Gamma function
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Implement one algorithm (or more) to compute the Gamma () function (in the real field only).
If your language has the function as built-in or you know a library which has it, compare your implementation's results with the results of the built-in/library function.
The Gamma function can be defined as:
This suggests a straightforward (but inefficient) way of computing the through numerical integration.
Better suggested methods:
11l
<lang 11l>V _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
]
F gamma(x)
V y = x - 1.0
V sm = :_a.last
L(n) (:_a.len-2 .. 0).step(-1)
sm = sm * y + :_a[n]
R 1.0 / sm
L(i) 1..10
print(‘#.14’.format(gamma(i / 3.0)))</lang>
- Output:
2.67893853470775 1.35411793942640 1.00000000000000 0.89297951156925 0.90274529295093 1.00000000000000 1.19063934875900 1.50457548825154 1.99999999999397 2.77815847933857
360 Assembly
For maximum compatibility, this program uses only the basic instruction set. <lang 360asm>GAMMAT CSECT
USING GAMMAT,R13
SAVEAR B STM-SAVEAR(R15)
DC 17F'0'
DC CL8'GAMMAT'
STM STM R14,R12,12(R13)
ST R13,4(R15)
ST R15,8(R13)
LR R13,R15
- ---- CODE
LE F4,=E'0'
LH R2,NI
LOOPI EQU *
AE F4,=E'1' xi=xi+1
LER F0,F4
DE F0,=E'10' x=xi/10
STE F0,X
LE F6,X
SE F6,=E'1' xx=x-1.0
LH R4,NT
BCTR R4,0
SLA R4,2
LE F0,T(R4)
STE F0,SUM sum=t(nt)
LH R3,NT
BCTR R3,0
SH R4,=H'4'
LOOPJ CH R3,=H'1' for j=nt-1 downto 1
BL ENDLOOPJ
LE F0,SUM
MER F0,F6 sum*xx
LE F2,T(R4) t(j)
AER F0,F2
STE F0,SUM sum=sum*xx+t(j)
BCTR R3,0
SH R4,=H'4'
B LOOPJ
ENDLOOPJ EQU *
LE F0,=E'1'
DE F0,SUM
STE F0,GAMMA gamma=1/sum
LE F0,X
BAL R14,CONVERT
MVC BUF(8),CONVERTM
LE F0,GAMMA
BAL R14,CONVERT
MVC BUF+9(13),CONVERTM
WTO MF=(E,WTOMSG)
BCT R2,LOOPI
- ---- END CODE
CNOP 0,4
L R13,4(0,R13)
LM R14,R12,12(R13)
XR R15,R15
BR R14
- ---- DATA
NI DC H'30' NT DC AL2((TEND-T)/4) T DC E'1.00000000000000000000'
DC E'0.57721566490153286061'
DC E'-0.65587807152025388108'
DC E'-0.04200263503409523553'
DC E'0.16653861138229148950'
DC E'-0.04219773455554433675'
DC E'-0.00962197152787697356'
DC E'0.00721894324666309954'
DC E'-0.00116516759185906511'
DC E'-0.00021524167411495097'
DC E'0.00012805028238811619'
DC E'-0.00002013485478078824'
DC E'-0.00000125049348214267'
DC E'0.00000113302723198170'
DC E'-0.00000020563384169776'
DC E'0.00000000611609510448'
DC E'0.00000000500200764447'
DC E'-0.00000000118127457049'
DC E'0.00000000010434267117'
DC E'0.00000000000778226344'
DC E'-0.00000000000369680562'
DC E'0.00000000000051003703'
DC E'-0.00000000000002058326'
DC E'-0.00000000000000534812'
DC E'0.00000000000000122678'
DC E'-0.00000000000000011813'
DC E'0.00000000000000000119'
DC E'0.00000000000000000141'
DC E'-0.00000000000000000023'
DC E'0.00000000000000000002'
TEND DS 0E X DS E SUM DS E GAMMA DS E WTOMSG DS 0F
DC AL2(L'BUF),XL2'0000'
BUF DC CL80' '
- Subroutine Convertion Float->Display
CONVERT CNOP 0,4
ME F0,CONVERTC
STE F0,CONVERTF
MVI CONVERTS,X'00'
L R9,CONVERTF
CH R9,=H'0'
BZ CONVERT7
BNL CONVERT1 is negative?
MVI CONVERTS,X'80'
N R9,=X'7FFFFFFF' sign bit
CONVERT1 LR R8,R9
N R8,=X'00FFFFFF'
BNZ CONVERT2
SR R9,R9
B CONVERT7
CONVERT2 LR R8,R9
N R8,=X'FF000000' characteristic
SRL R8,24
CH R8,=H'64'
BH CONVERT3
SR R9,R9
B CONVERT7
CONVERT3 CH R8,=H'72' 2**32
BNH CONVERT4
L R9,=X'7FFFFFFF' biggest
B CONVERT6
CONVERT4 SR R8,R8
SLDL R8,8
CH R8,=H'72'
BL CONVERT5
CH R9,=H'0'
BP CONVERT5
L R9,=X'7FFFFFFF'
B CONVERT6
CONVERT5 SH R8,=H'72'
LCR R8,R8
SLL R8,2
SRL R9,0(R8)
CONVERT6 SR R8,R8
IC R8,CONVERTS
CH R8,=H'0' sign bit set?
BZ CONVERT7
LCR R9,R9
CONVERT7 ST R9,CONVERTB
CVD R9,CONVERTP
MVC CONVERTD,=X'402020202120202020202020'
ED CONVERTD,CONVERTP+2
MVC CONVERTM(6),CONVERTD
MVI CONVERTM+6,C'.'
MVC CONVERTM+7(6),CONVERTD+6
BR R14
CONVERTC DC E'1E6' X'45F42400' CONVERTF DS F CONVERTB DS F CONVERTS DS X CONVERTM DS CL13 CONVERTD DS CL12 CONVERTP DS PL8
EQUREGS
EQUREGS REGS=FPR
END GAMMAT</lang>
- Output:
0.1 9.513504
0.2 4.590844
0.3 2.991569
0.4 2.218160
0.5 1.772453
0.6 1.489192
0.7 1.298056
0.8 1.164229
0.9 1.068628
1.0 1.000000
1.1 0.951350
1.2 0.918168
1.3 0.897470
1.4 0.887263
1.5 0.886227
1.6 0.893515
1.7 0.908638
1.8 0.931383
1.9 0.961766
2.0 1.000000
2.1 1.046486
2.2 1.101803
2.3 1.166712
2.4 1.242169
2.5 1.329341
2.6 1.429626
2.7 1.544686
2.8 1.676492
2.9 1.827354
3.0 1.999999
Ada
The implementation uses Taylor series coefficients of Γ(x+1)-1, |x| < ∞. The coefficients are taken from Mathematical functions and their approximations by Yudell L. Luke. <lang ada>function Gamma (X : Long_Float) return Long_Float is
A : constant array (0..29) of Long_Float :=
( 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
);
Y : constant Long_Float := X - 1.0;
Sum : Long_Float := A (A'Last);
begin
for N in reverse A'First..A'Last - 1 loop
Sum := Sum * Y + A (N);
end loop;
return 1.0 / Sum;
end Gamma;</lang> Test program: <lang ada>with Ada.Text_IO; use Ada.Text_IO; with Gamma;
procedure Test_Gamma is begin
for I in 1..10 loop
Put_Line (Long_Float'Image (Gamma (Long_Float (I) / 3.0)));
end loop;
end Test_Gamma;</lang>
- Output:
2.67893853470775E+00 1.35411793942640E+00 1.00000000000000E+00 8.92979511569249E-01 9.02745292950934E-01 1.00000000000000E+00 1.19063934875900E+00 1.50457548825154E+00 1.99999999999397E+00 2.77815847933858E+00
ALGOL 68
- Stirling & Spouge methods.
- Lanczos method.
