡ x 4.65 10 solve : for hyper 4 round vectorize 21 line :

Sol u idx el Sol augment :

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 :

sol Sol 0.125 - max 0.375 max 2 2 mat 3 1 sys

Σ U 010 0.5 range : i 1 U rows range Cumul i el x f x U 1 el U i el int eval : for U Cumul augment 31 line :

Continuous PDF Hypergeometric ... cumulative

x hypergeom x λ * x hypergeom Σ max 0 + 200 black 1 5 mat 4 1 sys

1 time t0 - min 13

At this point, you can: 1. interpolate each discrete inverser. 2. interpolate the cumulative. This document is demo of the techniques. Needs be done for each [N, n, p].