Gamma(x) Lanczos Davide

4a0aa0ce-8630-4116-9f56-078c5e6425d0 26 4 5 decimal

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

z Lanczos Davde Carpi 20200514 ε 11015 -^* : p 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - .33994649984811888699 104 -^* .46523628927048575665 104 -^* .98374475304879564677 - (104 -^* .15808870322491248884 103 -^* .21026444172410488319 - (103 -^* .21743961811521264320 103 -^* .16431810653676389022 - (103 -^* .84418223983852743293 104 -^* .26190838401581408670 - (104 -^* .36899182659531622704 105 -^* 141 mat : g 4.7421875 : z Re 0.5 < y π π z * sin 1 z - Lanczos * / : z z 1 - : x 0.99999999999999709182 : n 1 p length range x x p n el z n + / + : for t z g + 0.5 + : y 2 π * sqrt t z 0.5 +^* t - exp * x * : 51 line if y Im abs ε ≤ y Re y if 61 line :

g=4.7421875, n=15

x Lanczos X(Φ,α,β,ε) ..... RootSecant algo

"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 :

4 appVersion 0.98.6179.21440 ≡ Applications 33 s ≡ 21 mat

Twin companions [overlord] x Γ x Lanczos : 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 - : 51 line

z 2.5 i * :

z Lanczos 0.0161290540478 0.0267491089459 i * -

a 0.5 :

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 :

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

sol 11015 -^* 0.238739753660935 0.233859672474678 0.22735394621489 0.227469375974801 0.227468211772553 0.227468211559783 0.227468211559784 0.227468211559773 0.227468211559785 0.227468211559785 111 mat

a x Γ a x γ R a R Γ + 20 black 1 5 mat 3 1 sys

R a R Γ 12 mat 0.227 0.886 12 mat

x Γ

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

2.5 Γ 1.3293

ε 1017 -^:

Sanity check Mathcad 11

a 0.5 :

x λ 10 x ≤ (x 2 a * ≤ (& cases :

https://keisan.casio.com/exec/system/1180573447

1 ε Γ a ε Γ 21 mat 0.999999999999986 1.772453844580950 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

0.5 0.5 γ 0.5 0.5 Γ 14.5 Γ 31 mat 1.21003561931111 0.5624182315944 2.30923179223137 1010^* 31 mat

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

NOTE ... Smath plots at the kernel level but solves neither γ(a,x), Γ(a,x). It complains about 'k'... f(x) reconciliates the solver.

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

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 :

sol 11015 -^* 0.238739753660935 0.233859672474678 0.22735394621489 0.227469375974801 0.227468211772553 0.227468211559783 0.227468211559784 0.227468211559773 0.227468211559785 0.227468211559785 111 mat

Example_1: Multistage XFR

t0 1 time :

System TimeConstant T 1 :

Laplace UnitStep s 1 :

x λ 10 x ≤ (x 20 ≤ (& cases :

time 7 :

Laplace XFR multistage system
'n' stages, 'T' time constant n t f n 1 + Γ n 1 + t T / Γ - s n 1 + Γ * / :

3 x f x λ * 6 x f x λ * 9 x f x λ * time 1.25 time 0.25 - 2 2 mat 6 1 sys

3 time f 6 time f 9 time f 31 mat 0.92 0.55 0.17 31 mat

0.92 0.55 0.17 31 mat

XY 0 127.42 0.1 123.469 0.2 118.008 0.3 111.187 0.4 102.811 0.5 93.554 0.6 84.168 0.7 74.818 0.8 66.373 0.9 58.587 1 51.678 1.1 45.739 1.2 40.675 1.3 36.404 1.4 32.74 1.5 29.581 1.6 26.848 1.7 24.472 1.8 22.395 1.9 20.573 2 18.97 2.1 17.555 2.2 16.3 2.3 15.184 2.4 14.188 2.5 13.295 2.6 12.492 2.7 11.768 2.8 11.12 2.9 10.547 3 10.028 312 mat :

Example_2: Fitting Mathsoft data set.

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 :

pts XY o 5 black plot :

ε 1015 -^:

yo 10.028 : a 143 : b 1.77 : η 2.5 : 14 mat

x f yo a b x Γ η^* + :

Ω u ε 3 0.05 range : i 1 u rows range vy i el u i el f eval : for u vy augment 31 line :

X ε 3 0.05 range : i 1 X rows range Y i el X i el f eval : for x S X Y x cinterp : 31 line

... 'G' ▼  ordinate solve abscissa G 64 :

U x S G - 0 ≡ x ε 3 solve : 0.836

plot 0 G U G U 16 - 32 mat Ω pts 0.175 - G 7 + G 12 blue 15 mat U 0 U 12 blue 15 mat 51 sys :

plot

G U 12 mat 64 0.836 12 mat

1 time t0 - 32.7 s *

Example_3: Hypergeometric statistical function.

Hypergeom ...

much faster x Γ x Gamma :

x Γ x Lanczos :

HTDavis zapΓ ≡

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 :

Continuous PDF Hypergeometric

0 x ≤ x hypergeom outside lower bound if

In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of 'k' successes in 'n' draws, without replacement, from a finite population of size 'N' that contains exactly 'K' successes, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of 'k' successes in 'n' draws with replacement.

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 if/otherwise ... (x ≤ 0)&(x ≤ 15) suite IterativeProduct

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 :

RemToDo ▲ code recursive