<lang algol68># Coefficients used by the GNU Scientific Library # []LONG REAL p = ( LONG 0.99999 99999 99809 93,
LONG 676.52036 81218 851,
-LONG 1259.13921 67224 028,
LONG 771.32342 87776 5313,
-LONG 176.61502 91621 4059,
LONG 12.50734 32786 86905,
-LONG 0.13857 10952 65720 12,
LONG 9.98436 95780 19571 6e-6,
LONG 1.50563 27351 49311 6e-7);
PROC lanczos gamma = (LONG REAL in z)LONG REAL: (
# Reflection formula # LONG REAL z := in z; IF z < LONG 0.5 THEN long pi / (long sin(long pi*z)*lanczos gamma(1-z)) ELSE z -:= 1; LONG REAL x := p[1]; FOR i TO UPB p - 1 DO x +:= p[i+1]/(z+i) OD; LONG REAL t = z + UPB p - LONG 1.5; long sqrt(2*long pi) * t**(z+LONG 0.5) * long exp(-t) * x FI
);
PROC taylor gamma = (LONG REAL x)LONG REAL: BEGIN # good for values between 0 and 1 #
[]LONG REAL a =
( LONG 1.00000 00000 00000 00000,
LONG 0.57721 56649 01532 86061,
-LONG 0.65587 80715 20253 88108,
-LONG 0.04200 26350 34095 23553,
LONG 0.16653 86113 82291 48950,
-LONG 0.04219 77345 55544 33675,
-LONG 0.00962 19715 27876 97356,
LONG 0.00721 89432 46663 09954,
-LONG 0.00116 51675 91859 06511,
-LONG 0.00021 52416 74114 95097,
LONG 0.00012 80502 82388 11619,
-LONG 0.00002 01348 54780 78824,
-LONG 0.00000 12504 93482 14267,
LONG 0.00000 11330 27231 98170,
-LONG 0.00000 02056 33841 69776,
LONG 0.00000 00061 16095 10448,
LONG 0.00000 00050 02007 64447,
-LONG 0.00000 00011 81274 57049,
LONG 0.00000 00001 04342 67117,
LONG 0.00000 00000 07782 26344,
-LONG 0.00000 00000 03696 80562,
LONG 0.00000 00000 00510 03703,
-LONG 0.00000 00000 00020 58326,
-LONG 0.00000 00000 00005 34812,
LONG 0.00000 00000 00001 22678,
-LONG 0.00000 00000 00000 11813,
LONG 0.00000 00000 00000 00119,
LONG 0.00000 00000 00000 00141,
-LONG 0.00000 00000 00000 00023,
LONG 0.00000 00000 00000 00002
);
LONG REAL y = x - 1;
LONG REAL sum := a [UPB a];
FOR n FROM UPB a - 1 DOWNTO LWB a DO
sum := sum * y + a [n]
OD;
1/sum
END # taylor gamma #;
LONG REAL long e = long exp(1);
PROC sterling gamma = (LONG REAL n)LONG REAL: ( # improves for values much greater then 1 #
long sqrt(2*long pi/n)*(n/long e)**n
);
PROC factorial = (LONG INT n)LONG REAL: (
IF n=0 OR n=1 THEN 1 ELIF n=2 THEN 2 ELSE n*factorial(n-1) FI
);
REF[]LONG REAL fm := NIL;
PROC sponge gamma = (LONG REAL x)LONG REAL: (
INT a = 12; # alter to get required precision #
REF []LONG REAL fm := NIL;
LONG REAL res;
IF fm :=: REF[]LONG REAL(NIL) THEN
fm := HEAP[0:a-1]LONG REAL;
fm[0] := long sqrt(LONG 2*long pi);
FOR k TO a-1 DO
fm[k] := (((k-1) MOD 2=0) | 1 | -1) * long exp(a-k) *
(a-k) **(k-LONG 0.5) / factorial(k-1)
OD FI; res := fm[0]; FOR k TO a-1 DO res +:= fm[k] / ( x + k ) OD; res *:= long exp(-(x+a)) * (x+a)**(x + LONG 0.5); res/x
);
FORMAT real fmt = $g(-real width, real width - 2)$; FORMAT long real fmt16 = $g(-17, 17 - 2)$; # accurate to about 16 decimal places #
[]STRING methods = ("Genie", "Lanczos", "Sponge", "Taylor","Stirling");
printf(($11xg12xg12xg13xg13xgl$,methods));
FORMAT sample fmt = $"gamma("g(-3,1)")="f(real fmt)n(UPB methods-1)(", "f(long real fmt16))l$; FORMAT sqr sample fmt = $"gamma("g(-3,1)")**2="f(real fmt)n(UPB methods-1)(", "f(long real fmt16))l$; FORMAT sample exp fmt = $"gamma("g(-3)")="g(-15,11,0)n(UPB methods-1)(","g(-18,14,0))l$;
PROC sample = (LONG REAL x)[]LONG REAL:
(gamma(SHORTEN x), lanczos gamma(x), sponge gamma(x), taylor gamma(x), sterling gamma(x));
FOR i FROM 1 TO 20 DO
LONG REAL x = i / LONG 10;
printf((sample fmt, x, sample(x)));
IF i = 5 THEN # insert special case of a half #
printf((sqr sample fmt,
x, gamma(SHORTEN x)**2, lanczos gamma(x)**2, sponge gamma(x)**2,
taylor gamma(x)**2, sterling gamma(x)**2))
FI
OD; FOR x FROM 10 BY 10 TO 70 DO
printf((sample exp fmt, x, sample(x)))
OD</lang>
- Output:
Genie Lanczos Sponge Taylor Stirling gamma(0.1)=9.5135076986687, 9.513507698668730, 9.513507698668731, 9.513509522249043, 5.697187148977169 gamma(0.2)=4.5908437119988, 4.590843711998802, 4.590843711998803, 4.590843743037192, 3.325998424022393 gamma(0.3)=2.9915689876876, 2.991568987687590, 2.991568987687590, 2.991568988322729, 2.362530036269620 gamma(0.4)=2.2181595437577, 2.218159543757688, 2.218159543757688, 2.218159543764845, 1.841476335936235 gamma(0.5)=1.7724538509055, 1.772453850905517, 1.772453850905516, 1.772453850905353, 1.520346901066281 gamma(0.5)**2=3.1415926535898, 3.141592653589795, 3.141592653589793, 3.141592653589216, 2.311454699581843 gamma(0.6)=1.4891922488128, 1.489192248812817, 1.489192248812817, 1.489192248812758, 1.307158857448356 gamma(0.7)=1.2980553326476, 1.298055332647558, 1.298055332647558, 1.298055332647558, 1.159053292113920 gamma(0.8)=1.1642297137253, 1.164229713725304, 1.164229713725303, 1.164229713725303, 1.053370968425609 gamma(0.9)=1.0686287021193, 1.068628702119320, 1.068628702119319, 1.068628702119319, 0.977061507877695 gamma(1.0)=1.0000000000000, 1.000000000000000, 1.000000000000000, 1.000000000000000, 0.922137008895789 gamma(1.1)=0.9513507698669, 0.951350769866873, 0.951350769866873, 0.951350769866873, 0.883489953168704 gamma(1.2)=0.9181687423998, 0.918168742399761, 0.918168742399760, 0.918168742399761, 0.857755335396591 gamma(1.3)=0.8974706963063, 0.897470696306277, 0.897470696306277, 0.897470696306277, 0.842678259448392 gamma(1.4)=0.8872638175031, 0.887263817503076, 0.887263817503075, 0.887263817503064, 0.836744548637082 gamma(1.5)=0.8862269254528, 0.886226925452758, 0.886226925452758, 0.886226925452919, 0.838956552526496 gamma(1.6)=0.8935153492877, 0.893515349287691, 0.893515349287690, 0.893515349288799, 0.848693242152574 gamma(1.7)=0.9086387328533, 0.908638732853291, 0.908638732853290, 0.908638732822421, 0.865621471793884 gamma(1.8)=0.9313837709802, 0.931383770980243, 0.931383770980242, 0.931383769950169, 0.889639635287994 gamma(1.9)=0.9617658319074, 0.961765831907388, 0.961765831907387, 0.961765815012982, 0.920842721894229 gamma(2.0)=1.0000000000000, 1.000000000000000, 0.999999999999999, 1.000000010045742, 0.959502175744492 gamma( 10)= 3.6288000000e5, 3.6288000000000e5, 3.6288000000000e5, 4.051218760300e-7, 3.5986956187410e5 gamma( 20)= 1.216451004e17, 1.216451004088e17, 1.216451004088e17, 1.07701514977e-18, 1.211393423381e17 gamma( 30)= 8.841761994e30, 8.841761993740e30, 8.841761993739e30, 7.98891286318e-23, 8.817236530765e30 gamma( 40)= 2.039788208e46, 2.039788208120e46, 2.039788208120e46, 6.97946184592e-25, 2.035543161237e46 gamma( 50)= 6.082818640e62, 6.082818640343e62, 6.082818640342e62, 1.81016585713e-26, 6.072689187876e62 gamma( 60)= 1.386831185e80, 1.386831185457e80, 1.386831185457e80, 9.27306839649e-28, 1.384906385829e80 gamma( 70)= 1.711224524e98, 1.711224524281e98, 1.711224524281e98, 7.57303907062e-29, 1.709188578191e98
ANSI Standard BASIC
- Lanczos method.
<lang ANSI Standard BASIC>100 DECLARE EXTERNAL FUNCTION FNlngamma 110 120 DEF FNgamma(z) = EXP(FNlngamma(z)) 130 140 FOR x = 0.1 TO 2.05 STEP 0.1 150 PRINT USING$("#.#",x), USING$("##.############", FNgamma(x)) 160 NEXT x 170 END 180 190 EXTERNAL FUNCTION FNlngamma(z) 200 DIM lz(0 TO 6) 210 RESTORE 220 MAT READ lz 230 DATA 1.000000000190015, 76.18009172947146, -86.50532032941677, 24.01409824083091, -1.231739572450155, 0.0012086509738662, -0.000005395239385 240 IF z < 0.5 THEN 250 LET FNlngamma = LOG(PI / SIN(PI * z)) - FNlngamma(1.0 - z) 260 EXIT FUNCTION 270 END IF 280 LET z = z - 1.0 290 LET b = z + 5.5 300 LET a = lz(0) 310 FOR i = 1 TO 6 320 LET a = a + lz(i) / (z + i) 330 NEXT i 340 LET FNlngamma = (LOG(SQR(2*PI)) + LOG(a) - b) + LOG(b) * (z+0.5) 350 END FUNCTION</lang>
AutoHotkey
Search autohotkey.com: function
Source: AutoHotkey forum by Laszlo
<lang autohotkey>/*
Here is the upper incomplete Gamma function. Omitting or setting
the second parameter to 0 we get the (complete) Gamma function.
The code is based on: "Computation of Special Functions" Zhang and Jin,
John Wiley and Sons, 1996
- /
SetFormat FloatFast, 0.9e
Loop 10
MsgBox % GAMMA(A_Index/3) "`n" GAMMA(A_Index*10)
GAMMA(a,x=0) { ; upper incomplete gamma: Integral(t**(a-1)*e**-t, t = x..inf)
If (a > 171 || x < 0)
Return 2.e308 ; overflow
xam := x > 0 ? -x+a*ln(x) : 0
If (xam > 700)
Return 2.e308 ; overflow
If (x > 1+a) { ; no need for gamma(a)
t0 := 0, k := 60
Loop 60
t0 := (k-a)/(1+k/(x+t0)), --k
Return exp(xam) / (x+t0)
}
r := 1, ga := 1.0 ; compute ga = gamma(a) ...
