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Tina Müller 2016-12-05 23:44:36 +01:00
parent 9dc3c2bb62
commit bba7bfd280
13208 changed files with 134745 additions and 0 deletions
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47
Task/Gamma-function/FreeBASIC/gamma-function.freebasic Normal file
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' FB 1.05.0 Win64
Const pi = 3.1415926535897932
Const e = 2.7182818284590452
Function gammaStirling (x As Double) As Double
Return Sqr(2.0 * pi / x) * ((x / e) ^ x)
End Function
Function gammaLanczos (x As Double) As Double
Dim p(0 To 8) As Double = _
{ _
0.99999999999980993, _
676.5203681218851, _
-1259.1392167224028, _
771.32342877765313, _
-176.61502916214059, _
12.507343278686905, _
-0.13857109526572012, _
9.9843695780195716e-6, _
1.5056327351493116e-7 _
}
Dim As Integer g = 7
If x < 0.5 Then Return pi / (Sin(pi * x) * gammaLanczos(1-x))
x -= 1
Dim a As Double = p(0)
Dim t As Double = x + g + 0.5
For i As Integer = 1 To 7
a += p(i) / (x + i)
Next
Return Sqr(2.0 * pi) * (t ^ (x + 0.5)) * Exp(-t) * a
End Function
Print " x", " Stirling",, " Lanczos"
Print
For i As Integer = 1 To 20
Dim As Double d = i / 10.0
Print Using "#.##"; d;
Print , Using "#.###############"; gammaStirling(d);
Print , Using "#.###############"; gammaLanczos(d)
Next
Print
Print "Press any key to quit"
Sleep

21
Task/Gamma-function/Nim/gamma-function.nim Normal file
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const a = [
1.00000000000000000000, 0.57721566490153286061, -0.65587807152025388108,
-0.04200263503409523553, 0.16653861138229148950, -0.04219773455554433675,
-0.00962197152787697356, 0.00721894324666309954, -0.00116516759185906511,
-0.00021524167411495097, 0.00012805028238811619, -0.00002013485478078824,
-0.00000125049348214267, 0.00000113302723198170, -0.00000020563384169776,
0.00000000611609510448, 0.00000000500200764447, -0.00000000118127457049,
0.00000000010434267117, 0.00000000000778226344, -0.00000000000369680562,
0.00000000000051003703, -0.00000000000002058326, -0.00000000000000534812,
0.00000000000000122678, -0.00000000000000011813, 0.00000000000000000119,
0.00000000000000000141, -0.00000000000000000023, 0.00000000000000000002 ]
proc gamma(x): float =
let y = x.float - 1.0
result = a[a.high]
for n in countdown(high(a) - 1, low(a)):
result = result * y + a[n]
result = 1.0 / result
for i in 1..10:
echo gamma(i.float / 3.0)

17
Task/Gamma-function/Oforth/gamma-function.oforth Normal file
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import: math
[
676.5203681218851, -1259.1392167224028, 771.32342877765313,
-176.61502916214059, 12.507343278686905, -0.13857109526572012,
9.9843695780195716e-6, 1.5056327351493116e-7
] const: Gamma.Lanczos
: gamma(z)
| i t |
z 0.5 < ifTrue: [ Pi dup z * sin 1.0 z - gamma * / return ]
z 1.0 - ->z
0.99999999999980993 Gamma.Lanczos size loop: i [ i Gamma.Lanczos at z i + / + ]
z Gamma.Lanczos size + 0.5 - ->t
2 Pi * sqrt *
t z 0.5 + powf *
t neg exp * ;

18
Task/Gamma-function/Ring/gamma-function.ring Normal file
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decimals(3)
gamma = 0.577
coeff = -0.655
quad = -0.042
qui = 0.166
set = -0.042
for i=1 to 10
see gammafunc(i / 3.0) + nl
next
func recigamma z
return z + gamma * pow(z,2) + coeff * pow(z,3) + quad * pow(z,4) + qui * pow(z,5) + set * pow(z,6)
func gammafunc z
if z = 1 return 1
but fabs(z) <= 0.5 return 1 / recigamma(z)
else return (z - 1) * gammafunc(z-1) ok

