X(Φ,α,β,ε) ..... RootSecant algo

82e623be-0376-4245-b883-1a67226e9c8c 13 4 5 decimal

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"RootSecant" is viable  => guess wisely tial/error Φ α β ε X Root Secant u 1 el ε : u 2 el β β Φ α β - (* α Φ β 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 eval :

H. T. Davis Γa(x)

Abramowitz & Stegun, p. 256 c 1 0.5772156649015329 0.6558780715202538 - 0.0420026350340952 - 0.1665386113822915 0.0421977345555443 - 0.0096219715278770 - 0.0072189432466630 0.0011651675918591 - 0.0002152416741149 - 0.0001280502823882 0.0000201348547807 - 0.0000012504934821 - 0.0000011330272320 0.0000002056338417 - 0.0000000061160950 0.0000000050020075 0.0000000011812746 - 0.0000000001043427 0.0000000000077823 0.0000000000036968 - 0.0000000000005100 0.0000000000000206 - 0.0000000000000054 - 0.0000000000000014 0.0000000000000001 261 mat :

1 z Γ / c k el z k^* k 1 ∞ sum ≡

z Γpoly c k el z k^* k 126 sum eval :

z Γa 1 z Γpoly / :

x c 16 - x ≤ (x 1 ≤ (& cases :

x Γa x c * 1 1 + 15 red 1 5 mat 2 1 sys

x U x Γa x 0 ≤ x Γa 0 x ≤ (x 1 ≤ (& j 1 x Ceil range vY j el x j - (: for vY k el k 1 x Ceil product x x Ceil - Γa * 21 line cases :

Lower Incomplete Gamma Function a x γ x a^x - exp * x k^a j + (j 0 k product / k 050 sum * : Upper Incomplete gamma Function a x Γ a Γ a x γ - : 41 line

x Γ x 1 - (! x x Floor - 0 ≡ x U cases :

a 0.5 :

164.56789 Γ 3.62616481361685 10292^*

x λ 10 x ≤ (x 2 a * ≤ (& cases :

x 6.123456789 :

x Γ x Gamma 21 mat 148.33825986845 148.338490665811 21 mat

Jnct a a Γ a a γ + 2 / : 0.8862

X 0.1 a 0.11 range : i 1 X rows range Y i el a X i el Γ : for x f X Y x cinterp : 31 line

x Φ a x γ a x Γ - eval :

α 0.2 : β 0.5 : ε 1015 -^: 13 mat

sol Φ α β ε X :

U x f Jnct - 0 ≡ x X 1 el a solve : 0.2274682

sol 11015 -^* 0.238739753660937 0.233859672474679 0.227353946214895 0.227469375974804 0.227468211772555 0.227468211559784 0.227468211559786 0.227468211559789 0.227468211559788 0.227468211559787 111 mat

a x Γ x λ * a x γ x λ * U a U Γ + 50 black 1 5 mat a a a Γ . 14 blue 1 5 mat a a a γ . 14 red 1 5 mat 5 1 sys

Explode the nested 'sol' R sol sol rows row 1 el :

R a R Γ 12 mat 0.227 0.886 12 mat

N n p x 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 N n p x hypergeom :

much faster for convenience x Γ x Gamma ≡

Continuous PDF Hypergeometric

0 x ≤ x hypergeom outside lower bound if if/otherwise ... (x ≤ 0)&(x ≤ 15) Reconciliate the Solver

Gamma(x)

constant 15 D accuracy @ integers ... can be made iterated product(s) x Γ x Γa x 0 ≤ x Γa 0 x ≤ (x 1 ≤ (& x 1 - (x 1 - Γa * 1 x ≤ (x 2 ≤ (& x 2 - (x 1 - (* x 2 - Γa * 2 x ≤ (x 3 ≤ (& x 3 - (x 2 - (* x 1 - (* x 3 - Γa * 3 x ≤ (x 4 ≤ (& x 4 - (x 3 - (* x 2 - (* x 1 - (* x 4 - Γa * 4 x ≤ (x 5 ≤ (& x 5 - (x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 5 - Γa * 5 x ≤ (x 6 ≤ (& x 6 - (x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 6 - Γa * 6 x ≤ (x 7 ≤ (& x 7 - (x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 7 - Γa * 7 x ≤ (x 8 ≤ (& x 8 - (x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 8 - Γa * 8 x ≤ (x 9 ≤ (& x 9 - (x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 9 - Γa * 9 x ≤ (x 10 ≤ (& x 10 - (x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 10 - Γa * 10 x ≤ (x 11 ≤ (& x 11 - (x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 11 - Γa * 11 x ≤ (x 12 ≤ (& x 12 - (x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 12 - Γa * 12 x ≤ (x 13 ≤ (& x 13 - (x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 13 - Γa * 13 x ≤ (x 14 ≤ (& x 14 - (x 13 - (* x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 14 - Γa * 14 x ≤ (x 15 ≤ (& x 15 - (x 14 - (* x 13 - (* x 12 - (* x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 15 - Γa * cases :

Ω 2.5 :

x U 1 - Ω 10 Ω 2 2 mat 2 1 sys

x Γ Ω - 0 ≡ x 04 solve 0.3562 3.2307 21 mat