If (a = round(a)) ; if integer: factorial
If (a > 0)
Loop % a-1
ga *= A_Index
Else ; negative integer
ga := 1.7976931348623157e+308 ; Dmax
Else { ; not integer
If (abs(a) > 1) {
z := abs(a)
m := floor(z)
Loop %m%
r *= (z-A_Index)
z -= m
}
Else
z := a
gr := ((((((((((((((((((((((( 0.14e-14
*z - 0.54e-14) *z - 0.206e-13) *z + 0.51e-12)
*z - 0.36968e-11) *z + 0.77823e-11) *z + 0.1043427e-9)
*z - 0.11812746e-8) *z + 0.50020075e-8) *z + 0.6116095e-8)
*z - 0.2056338417e-6) *z + 0.1133027232e-5) *z - 0.12504934821e-5)
*z - 0.201348547807e-4) *z + 0.1280502823882e-3) *z - 0.2152416741149e-3)
*z - 0.11651675918591e-2) *z + 0.7218943246663e-2) *z - 0.9621971527877e-2)
*z - 0.421977345555443e-1) *z + 0.1665386113822915) *z - 0.420026350340952e-1)
*z - 0.6558780715202538) *z + 0.5772156649015329) *z + 1
ga := 1.0/(gr*z) * r
If (a < -1)
ga := -3.1415926535897931/(a*ga*sin(3.1415926535897931*a))
}
If (x = 0) ; complete gamma requested
Return ga
s := 1/a ; here x <= 1+a
r := s
Loop 60 {
r *= x/(a+A_Index)
s += r
If (abs(r/s) < 1.e-15)
break
}
Return ga - exp(xam)*s
}
/* The 10 results shown: 2.678938535e+000 1.354117939e+000 1.0 8.929795115e-001 9.027452930e-001 3.628800000e+005 1.216451004e+017 8.841761994e+030 2.039788208e+046 6.082818640e+062
1.000000000e+000 1.190639348e+000 1.504575489e+000 2.000000000e+000 2.778158479e+000 1.386831185e+080 1.711224524e+098 8.946182131e+116 1.650795516e+136 9.332621544e+155
- /</lang>
AWK
<lang AWK>
- syntax: GAWK -f GAMMA_FUNCTION.AWK
BEGIN {
e = (1+1/100000)^100000
pi = atan2(0,-1)
leng = split("0.99999999999980993,676.5203681218851,-1259.1392167224028,771.32342877765313,-176.61502916214059,12.507343278686905,-0.13857109526572012,9.9843695780195716e-6,1.5056327351493116e-7",p,",")
print("X Stirling")
for (i=1; i<=20; i++) {
d = i / 10
printf("%4.2f %9.5f\n",d,gamma_stirling(d))
}
exit(0)
} function gamma_stirling(x) {
return sqrt(2*pi/x) * pow(x/e,x)
} function pow(a,b) {
return exp(b*log(a))
} </lang>
- Output:
X Stirling 0.10 5.69719 0.20 3.32600 0.30 2.36253 0.40 1.84148 0.50 1.52035 0.60 1.30716 0.70 1.15906 0.80 1.05338 0.90 0.97707 1.00 0.92214 1.10 0.88349 1.20 0.85776 1.30 0.84268 1.40 0.83675 1.50 0.83896 1.60 0.84870 1.70 0.86563 1.80 0.88965 1.90 0.92085 2.00 0.95951
BBC BASIC
Uses the Lanczos approximation. <lang bbcbasic> *FLOAT64
INSTALL @lib$+"FNUSING"
FOR x = 0.1 TO 2.05 STEP 0.1
PRINT FNusing("#.#",x), FNusing("##.############", FNgamma(x))
NEXT
END
DEF FNgamma(z) = EXP(FNlngamma(z))
DEF FNlngamma(z)
LOCAL a, b, i%, lz()
DIM lz(6)
lz() = 1.000000000190015, 76.18009172947146, -86.50532032941677, \
\ 24.01409824083091, -1.231739572450155, 0.0012086509738662, -0.000005395239385
IF z < 0.5 THEN = LN(PI / SIN(PI * z)) - FNlngamma(1.0 - z)
z -= 1.0
b = z + 5.5
a = lz(0)
FOR i% = 1 TO 6
a += lz(i%) / (z + i%)
NEXT
= (LNSQR(2*PI) + LN(a) - b) + LN(b) * (z+0.5)</lang>
Output:
0.1 9.513507698670 0.2 4.590843712000 0.3 2.991568987689 0.4 2.218159543760 0.5 1.772453850902 0.6 1.489192248811 0.7 1.298055332647 0.8 1.164229713725 0.9 1.068628702119 1.0 1.000000000000 1.1 0.951350769867 1.2 0.918168742400 1.3 0.897470696306 1.4 0.887263817503 1.5 0.886226925453 1.6 0.893515349288 1.7 0.908638732853 1.8 0.931383770980 1.9 0.961765831907 2.0 1.000000000000
C
This implements Stirling's approximation and Spouge's approximation.
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
- include <gsl/gsl_sf_gamma.h>
- ifndef M_PI
- define M_PI 3.14159265358979323846
- endif
/* very simple approximation */ double st_gamma(double x) {
return sqrt(2.0*M_PI/x)*pow(x/M_E, x);
}
- define A 12
double sp_gamma(double z) {
const int a = A;
static double c_space[A];
static double *c = NULL;
int k;
double accm;
if ( c == NULL ) {
double k1_factrl = 1.0; /* (k - 1)!*(-1)^k with 0!==1*/
c = c_space;
c[0] = sqrt(2.0*M_PI);
for(k=1; k < a; k++) {
c[k] = exp(a-k) * pow(a-k, k-0.5) / k1_factrl;
k1_factrl *= -k;
}
}
accm = c[0];
for(k=1; k < a; k++) {
accm += c[k] / ( z + k );
}
accm *= exp(-(z+a)) * pow(z+a, z+0.5); /* Gamma(z+1) */
return accm/z;
}
int main() {
double x;
printf("%15s%15s%15s%15s\n", "Stirling", "Spouge", "GSL", "libm");
for(x=1.0; x <= 10.0; x+=1.0) {
printf("%15.8lf%15.8lf%15.8lf%15.8lf\n", st_gamma(x/3.0), sp_gamma(x/3.0),
gsl_sf_gamma(x/3.0), tgamma(x/3.0));
} return 0;
}</lang>
- Output:
Stirling Spouge GSL libm
2.15697602 2.67893853 2.67893853 2.67893853
1.20285073 1.35411794 1.35411794 1.35411794
0.92213701 1.00000000 1.00000000 1.00000000
0.83974270 0.89297951 0.89297951 0.89297951
0.85919025 0.90274529 0.90274529 0.90274529
0.95950218 1.00000000 1.00000000 1.00000000
1.14910642 1.19063935 1.19063935 1.19063935
1.45849038 1.50457549 1.50457549 1.50457549
1.94540320 2.00000000 2.00000000 2.00000000
2.70976382 2.77815848 2.77815848 2.77815848
C#
This is just rewritten from the Wikipedia Lanczos article. Works with complex numbers as well as reals. <lang csharp>using System; using System.Numerics;
static int g = 7; static double[] p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7};
Complex Gamma(Complex z) {
// Reflection formula if (z.Real < 0.5)
{
return Math.PI / (Complex.Sin( Math.PI * z) * Gamma(1 - z));
}
else
{
z -= 1;
Complex x = p[0];
for (var i = 1; i < g + 2; i++)
{
x += p[i]/(z+i);
}
Complex t = z + g + 0.5;
return Complex.Sqrt(2 * Math.PI) * (Complex.Pow(t, z + 0.5)) * Complex.Exp(-t) * x;
} } </lang>
Clojure
<lang clojure>(defn gamma
"Returns Gamma(z + 1 = number) using Lanczos approximation."