19
Task/Gamma-function/Sidef/gamma-function-1.sidef Normal file
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var a = [ 1.00000_00000_00000_00000, 0.57721_56649_01532_86061, -0.65587_80715_20253_88108,
-0.04200_26350_34095_23553, 0.16653_86113_82291_48950, -0.04219_77345_55544_33675,
-0.00962_19715_27876_97356, 0.00721_89432_46663_09954, -0.00116_51675_91859_06511,
-0.00021_52416_74114_95097, 0.00012_80502_82388_11619, -0.00002_01348_54780_78824,
-0.00000_12504_93482_14267, 0.00000_11330_27231_98170, -0.00000_02056_33841_69776,
0.00000_00061_16095_10448, 0.00000_00050_02007_64447, -0.00000_00011_81274_57049,
0.00000_00001_04342_67117, 0.00000_00000_07782_26344, -0.00000_00000_03696_80562,
0.00000_00000_00510_03703, -0.00000_00000_00020_58326, -0.00000_00000_00005_34812,
0.00000_00000_00001_22678, -0.00000_00000_00000_11813, 0.00000_00000_00000_00119,
0.00000_00000_00000_00141, -0.00000_00000_00000_00023, 0.00000_00000_00000_00002 ];
func gamma(x) {
var y = (x - 1);
1 / a.reverse.reduce {|sum, an| sum*y + an};
}
for i in 1..10 {
say ("%.14e" % gamma(i/3));
}

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Task/Gamma-function/Sidef/gamma-function-2.sidef Normal file
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func gamma(z) {
var epsilon = 0.0000001
func withinepsilon(x) {
abs(x - abs(x)) <= epsilon
}
var p = [
676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059,
12.507343278686905, -0.13857109526572012,
9.9843695780195716e-6, 1.5056327351493116e-7,
]
var result;
z = Complex(z)
const pi = Complex.pi
if (z.real < 0.5) {
result = (pi / (sin(pi * z) * gamma(Complex(1) - z)))
}
else {
z -= 1
var x = 0.99999999999980993
p.each_kv { |i, v|
x += v/(z + i + 1)
}
var t = (z + p.len - 0.5)
result = (sqrt(pi*2) * t**(z+0.5) * exp(-t) * x)
}
withinepsilon(result.im) ? result.real : result
}
for i in 1..10 {
say ("%.14e" % gamma(i/3));
}

11
Task/Gamma-function/Sidef/gamma-function-3.sidef Normal file
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define e = Math.e;
define pi = Math.pi;
func Γ(t) {
t < 20 ? (__FUNC__(t + 1) / t)
: (Math.sqrt(2*pi*t) * Math.pow(t/e + 1/(12*e*t), t) / t);
}
for i in 1..10 {
say ("%.14e" % Γ(i/3));
}

60
Task/Gamma-function/Visual-FoxPro/gamma-function.visual Normal file
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LOCAL i As Integer, x As Double, o As lanczos
CLOSE DATABASES ALL
CLEAR
CREATE CURSOR results (ZVal B(1), GamVal B(15))
INDEX ON zval TAG ZVal COLLATE "Machine"
SET ORDER TO 0
o = CREATEOBJECT("lanczos")
FOR i = 1 TO 20
x = i/10
INSERT INTO results VALUES (x, o.Gamma(x))
ENDFOR
UPDATE results SET GamVal = ROUND(GamVal, 0) WHERE ZVal = INT(ZVal)
*!* This just creates the output text - it is not part of the algorithm
DO cursor2txt.prg WITH "results", .T.
DEFINE CLASS lanczos As Session
#DEFINE FPF 5.5
#DEFINE HALF 0.5
#DEFINE PY PI()
DIMENSION LanCoeff[7]
nSize = 0
PROCEDURE Init
DODEFAULT()
WITH THIS
.LanCoeff[1] = 1.000000000190015
.LanCoeff[2] = 76.18009172947146
.LanCoeff[3] = -86.50532032941677
.LanCoeff[4] = 24.01409824083091
.LanCoeff[5] = -1.231739572450155
.LanCoeff[6] = 0.0012086509738662
.LanCoeff[7] = -0.000005395239385
.nSize = ALEN(.LanCoeff)
ENDWITH
ENDPROC
FUNCTION Gamma(z)
RETURN EXP(THIS.LogGamma(z))
ENDFUNC
FUNCTION LogGamma(z)
LOCAL a As Double, b As Double, i As Integer, j As Integer, lg As Double
IF z < 0.5
lg = LOG(PY/SIN(PY*z)) - THIS.LogGamma(1 - z)
ELSE
WITH THIS
z = z - 1
b = z + FPF
a = .LanCoeff[1]
FOR i = 2 TO .nSize
j = i - 1
a = a + .LanCoeff[i]/(z + j)
ENDFOR
lg = (LOG(SQRT(2*PY)) + LOG(a) - b) + LOG(b)*(z + HALF)
ENDWITH
ENDIF
RETURN lg
ENDFUNC
ENDDEFINE