[number]
(if (< number 0.5)
(/ Math/PI (* (Math/sin (* Math/PI number))
(gamma (- 1 number))))
(let [n (dec number)
c [0.99999999999980993 676.5203681218851 -1259.1392167224028
771.32342877765313 -176.61502916214059 12.507343278686905 -0.13857109526572012 9.9843695780195716e-6 1.5056327351493116e-7]]
(* (Math/sqrt (* 2 Math/PI))
(Math/pow (+ n 7 0.5) (+ n 0.5)) (Math/exp (- (+ n 7 0.5)))
(+ (first c)
(apply + (map-indexed #(/ %2 (+ n %1 1)) (next c))))))))</lang>
- Output:
<lang clojure>(map #(printf "%.1f %.4f\n" % (gamma %)) (map #(float (/ % 10)) (range 1 31)))</lang>
0.1 9.5135 0.2 4.5908 0.3 2.9916 0.4 2.2182 0.5 1.7725 0.6 1.4892 0.7 1.2981 0.8 1.1642 0.9 1.0686 1.0 1.0000 1.1 0.9514 1.2 0.9182 1.3 0.8975 1.4 0.8873 1.5 0.8862 1.6 0.8935 1.7 0.9086 1.8 0.9314 1.9 0.9618 2.0 1.0000 2.1 1.0465 2.2 1.1018 2.3 1.1667 2.4 1.2422 2.5 1.3293 2.6 1.4296 2.7 1.5447 2.8 1.6765 2.9 1.8274 3.0 2.0000
Common Lisp
<lang lisp>; Taylor series coefficients (defconstant tcoeff
'( 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))
- number of coefficients
(defconstant numcoeff (length tcoeff))
(defun gamma (x)
(let ((y (- x 1.0))
(sum (nth (- numcoeff 1) tcoeff)))
(loop for i from (- numcoeff 2) downto 0 do
(setf sum (+ (* sum y) (nth i tcoeff))))
(/ 1.0 sum)))
(loop for i from 1 to 10
do (
format t "~12,10f~%" (gamma (/ i 3.0))))</lang>
- Produces:
2.6789380000 1.3541179000 1.0000000000 0.8929794500 0.9027453000 1.0000000000 1.1906393000 1.5045753000 1.9999995000 2.7781580000
Crystal
Taylor Series | Lanczos Method | Builtin Function
<lang ruby># Taylor Series def 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 ]
end
def taylor_gamma(x)
y = x.to_f - 1
1.0 / a.reverse.reduce(0) { |sum, an| sum * y + an }
end
- Lanczos Method
def p
[ 0.99999_99999_99809_93, 676.52036_81218_851, -1259.13921_67224_028, 771.32342_87776_5313, -176.61502_91621_4059, 12.50734_32786_86905, -0.13857_10952_65720_12, 9.98436_95780_19571_6e-6, 1.50563_27351_49311_6e-7 ]
end
def lanczos_gamma(z)
# Reflection formula
z = z.to_f
if z < 0.5
Math::PI / (Math.sin(Math::PI * z) * lanczos_gamma(1 - z))
else
z -= 1
x = p[0]
(1..p.size - 1).each { |i| x += p[i] / (z + i) }
t = z + p.size - 1.5
Math.sqrt(2 * Math::PI) * t**(z + 0.5) * Math.exp(-t) * x
end
end
puts " Taylor Series Lanczos Method Builtin Function" (1..27).each { |i| n = i/3.0; puts "gamma(%.2f) = %.14e %.14e %.14e" % [n, taylor_gamma(n), lanczos_gamma(n), Math.gamma(n)] } </lang>
- Output:
Taylor Series Lanczos Method Builtin Function
gamma(0.33) = 2.67893853470775e+00 2.67893853470775e+00 2.67893853470775e+00
gamma(0.67) = 1.35411793942640e+00 1.35411793942640e+00 1.35411793942640e+00
gamma(1.00) = 1.00000000000000e+00 1.00000000000000e+00 1.00000000000000e+00
gamma(1.33) = 8.92979511569249e-01 8.92979511569249e-01 8.92979511569249e-01
gamma(1.67) = 9.02745292950934e-01 9.02745292950935e-01 9.02745292950934e-01
gamma(2.00) = 1.00000000000000e+00 1.00000000000000e+00 1.00000000000000e+00
gamma(2.33) = 1.19063934875900e+00 1.19063934875900e+00 1.19063934875900e+00
gamma(2.67) = 1.50457548825154e+00 1.50457548825156e+00 1.50457548825156e+00
gamma(3.00) = 1.99999999999397e+00 2.00000000000000e+00 2.00000000000000e+00
gamma(3.33) = 2.77815847933857e+00 2.77815848043767e+00 2.77815848043766e+00
gamma(3.67) = 4.01220118377482e+00 4.01220130200415e+00 4.01220130200415e+00
gamma(4.00) = 5.99999141007240e+00 6.00000000000001e+00 6.00000000000000e+00
gamma(4.33) = 9.26006653812473e+00 9.26052826812555e+00 9.26052826812554e+00
gamma(4.67) = 1.46918499266721e+01 1.47114047740152e+01 1.47114047740152e+01
gamma(5.00) = 2.33327665969918e+01 2.40000000000000e+01 2.40000000000000e+01
gamma(5.33) = 2.65211050660964e+01 4.01289558285441e+01 4.01289558285440e+01
gamma(5.67) = 7.70471336505311e+00 6.86532222787379e+01 6.86532222787377e+01
gamma(6.00) = 1.10934146590517e+00 1.20000000000000e+02 1.20000000000000e+02
gamma(6.33) = 1.64621072447163e-01 2.14021097752236e+02 2.14021097752235e+02
gamma(6.67) = 2.72102446536397e-02 3.89034926246181e+02 3.89034926246180e+02
gamma(7.00) = 4.98014348954507e-03 7.20000000000002e+02 7.20000000000000e+02
gamma(7.33) = 9.98845907123850e-04 1.35546695243082e+03 1.35546695243082e+03
gamma(7.67) = 2.17513475446479e-04 2.59356617497454e+03 2.59356617497454e+03
gamma(8.00) = 5.10217006678528e-05 5.04000000000001e+03 5.04000000000000e+03
gamma(8.33) = 1.28035516395359e-05 9.94009098449271e+03 9.94009098449270e+03
gamma(8.67) = 3.41689149138074e-06 1.98840073414715e+04 1.98840073414714e+04
gamma(9.00) = 9.64721467591131e-07 4.03200000000001e+04 4.03200000000000e+04
D
<lang d>import std.stdio, std.math, std.mathspecial;
real taylorGamma(in real x) pure nothrow @safe @nogc {
static immutable real[30] table = [
0x1p+0, 0x1.2788cfc6fb618f4cp-1,
-0x1.4fcf4026afa2dcecp-1, -0x1.5815e8fa27047c8cp-5,
0x1.5512320b43fbe5dep-3, -0x1.59af103c340927bep-5,
-0x1.3b4af28483e214e4p-7, 0x1.d919c527f60b195ap-8,
-0x1.317112ce3a2a7bd2p-10, -0x1.c364fe6f1563ce9cp-13,
0x1.0c8a78cd9f9d1a78p-13, -0x1.51ce8af47eabdfdcp-16,
-0x1.4fad41fc34fbb2p-20, 0x1.302509dbc0de2c82p-20,
-0x1.b9986666c225d1d4p-23, 0x1.a44b7ba22d628acap-28,
0x1.57bc3fc384333fb2p-28, -0x1.44b4cedca388f7c6p-30,
0x1.cae7675c18606c6p-34, 0x1.11d065bfaf06745ap-37,
-0x1.0423bac8ca3faaa4p-38, 0x1.1f20151323cd0392p-41,
-0x1.72cb88ea5ae6e778p-46, -0x1.815f72a05f16f348p-48,
0x1.6198491a83bccbep-50, -0x1.10613dde57a88bd6p-53,
0x1.5e3fee81de0e9c84p-60, 0x1.a0dc770fb8a499b6p-60,
-0x1.0f635344a29e9f8ep-62, 0x1.43d79a4b90ce8044p-66];
immutable real y = x - 1.0L;
real sm = table[$ - 1];
foreach_reverse (immutable an; table[0 .. $ - 1])
sm = sm * y + an;
return 1.0L / sm;
}
real lanczosGamma(real z) pure nothrow @safe @nogc {
// Coefficients used by the GNU Scientific Library. // http://en.wikipedia.org/wiki/Lanczos_approximation enum g = 7; static immutable real[9] table = [ 0.99999_99999_99809_93, 676.52036_81218_851, -1259.13921_67224_028, 771.32342_87776_5313, -176.61502_91621_4059, 12.50734_32786_86905, -0.13857_10952_65720_12, 9.98436_95780_19571_6e-6, 1.50563_27351_49311_6e-7];
// Reflection formula.
if (z < 0.5L) {
return PI / (sin(PI * z) * lanczosGamma(1 - z));
} else {
z -= 1;
real x = table[0];
foreach (immutable i; 1 .. g + 2)
x += table[i] / (z + i);
immutable real t = z + g + 0.5L;
return sqrt(2 * PI) * t ^^ (z + 0.5L) * exp(-t) * x;
}
}
void main() {
foreach (immutable i; 1 .. 11) {
immutable real x = i / 3.0L;
writefln("%f: %20.19e %20.19e %20.19e", x,
x.taylorGamma, x.lanczosGamma, x.gamma);
}
}</lang>
- Output:
0.333333: 2.6789385347077476335e+00 2.6789385347077470551e+00 2.6789385347077476339e+00 0.666667: 1.3541179394264004169e+00 1.3541179394264007092e+00 1.3541179394264004170e+00 1.000000: 1.0000000000000000000e+00 1.0000000000000002126e+00 1.0000000000000000000e+00 1.333333: 8.9297951156924921124e-01 8.9297951156924947465e-01 8.9297951156924921132e-01 1.666667: 9.0274529295093361132e-01 9.0274529295093396555e-01 9.0274529295093361123e-01 2.000000: 1.0000000000000000000e+00 1.0000000000000004903e+00 1.0000000000000000000e+00 2.333333: 1.1906393487589989474e+00 1.1906393487589996490e+00 1.1906393487589989482e+00 2.666667: 1.5045754882515545787e+00 1.5045754882515570474e+00 1.5045754882515560190e+00 3.000000: 1.9999999999992207405e+00 2.0000000000000015575e+00 2.0000000000000000000e+00 3.333333: 2.7781584802531739378e+00 2.7781584804376666336e+00 2.7781584804376642124e+00
Elixir
<lang elixir>defmodule Gamma do
@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 ]
|> Enum.reverse
def taylor(x) do
y = x - 1
1 / Enum.reduce(@a, 0, fn a,sum -> sum * y + a end)
end
end
Enum.each(Enum.map(1..10, &(&1/3)), fn x ->
:io.format "~f ~18.15f~n", [x, Gamma.taylor(x)]
end)</lang>
- Output:
0.333333 2.678938534707748 0.666667 1.354117939426401 1.000000 1.000000000000000 1.333333 0.892979511569249 1.666667 0.902745292950934 2.000000 1.000000000000000 2.333333 1.190639348758999 2.666667 1.504575488251540 3.000000 1.999999999993968 3.333333 2.778158479338573
F#
Solved using the Lanczos Coefficients described in Numerical Recipes (Press et al)
<lang F Sharp>
open System
let gamma z =
let lanczosCoefficients = [76.18009172947146;-86.50532032941677;24.01409824083091;-1.231739572450155;0.1208650973866179e-2;-0.5395239384953e-5]
let rec sumCoefficients acc i coefficients =
match coefficients with
| [] -> acc
| h::t -> sumCoefficients (acc + (h/i)) (i+1.0) t
let gamma = 5.0
let x = z - 1.0
Math.Pow(x + gamma + 0.5, x + 0.5) * Math.Exp( -(x + gamma + 0.5) ) * Math.Sqrt( 2.0 * Math.PI ) * sumCoefficients 1.000000000190015 (x + 1.0) lanczosCoefficients
seq { for i in 1 .. 20 do yield ((double)i/10.0) } |> Seq.iter ( fun v -> System.Console.WriteLine("{0} : {1}", v, gamma v ) ) seq { for i in 1 .. 10 do yield ((double)i*10.0) } |> Seq.iter ( fun v -> System.Console.WriteLine("{0} : {1}", v, gamma v ) )
</lang>
- Output:
0.1 : 9.51350769855015 0.2 : 4.59084371196153 0.3 : 2.99156898767207 0.4 : 2.21815954375051 0.5 : 1.77245385090205 0.6 : 1.48919224881114 0.7 : 1.29805533264677 0.8 : 1.16422971372497 0.9 : 1.06862870211921 1 : 1 1.1 : 0.951350769866919 1.2 : 0.91816874239982 1.3 : 0.897470696306335 1.4 : 0.887263817503124 1.5 : 0.886226925452797 1.6 : 0.893515349287718 1.7 : 0.908638732853309 1.8 : 0.931383770980253 1.9 : 0.961765831907391 2 : 1 10 : 362880.000000085 20 : 1.21645100409886E+17 30 : 8.84176199395902E+30 40 : 2.03978820820436E+46 50 : 6.08281864068541E+62 60 : 1.38683118555266E+80 70 : 1.71122452441801E+98 80 : 8.94618213157899E+116 90 : 1.65079551625067E+136 100 : 9.33262154536104E+155
Factor
<lang factor>! built in USING: picomath prettyprint ; 0.1 gamma . ! 9.513507698668723 2.0 gamma . ! 1.0 10. gamma . ! 362880.0</lang>
Forth
Cristinel Mortici describes this method in Applied Mathematics Letters. "A substantial improvement of the Stirling formula". This algorithm is said to give about 7 good digits, but becomes more inaccurate close to zero. Therefore, a "shift" is performed to move the value returned into the more accurate domain. <lang forth>8 constant (gamma-shift)
- (mortici) ( f1 -- f2)
-1 s>f f+ 1 s>f fover 271828183e-8 f* 12 s>f f* f/ fover 271828183e-8 f/ f+ fover f** fswap 628318530e-8 f* fsqrt f* \ 2*pi
- gamma ( f1 -- f2)
fdup f0< >r fdup f0= r> or abort" Gamma less or equal to zero" fdup (gamma-shift) s>f f+ (mortici) fswap 1 s>f (gamma-shift) 0 do fover i s>f f+ f* loop fswap fdrop f/
- </lang>
0.1e gamma f. 9.51348888533932 ok 2e gamma f. 0.999999031674546 ok 10e gamma f. 362879.944850072 ok 70e gamma fe. 171.122444600510E96 ok
This is a word, based on a formula of Ramanujan's famous "lost notebook", which was rediscovered in 1976. His formula contained a constant, which had a value between 1/100 and 1/30. In 2008, E.A. Karatsuba described the function, which determines the value of this constant. Since it contains the gamma function itself, it can't be used in a word calculating the gamma function, so here it is emulated by two symmetrical sigmoidals. <lang forth>2 constant (gamma-shift) \ don't change this
\ an approximation of the d(x) function
- ~d(x) ( f1 -- f2)
fdup 10 s>f f< \ use first symmetrical sigmoidal if \ for range 1-10 -2705443e-8 fswap 2280802e-6 f/ 1428045e-6 f** 1 s>f f+ f/ 3187831e-8 f+ else \ use second symmetrical sigmoidal -29372563e-9 fswap 1841693e-6 f/ 1052779e-6 f** 1 s>f f+ f/ 3330828e-8 f+ then 333333333e-10 fover f< if fdrop 1 s>f 30 s>f f/ then
- \ perform some sane clipping to infinity
- (ramanujan) ( f1 -- f2)
fdup fdup f* 4 s>f f* ( n 4n2) fover fover f* fdup f+ f+ fover f+ ( n 8n3+4n2+n) fover ~d(x) f+ ( n 8n3+4n2+n+d[x]) 1 s>f 6 s>f f/ f** ( n 8n3+4n2+n+d[x]^1/6) fswap fdup 2.7182818284590452353e f/ ( 8n3+4n2+n+d[x]^1/6 n n/e) fswap f** f* pi fsqrt f* ( f)
- gamma ( f1 -- f2)
fdup f0< >r fdup f0= r> or abort" Gamma less or equal to zero" fdup (gamma-shift) 1- s>f f+ (ramanujan) fswap 1 s>f (gamma-shift) 0 do fover i s>f f+ f* loop fswap fdrop f/
- </lang>
0.1e gamma f. 9.51351721918848 ok 2e gamma f. 0.999999966026125 ok 10e gamma f. 362879.999559333 ok 70e gamma fe. 171.122452428147E96 ok
Fortran
This code shows two methods: Numerical Integration through Simpson formula, and Lanczos approximation. The results of testing are printed altogether with the values given by the function gamma; this function is defined in the Fortran 2008 standard, and supported by GNU Fortran (and other vendors) as extension; if not present in your compiler, you can remove the last part of the print in order to get it compiled with any Fortran 95 compliant compiler.
<lang fortran>program ComputeGammaInt
implicit none
integer :: i
write(*, "(3A15)") "Simpson", "Lanczos", "Builtin"
do i=1, 10
write(*, "(3F15.8)") my_gamma(i/3.0), lacz_gamma(i/3.0), gamma(i/3.0)
end do
contains
pure function intfuncgamma(x, y) result(z) real :: z real, intent(in) :: x, y z = x**(y-1.0) * exp(-x) end function intfuncgamma
function my_gamma(a) result(g) real :: g real, intent(in) :: a
real, parameter :: small = 1.0e-4 integer, parameter :: points = 100000
real :: infty, dx, p, sp(2, points), x integer :: i logical :: correction
x = a
correction = .false.
! value with x<1 gives \infty, so we use
! \Gamma(x+1) = x\Gamma(x)
! to avoid the problem
if ( x < 1.0 ) then
correction = .true.
x = x + 1
end if
! find a "reasonable" infinity...
! we compute this integral indeed
! \int_0^M dt t^{x-1} e^{-t}
! where M is such that M^{x-1} e^{-M} ≤ \epsilon
infty = 1.0e4
do while ( intfuncgamma(infty, x) > small )
infty = infty * 10.0
end do
! using simpson
dx = infty/real(points)
sp = 0.0
forall(i=1:points/2-1) sp(1, 2*i) = intfuncgamma(2.0*(i)*dx, x)
forall(i=1:points/2) sp(2, 2*i - 1) = intfuncgamma((2.0*(i)-1.0)*dx, x)
g = (intfuncgamma(0.0, x) + 2.0*sum(sp(1,:)) + 4.0*sum(sp(2,:)) + &
intfuncgamma(infty, x))*dx/3.0
if ( correction ) g = g/a
end function my_gamma
recursive function lacz_gamma(a) result(g) real, intent(in) :: a real :: g
real, parameter :: pi = 3.14159265358979324 integer, parameter :: cg = 7
! these precomputed values are taken by the sample code in Wikipedia,
! and the sample itself takes them from the GNU Scientific Library
real, dimension(0:8), parameter :: p = &
(/ 0.99999999999980993, 676.5203681218851, -1259.1392167224028, &
771.32342877765313, -176.61502916214059, 12.507343278686905, &
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 /)
real :: t, w, x integer :: i
x = a
if ( x < 0.5 ) then
g = pi / ( sin(pi*x) * lacz_gamma(1.0-x) )
else
x = x - 1.0
t = p(0)
do i=1, cg+2
t = t + p(i)/(x+real(i))
end do
w = x + real(cg) + 0.5
g = sqrt(2.0*pi) * w**(x+0.5) * exp(-w) * t
end if
end function lacz_gamma
end program ComputeGammaInt</lang>
- Output:
Simpson Lanczos Builtin
2.65968132 2.67893744 2.67893839
1.35269761 1.35411859 1.35411787
1.00000060 1.00000024 1.00000000
0.88656044 0.89297968 0.89297950
0.90179849 0.90274525 0.90274531
0.99999803 1.00000036 1.00000000
1.19070935 1.19063985 1.19063926
1.50460517 1.50457609 1.50457561
2.00000286 2.00000072 2.00000000
2.77815390 2.77816010 2.77815843
FreeBASIC
<lang freebasic>' 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 8
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</lang>
- Output:
x Stirling Lanczos 0.10 5.697187148977170 9.513507698668738 0.20 3.325998424022393 4.590843711998803 0.30 2.362530036269620 2.991568987687590 0.40 1.841476335936235 2.218159543757687 0.50 1.520346901066281 1.772453850905516 0.60 1.307158857448356 1.489192248812818 0.70 1.159053292113920 1.298055332647558 0.80 1.053370968425609 1.164229713725303 0.90 0.977061507877695 1.068628702119319 1.00 0.922137008895789 1.000000000000000 1.10 0.883489953168704 0.951350769866874 1.20 0.857755335396591 0.918168742399761 1.30 0.842678259448392 0.897470696306278 1.40 0.836744548637082 0.887263817503076 1.50 0.838956552526496 0.886226925452759 1.60 0.848693242152574 0.893515349287691 1.70 0.865621471793884 0.908638732853291 1.80 0.889639635287995 0.931383770980243 1.90 0.920842721894229 0.961765831907388 2.00 0.959502175744492 1.000000000000000
Go
<lang go>package main
import (
"fmt" "math"
)
func main() {
fmt.Println(" x math.Gamma Lanczos7")
for _, x := range []float64{-.5, .1, .5, 1, 1.5, 2, 3, 10, 140, 170} {
fmt.Printf("%5.1f %24.16g %24.16g\n", x, math.Gamma(x), lanczos7(x))
}
}
func lanczos7(z float64) float64 {
t := z + 6.5
x := .99999999999980993 +
676.5203681218851/z -
1259.1392167224028/(z+1) +
771.32342877765313/(z+2) -
176.61502916214059/(z+3) +
12.507343278686905/(z+4) -
.13857109526572012/(z+5) +
9.9843695780195716e-6/(z+6) +
1.5056327351493116e-7/(z+7)
return math.Sqrt2 * math.SqrtPi * math.Pow(t, z-.5) * math.Exp(-t) * x
}</lang>
- Output:
x math.Gamma Lanczos7 -0.5 -3.544907701811032 -3.544907701811087 0.1 9.513507698668732 9.513507698668752 0.5 1.772453850905516 1.772453850905517 1.0 1 1 1.5 0.8862269254527579 0.8862269254527587 2.0 1 1 3.0 2 2 10.0 362880 362880.0000000015 140.0 9.61572319694107e+238 9.615723196940201e+238 170.0 4.269068009004746e+304 +Inf
Groovy
<lang groovy>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].reverse()
def gamma = { 1.0 / a.inject(0) { sm, a_i -> sm * (it - 1) + a_i } }
(1..10).each{ printf("% 1.9e\n", gamma(it / 3.0)) } </lang>
- Output:
2.678938535e+00 1.354117939e+00 1.000000000e+00 8.929795116e-01 9.027452930e-01 1.000000000e+00 1.190639349e+00 1.504575488e+00 2.000000000e+00 2.778158479e+00
Haskell
Based on HaskellWiki (compatible license):
- The Gamma and Beta function as described in 'Numerical Recipes in C++', the approximation is taken from [Lanczos, C. 1964 SIAM Journal on Numerical Analysis, ser. B, vol. 1, pp. 86-96]
<lang haskell>cof :: [Double] cof =
[ 76.18009172947146 , -86.50532032941677 , 24.01409824083091 , -1.231739572450155 , 0.001208650973866179 , -0.000005395239384953 ]
ser :: Double ser = 1.000000000190015
gammaln :: Double -> Double gammaln xx =
let tmp_ = (xx + 5.5) - (xx + 0.5) * log (xx + 5.5)
ser_ = ser + sum (zipWith (/) cof [xx + 1 ..])