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Task/Gamma-function/jq/gamma-function-1.jq Normal file
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def gamma:
[
1.00000000000000000000, 0.57721566490153286061, -0.65587807152025388108, -0.04200263503409523553,
0.16653861138229148950, -0.04219773455554433675, -0.00962197152787697356, 0.00721894324666309954,
-0.00116516759185906511, -0.00021524167411495097, 0.00012805028238811619, -0.00002013485478078824,
-0.00000125049348214267, 0.00000113302723198170, -0.00000020563384169776, 0.00000000611609510448,
0.00000000500200764447, -0.00000000118127457049, 0.00000000010434267117, 0.00000000000778226344,
-0.00000000000369680562, 0.00000000000051003703, -0.00000000000002058326, -0.00000000000000534812,
0.00000000000000122678, -0.00000000000000011813, 0.00000000000000000119, 0.00000000000000000141,
-0.00000000000000000023, 0.00000000000000000002
] as $a
| (. - 1) as $y
| ($a|length) as $n
| reduce range(2; 1+$n) as $an
($a[$n-1]; (. * $y) + $a[$n - $an])
| 1 / . ;

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Task/Gamma-function/jq/gamma-function-2.jq Normal file
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# for reals, but easily extended to complex values
def gamma_by_lanczos:
def pow(x): if x == 0 then 1 elif x == 1 then . else x * log | exp end;
. as $x
| ((1|atan) * 4) as $pi
| if $x < 0.5 then $pi / ((($pi * $x) | sin) * ((1-$x)|gamma_by_lanczos ))
else
[ 0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7] as $p
| ($x - 1) as $x
| ($x + 7.5) as $t
| reduce range(1; $p|length) as $i
($p[0]; . + ($p[$i]/($x + $i) ))
* ((2*$pi) | sqrt) * ($t | pow($x+0.5)) * ((-$t)|exp)
end;

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Task/Gamma-function/jq/gamma-function-3.jq Normal file
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def gamma_by_stirling:
def pow(x): if x == 0 then 1 elif x == 1 then . else x * log | exp end;
((1|atan) * 8) as $twopi
| . as $x
| (($twopi/$x) | sqrt) * ( ($x / (1|exp)) | pow($x));

6
Task/Gamma-function/jq/gamma-function-4.jq Normal file
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def pad(n): tostring | . + (n - length) * " ";
" i: gamma lanczos tgamma",
(range(1;11)
| . / 3.0
| "\(pad(18)): \(gamma|pad(18)) : \(gamma_by_lanczos|pad(18)) : \(tgamma)")

12
Task/Gamma-function/jq/gamma-function-5.jq Normal file
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$ jq -M -r -n -f Gamma_function_Stirling.jq
i: gamma lanczos tgamma
0.3333333333333333: 2.6789385347077483 : 2.6789385347077483 : 2.678938534707748
0.6666666666666666: 1.3541179394264005 : 1.3541179394263998 : 1.3541179394264005
1 : 1 : 0.9999999999999998 : 1
1.3333333333333333: 0.8929795115692493 : 0.8929795115692494 : 0.8929795115692493
1.6666666666666667: 0.9027452929509336 : 0.9027452929509342 : 0.9027452929509336
2 : 1 : 1.0000000000000002 : 1
2.3333333333333335: 1.190639348758999 : 1.1906393487589995 : 1.190639348758999
2.6666666666666665: 1.5045754882515399 : 1.5045754882515576 : 1.5045754882515558
3 : 1.9999999999939684 : 2.0000000000000013 : 2
3.3333333333333335: 2.778158479338573 : 2.778158480437665 : 2.7781584804376647