in -tmp_ + log (2.5066282746310005 * ser_ / xx)
main :: IO () main = mapM_ print $ gammaln <$> [0.1,0.2 .. 1.0]</lang>
Or equivalently, as a point-free applicative expression: <lang haskell>import Control.Applicative
cof :: [Double] cof =
[ 76.18009172947146 , -86.50532032941677 , 24.01409824083091 , -1.231739572450155 , 0.001208650973866179 , -0.000005395239384953 ]
gammaln :: Double -> Double gammaln =
((+) . negate . (((-) . (5.5 +)) <*> (((*) . (0.5 +)) <*> (log . (5.5 +))))) <*> (log . ((/) =<< (2.5066282746310007 *) . (1.000000000190015 +) . sum . zipWith (/) cof . enumFrom . (1 +)))
main :: IO () main = mapM_ print $ gammaln <$> [0.1,0.2 .. 1.0]</lang>
- Output:
2.252712651734255 1.5240638224308496 1.09579799481814 0.7966778177018394 0.572364942924743 0.3982338580692666 0.2608672465316877 0.15205967839984869 6.637623973474716e-2 -4.440892098500626e-16
Icon and Unicon
This works in Unicon. Changing the !10 into (1 to 10) would enable it to work in Icon. <lang unicon>procedure main()
every write(left(i := !10/10.0,5),gamma(.i))
end
procedure gamma(z) # Stirling's approximation
return (2*&pi/z)^0.5 * (z/&e)^z
end</lang>
- Output:
->gamma 0.1 5.69718714897717 0.2 3.325998424022393 0.3 2.36253003626962 0.4 1.841476335936235 0.5 1.520346901066281 0.6 1.307158857448356 0.7 1.15905329211392 0.8 1.053370968425609 0.9 0.9770615078776954 1.0 0.9221370088957891 ->
J
This code shows the built-in method, which works for any value (positive, negative and complex numbers -- but note that negative integer arguments give infinite results).
<lang j>gamma=: !@<:</lang>
Note that <: subtracts one from a number. It's sort of like --lvalue in C, except it always accepts an "rvalue" as an argument (which means it does not modify that argument). And !value finds the factorial of value if value is a positive integer. This illustrates the close relationship between the factorial and gamma functions.
The following direct coding of the task comes from the Stirling's approximation essay on the J wiki:
<lang j>sbase =: %:@(2p1&%) * %&1x1 ^ ]
scorr =: 1 1r12 1r288 _139r51840 _571r2488320&p.@%
stirlg=: sbase * scorr</lang>
Checking against !@<: we can see that this approximation loses accuracy for small arguments
<lang j> (,. stirlg ,. gamma) 10 1p1 1x1 1.5 1
10 362880 362880
3.14159 2.28803 2.28804 2.71828 1.56746 1.56747
1.5 0.886155 0.886227
1 0.999499 1</lang>
(Column 1 is the argument, column 2 is the stirling approximation and column 3 uses the builtin support for gamma.)
Java
Implementation of Stirling's approximation and Lanczos approximation. <lang java>public class GammaFunction {
public double st_gamma(double x){ return Math.sqrt(2*Math.PI/x)*Math.pow((x/Math.E), x); }
public double la_gamma(double x){ double[] p = {0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7}; int g = 7; if(x < 0.5) return Math.PI / (Math.sin(Math.PI * x)*la_gamma(1-x));
x -= 1; double a = p[0]; double t = x+g+0.5; for(int i = 1; i < p.length; i++){ a += p[i]/(x+i); }
return Math.sqrt(2*Math.PI)*Math.pow(t, x+0.5)*Math.exp(-t)*a; }
public static void main(String[] args) { GammaFunction test = new GammaFunction(); System.out.println("Gamma \t\tStirling \t\tLanczos"); for(double i = 1; i <= 20; i += 1){ System.out.println("" + i/10.0 + "\t\t" + test.st_gamma(i/10.0) + "\t" + test.la_gamma(i/10.0)); } } }</lang>
- Output:
Gamma Stirling Lanczos 0.1 5.697187148977169 9.513507698668734 0.2 3.3259984240223925 4.590843711998803 0.3 2.3625300362696198 2.9915689876875904 0.4 1.8414763359362354 2.218159543757687 0.5 1.5203469010662807 1.7724538509055159 0.6 1.307158857448356 1.489192248812818 0.7 1.15905329211392 1.2980553326475577 0.8 1.0533709684256085 1.1642297137253035 0.9 0.9770615078776954 1.0686287021193193 1.0 0.9221370088957891 0.9999999999999998 1.1 0.8834899531687038 0.9513507698668735 1.2 0.8577553353965909 0.9181687423997607 1.3 0.8426782594483921 0.8974706963062777 1.4 0.8367445486370817 0.8872638175030757 1.5 0.8389565525264963 0.8862269254527586 1.6 0.8486932421525738 0.8935153492876909 1.7 0.865621471793884 0.9086387328532916 1.8 0.8896396352879945 0.9313837709802425 1.9 0.9208427218942293 0.9617658319073877 2.0 0.9595021757444916 1.0000000000000002
JavaScript
Implementation of Lanczos approximation. <lang javascript>function gamma(x) {
var p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7
];
var g = 7;
if (x < 0.5) {
return Math.PI / (Math.sin(Math.PI * x) * gamma(1 - x));
}
x -= 1;
var a = p[0];
var t = x + g + 0.5;
for (var i = 1; i < p.length; i++) {
a += p[i] / (x + i);
}
return Math.sqrt(2 * Math.PI) * Math.pow(t, x + 0.5) * Math.exp(-t) * a;
}</lang>
jq
Taylor Series
<lang jq>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 / . ;</lang>
Lanczos Approximation
<lang jq># 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;</lang>
Stirling's Approximation
<lang jq>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));</lang>
Examples
Stirling's method produces poor results, so to save space, the examples contrast the Taylor series and Lanczos methods with built-in tgamma: <lang jq>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)")</lang>
- Output:
<lang sh>$ 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</lang>
Jsish
<lang javascript>#!/usr/bin/env jsish /* Gamma function, in Jsish, using the Lanczos approximation */ function gamma(x) {
var p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7
];
var g = 7;
if (x < 0.5) {
return Math.PI / (Math.sin(Math.PI * x) * gamma(1 - x));
}
x -= 1;
var a = p[0];
var t = x + g + 0.5;
for (var i = 1; i < p.length; i++) {
a += p[i] / (x + i);
}
return Math.sqrt(2 * Math.PI) * Math.pow(t, x + 0.5) * Math.exp(-t) * a;
}
if (Interp.conf('unitTest')) {
for (var i=-5.5; i <= 5.5; i += 0.5) {
printf('%2.1f %+e\n', i, gamma(i));
}
}
/*
!EXPECTSTART!
-5.5 +1.091265e-02 -5.0 -4.275508e+13 -4.5 -6.001960e-02 -4.0 +2.672193e+14 -3.5 +2.700882e-01 -3.0 -1.425169e+15 -2.5 -9.453087e-01 -2.0 +6.413263e+15 -1.5 +2.363272e+00 -1.0 -2.565305e+16 -0.5 -3.544908e+00 0.0 +inf 0.5 +1.772454e+00 1.0 +1.000000e+00 1.5 +8.862269e-01 2.0 +1.000000e+00 2.5 +1.329340e+00 3.0 +2.000000e+00 3.5 +3.323351e+00 4.0 +6.000000e+00 4.5 +1.163173e+01 5.0 +2.400000e+01 5.5 +5.234278e+01
!EXPECTEND!
- /</lang>
- Output:
prompt$ jsish --U gammaFunction.jsi -5.5 +1.091265e-02 -5.0 -4.275508e+13 -4.5 -6.001960e-02 -4.0 +2.672193e+14 -3.5 +2.700882e-01 -3.0 -1.425169e+15 -2.5 -9.453087e-01 -2.0 +6.413263e+15 -1.5 +2.363272e+00 -1.0 -2.565305e+16 -0.5 -3.544908e+00 0.0 +inf 0.5 +1.772454e+00 1.0 +1.000000e+00 1.5 +8.862269e-01 2.0 +1.000000e+00 2.5 +1.329340e+00 3.0 +2.000000e+00 3.5 +3.323351e+00 4.0 +6.000000e+00 4.5 +1.163173e+01 5.0 +2.400000e+01 5.5 +5.234278e+01 prompt$ jsish -u gammaFunction.jsi [PASS] gammaFunction.jsi
Julia
Built-in function: <lang julia>@show gamma(1)</lang>
By adaptive Gauss-Kronrod integration: <lang julia>using QuadGK gammaquad(t::Float64) = first(quadgk(x -> x ^ (t - 1) * exp(-x), zero(t), Inf, reltol = 100eps(t))) @show gammaquad(1.0)</lang>
- Output:
gamma(1) = 1.0 gammaquad(1.0) = 0.9999999999999999
Library function: <lang julia>using SpecialFunctions gamma(1/2) - sqrt(pi)</lang>
- Output:
2.220446049250313e-16
Kotlin
<lang scala>// version 1.0.6
fun gammaStirling(x: Double): Double = Math.sqrt(2.0 * Math.PI / x) * Math.pow(x / Math.E, x)
fun gammaLanczos(x: Double): Double {
var xx = x
val p = doubleArrayOf(
0.99999999999980993,
676.5203681218851,
-1259.1392167224028,
771.32342877765313,
-176.61502916214059,
12.507343278686905,
-0.13857109526572012,
9.9843695780195716e-6,
1.5056327351493116e-7
)
val g = 7
if (xx < 0.5) return Math.PI / (Math.sin(Math.PI * xx) * gammaLanczos(1.0 - xx))
xx--
var a = p[0]
val t = xx + g + 0.5
for (i in 1 until p.size) a += p[i] / (xx + i)
return Math.sqrt(2.0 * Math.PI) * Math.pow(t, xx + 0.5) * Math.exp(-t) * a
}
fun main(args: Array<String>) {
println(" x\tStirling\t\tLanczos\n")
for (i in 1 .. 20) {
val d = i / 10.0
print("%4.2f\t".format(d))
print("%17.15f\t".format(gammaStirling(d)))
println("%17.15f".format(gammaLanczos(d)))
}
}</lang>
- Output:
x Stirling Lanczos 0.10 5.697187148977170 9.513507698668736 0.20 3.325998424022393 4.590843711998803 0.30 2.362530036269620 2.991568987687590 0.40 1.841476335936235 2.218159543757687 0.50 1.520346901066281 1.772453850905516 0.60 1.307158857448356 1.489192248812818 0.70 1.159053292113920 1.298055332647558 0.80 1.053370968425609 1.164229713725304 0.90 0.977061507877695 1.068628702119319 1.00 0.922137008895789 1.000000000000000 1.10 0.883489953168704 0.951350769866874 1.20 0.857755335396591 0.918168742399761 1.30 0.842678259448392 0.897470696306278 1.40 0.836744548637082 0.887263817503076 1.50 0.838956552526496 0.886226925452759 1.60 0.848693242152574 0.893515349287691 1.70 0.865621471793884 0.908638732853292 1.80 0.889639635287995 0.931383770980243 1.90 0.920842721894229 0.961765831907388 2.00 0.959502175744492 1.000000000000000
Limbo
A fairly straightforward port of the Go code. (It could almost have been done with sed). A few small differences are in the use of a tuple as a return value for the builtin gamma function, and we import a few functions from the math library so that we don't have to qualify them.
<lang Limbo>implement Lanczos7;
include "sys.m"; sys: Sys; include "draw.m"; include "math.m"; math: Math; lgamma, exp, pow, sqrt: import math;
Lanczos7: module { init: fn(nil: ref Draw->Context, nil: list of string); };
init(nil: ref Draw->Context, nil: list of string) { sys = load Sys Sys->PATH; math = load Math Math->PATH; # We ignore some floating point exceptions: math->FPcontrol(0, Math->OVFL|Math->UNFL); ns : list of real = -0.5 :: 0.1 :: 0.5 :: 1.0 :: 1.5 :: 2.0 :: 3.0 :: 10.0 :: 140.0 :: 170.0 :: nil;
sys->print("%5s %24s %24s\n", "x", "math->lgamma", "lanczos7"); while(ns != nil) { x := hd ns; ns = tl ns; # math->lgamma returns a tuple. (i, r) := lgamma(x); g := real i * exp(r); sys->print("%5.1f %24.16g %24.16g\n", x, g, lanczos7(x)); } }
lanczos7(z: real): real { t := z + 6.5; x := 0.99999999999980993 + 676.5203681218851/z - 1259.1392167224028/(z+1.0) + 771.32342877765313/(z+2.0) - 176.61502916214059/(z+3.0) + 12.507343278686905/(z+4.0) - 0.13857109526572012/(z+5.0) + 9.9843695780195716e-6/(z+6.0) + 1.5056327351493116e-7/(z+7.0); return sqrt(2.0) * sqrt(Math->Pi) * pow(t, z - 0.5) * exp(-t) * x; } </lang>
- Output:
x math->lgamma lanczos7 -0.5 -3.544907701811032 -3.544907701811089 0.1 9.513507698668729 9.51350769866875 0.5 1.772453850905516 1.772453850905516 1.0 1 0.9999999999999999 1.5 0.8862269254527581 0.8862269254527587 2.0 1 1 3.0 2 2.000000000000001 10.0 362880.0000000005 362880.0000000015 140.0 9.615723196940553e+238 9.615723196940235e+238 170.0 4.269068009004526e+304 Infinity
Lua
Uses the wp:Reciprocal gamma function to calculate small values. <lang lua>gamma, coeff, quad, qui, set = 0.577215664901, -0.65587807152056, -0.042002635033944, 0.16653861138228, -0.042197734555571 function recigamma(z)
return z + gamma * z^2 + coeff * z^3 + quad * z^4 + qui * z^5 + set * z^6
end
function gammafunc(z)
if z == 1 then return 1 elseif math.abs(z) <= 0.5 then return 1 / recigamma(z) else return (z - 1) * gammafunc(z-1) end
end</lang>
M2000 Interpreter
<lang M2000 Interpreter> Module PrepareLambdaFunctions {
Const e = 2.7182818284590452@
Exp= Lambda e (x) -> e^x
gammaStirling=lambda e (x As decimal)->Sqrt(2.0 * pi / x) * ((x / e) ^ x)
Rad2Deg =Lambda pidivby180=pi/180 (RadAngle)->RadAngle / pidivby180
Dim p(9)
p(0)=0.99999999999980993@, 676.5203681218851@, -1259.1392167224028@, 771.32342877765313@
p(4)=-176.61502916214059@, 12.507343278686905@, -0.13857109526572012@, 0.0000099843695780195716@
p(8)=0.00000015056327351493116@
gammaLanczos =Lambda p(), Rad2Deg, Exp (x As decimal) -> {
Def Decimal a, t
If x < 0.5 Then =pi / (Sin(Rad2Deg(pi * x)) *Lambda(1-x)) : Exit
x -= 1@
a=p(0)
t = x + 7.5@
For i= 1@ To 8@ {
a += p(i) / (x + i)
}
= Sqrt(2.0 * pi) * (t ^ (x + 0.5)) * Exp(-t) * a
}
Push gammaStirling, gammaLanczos
} Call PrepareLambdaFunctions Read gammaLanczos, gammaStirling Font "Courier New" Form 120, 40 document doc$=" χ Stirling Lanczos"+{ } Print $(2,20),"x", "Stirling",@(55),"Lanczos", $(0) Print For d = 0.1 To 2 step 0.1
Print $("0.00"), d,
Print $("0.000000000000000"), gammaStirling(d),
Print $("0.0000000000000000000000000000"), gammaLanczos(d)
doc$=format$("{0:-10} {1:-30} {2:-34}",str$(d,"0.00"), str$(gammaStirling(d),"0.000000000000000"), str$(gammaLanczos(d),"0.0000000000000000000000000000"))+{
}
Next d Print $("") clipboard doc$ </lang>
χ Stirling Lanczos
0.10 5.697187148977170 9.5135076986687024462927178610
0.20 3.325998424022390 4.5908437119987955107204909409
0.30 2.362530036269620 2.9915689876875914865114179656
0.40 1.841476335936240 2.2181595437576816416854441034
0.50 1.520346901066280 1.7724538509055147387430498835
0.60 1.307158857448360 1.4891922488128208508983507496
0.70 1.159053292113920 1.2980553326475564892857625396
0.80 1.053370968425610 1.1642297137253055422419914101
0.90 0.977061507877695 1.0686287021193206646594133376
1.00 0.922137008895789 1.0000000000000007024882980221
1.10 0.883489953168704 0.9513507698668745807357371716
1.20 0.857755335396591 0.9181687423997605348002977483
1.30 0.842678259448392 0.8974706963062785326402091223
1.40 0.836744548637082 0.8872638175030748314253582066
1.50 0.838956552526496 0.8862269254527587632845492097
1.60 0.848693242152574 0.8935153492876912865293624528
1.70 0.865621471793884 0.9086387328532921150064803085
1.80 0.889639635287995 0.9313837709802428420608295699
1.90 0.920842721894229 0.9617658319073891431109375442
2.00 0.959502175744492 1.0000000000000015609456469406
Maple
Built-in method that accepts any value. <lang Maple>GAMMA(17/2); GAMMA(7*I); M := Matrix(2, 3, 'fill' = -3.6); MTM:-gamma(M);</lang>
- Output:
2027025*sqrt(Pi)*(1/256) GAMMA(7*I) Matrix(2, 3, [[.2468571430, .2468571430, .2468571430], [.2468571430, .2468571430, .2468571430]])
Mathematica
This code shows the built-in method, which works for any value (positive, negative and complex numbers). <lang mathematica>Gamma[x]</lang> Output integers and half-integers (a space is multiplication in Mathematica):
1/2 Sqrt[pi] 1 1 3/2 Sqrt[pi]/2 2 1 5/2 (3 Sqrt[pi])/4 3 2 7/2 (15 Sqrt[pi])/8 4 6 9/2 (105 Sqrt[pi])/16 5 24 11/2 (945 Sqrt[pi])/32 6 120 13/2 (10395 Sqrt[pi])/64 7 720 15/2 (135135 Sqrt[pi])/128 8 5040 17/2 (2027025 Sqrt[pi])/256 9 40320 19/2 (34459425 Sqrt[pi])/512 10 362880
Output approximate numbers:
0.1 9.51351 0.2 4.59084 0.3 2.99157 0.4 2.21816 0.5 1.77245 0.6 1.48919 0.7 1.29806 0.8 1.16423 0.9 1.06863 1. 1.
Output complex numbers:
I -0.15495-0.498016 I 2 I 0.00990244-0.075952 I 3 I 0.0112987-0.00643092 I 4 I 0.00173011+0.00157627 I 5 I -0.000271704+0.000339933 I
Maxima
<lang maxima>fpprec: 30$
gamma_coeff(n) := block([a: makelist(1, n)],
a[2]: bfloat(%gamma),
for k from 3 thru n do
a[k]: bfloat((sum((-1)^j * zeta(j) * a[k - j], j, 2, k - 1) - a[2] * a[k - 1]) / (1 - k * a[1])),
a)$
poleval(a, x) := block([y: 0],
for k from length(a) thru 1 step -1 do
y: y * x + a[k],
y)$
gc: gamma_coeff(20)$
gamma_approx(x) := block([y: 1],
while x > 2 do (x: x - 1, y: y * x), y / (poleval(gc, x - 1)))$
gamma_approx(12.3b0) - gamma(12.3b0); /* -9.25224705314470500985141176997b-15 */</lang>
МК-61/52
П9 9 П0 ИП9 ИП9 1 + * Вx L0 05 1 + П9 ^ ln 1 - * ИП9 1 2 * 1/x + e^x <-> / 2 пи * ИП9 / КвКор * ^ ВП 3 + Вx - С/П
Modula-3
<lang modula3>MODULE Gamma EXPORTS Main;
FROM IO IMPORT Put; FROM Fmt IMPORT Extended, Style;
PROCEDURE Taylor(x: EXTENDED): EXTENDED =
CONST a = ARRAY [0..29] OF EXTENDED {
1.00000000000000000000X0, 0.57721566490153286061X0,
-0.65587807152025388108X0, -0.04200263503409523553X0,
0.16653861138229148950X0, -0.04219773455554433675X0,
-0.00962197152787697356X0, 0.00721894324666309954X0,
-0.00116516759185906511X0, -0.00021524167411495097X0,
0.00012805028238811619X0, -0.00002013485478078824X0,
-0.00000125049348214267X0, 0.00000113302723198170X0,
-0.00000020563384169776X0, 0.00000000611609510448X0,
0.00000000500200764447X0, -0.00000000118127457049X0,
0.00000000010434267117X0, 0.00000000000778226344X0,
-0.00000000000369680562X0, 0.00000000000051003703X0,
-0.00000000000002058326X0, -0.00000000000000534812X0,
0.00000000000000122678X0, -0.00000000000000011813X0,
0.00000000000000000119X0, 0.00000000000000000141X0,
-0.00000000000000000023X0, 0.00000000000000000002X0 };
VAR y := x - 1.0X0;
sum := a[LAST(a)];
BEGIN
FOR i := LAST(a) - 1 TO FIRST(a) BY -1 DO
sum := sum * y + a[i];
END;
RETURN 1.0X0 / sum;
END Taylor;
BEGIN
FOR i := 1 TO 10 DO Put(Extended(Taylor(FLOAT(i, EXTENDED) / 3.0X0), style := Style.Sci) & "\n"); END;
END Gamma.</lang>
- Output:
2.6789385347077490e+000 1.3541179394264005e+000 1.0000000000000000e+000 8.9297951156924930e-001 9.0274529295093360e-001 1.0000000000000000e+000 1.1906393487589992e+000 1.5045754882515399e+000 1.9999999999939684e+000 2.7781584793385790e+000
Nim
The algorithm is a translation of that from the Ada solution. We have added a comparison with the gamma function provided by the “math” module from Nim standard library (which is, in fact, the C “tgamma” function). <lang nim>import math, strformat
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): float =
let y = x - 1 result = A[^1] for n in countdown(A.high - 1, A.low): result = result * y + A[n] result = 1 / result
echo "Our gamma function Nim gamma function Difference" echo "------------------ ------------------ ----------" for i in 1..10:
let val1 = gamma(i.toFloat / 3)
let val2 = math.gamma(i.toFloat / 3)
echo &"{val1:18.16f} {val2:18.16f} {val1 - val2:11.4e}"</lang>
- Output:
Our gamma function Nim gamma function Difference ------------------ ------------------ ---------- 2.6789385347077483 2.6789385347077479 4.4409e-16 1.3541179394264005 1.3541179394264005 0.0000e+00 1.0000000000000000 1.0000000000000000 0.0000e+00 0.8929795115692493 0.8929795115692493 0.0000e+00 0.9027452929509336 0.9027452929509336 0.0000e+00 1.0000000000000000 1.0000000000000000 0.0000e+00 1.1906393487589990 1.1906393487589990 0.0000e+00 1.5045754882515399 1.5045754882515558 -1.5987e-14 1.9999999999939684 2.0000000000000000 -6.0316e-12 2.7781584793385732 2.7781584804376647 -1.0991e-09
OCaml
<lang ocaml>let e = exp 1. let pi = 4. *. atan 1. let sqrttwopi = sqrt (2. *. pi)
module Lanczos = struct
(* Lanczos method *) (* Coefficients used by the GNU Scientific Library *) let g = 7. let c = [|0.99999999999980993; 676.5203681218851; -1259.1392167224028;
771.32342877765313; -176.61502916214059; 12.507343278686905; -0.13857109526572012; 9.9843695780195716e-6; 1.5056327351493116e-7|]
let rec ag z d = if d = 0 then c.(0) +. ag z 1 else if d < 8 then c.(d) /. (z +. float d) +. ag z (succ d) else c.(d) /. (z +. float d)
let gamma z = let z = z -. 1. in let p = z +. g +. 0.5 in sqrttwopi *. p ** (z +. 0.5) *. exp (-. p) *. ag z 0
end
module Stirling = struct
(* Stirling method *) let gamma z = sqrttwopi /. sqrt z *. (z /. e) ** z
end
module Stirling2 = struct
(* Extended Stirling method seen in Abramowitz and Stegun *) let d = [|1./.12.; 1./.288.; -139./.51840.; -571./.2488320.|]
let rec corr z x n = if n < Array.length d - 1 then d.(n) /. x +. corr z (x *. z) (succ n) else d.(n) /. x
let gamma z = Stirling.gamma z *. (1. +. corr z z 0)
end
let mirror gma z =
if z > 0.5 then gma z else pi /. sin (pi *. z) /. gma (1. -. z)
let _ =
Printf.printf "z\t\tLanczos\t\tStirling\tStirling2\n"; for i = 1 to 20 do let z = float i /. 10. in Printf.printf "%-10.8g\t%10.8e\t%10.8e\t%10.8e\n" z
(mirror Lanczos.gamma z) (mirror Stirling.gamma z) (mirror Stirling2.gamma z)
done; for i = 1 to 7 do let z = 10. *. float i in Printf.printf "%-10.8g\t%10.8e\t%10.8e\t%10.8e\n" z
(Lanczos.gamma z) (Stirling.gamma z) (Stirling2.gamma z)
done</lang>
- Output:
z Lanczos Stirling Stirling2 0.1 9.51350770e+00 1.04050843e+01 9.52104183e+00 0.2 4.59084371e+00 5.07399275e+00 4.59686230e+00 0.3 2.99156899e+00 3.35033954e+00 2.99844028e+00 0.4 2.21815954e+00 2.52705781e+00 2.22775889e+00 0.5 1.77245385e+00 2.06636568e+00 1.78839014e+00 0.6 1.48919225e+00 1.30715886e+00 1.48277536e+00 0.7 1.29805533e+00 1.15905329e+00 1.29508068e+00 0.8 1.16422971e+00 1.05337097e+00 1.16270541e+00 0.9 1.06862870e+00 9.77061508e-01 1.06778308e+00 1 1.00000000e+00 9.22137009e-01 9.99499469e-01 1.1 9.51350770e-01 8.83489953e-01 9.51037997e-01 1.2 9.18168742e-01 8.57755335e-01 9.17964058e-01 1.3 8.97470696e-01 8.42678259e-01 8.97331287e-01 1.4 8.87263818e-01 8.36744549e-01 8.87165485e-01 1.5 8.86226925e-01 8.38956553e-01 8.86155384e-01 1.6 8.93515349e-01 8.48693242e-01 8.93461840e-01 1.7 9.08638733e-01 8.65621472e-01 9.08597702e-01 1.8 9.31383771e-01 8.89639635e-01 9.31351590e-01 1.9 9.61765832e-01 9.20842722e-01 9.61740068e-01 2 1.00000000e+00 9.59502176e-01 9.99978981e-01 10 3.62880000e+05 3.59869562e+05 3.62879997e+05 20 1.21645100e+17 1.21139342e+17 1.21645100e+17 30 8.84176199e+30 8.81723653e+30 8.84176199e+30 40 2.03978821e+46 2.03554316e+46 2.03978821e+46 50 6.08281864e+62 6.07268919e+62 6.08281864e+62 60 1.38683119e+80 1.38490639e+80 1.38683119e+80 70 1.71122452e+98 1.70918858e+98 1.71122452e+98
Octave
<lang octave>function g = lacz_gamma(a, cg=7)
p = [ 0.99999999999980993, 676.5203681218851, -1259.1392167224028, \
771.32342877765313, -176.61502916214059, 12.507343278686905, \
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 ];
x=a;
if ( x < 0.5 )
g = pi / ( sin(pi*x) * lacz_gamma(1.0-x) );
else
x = x - 1.0;
t = p(1);
for i=1:(cg+1)
t = t + p(i+1)/(x+double(i));
endfor
w = x + double(cg) + 0.5;
g = sqrt(2.0*pi) * w**(x+0.5) * exp(-w) * t;
endif
endfunction
for i = 1:10
printf("%f %f\n", gamma(i/3.0), lacz_gamma(i/3.0));
endfor</lang>
- Output:
2.678939 2.678939 1.354118 1.354118 1.000000 1.000000 0.892980 0.892980 0.902745 0.902745 1.000000 1.000000 1.190639 1.190639 1.504575 1.504575 2.000000 2.000000 2.778158 2.778158
Which suggests that the built-in gamma uses the same approximation.
Oforth
<lang oforth>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 * ;</lang>
- Output:
>20 seq apply(#[ 10.0 / dup . gamma .cr ]) 0.1 9.51350769866874 0.2 4.5908437119988 0.3 2.99156898768759 0.4 2.21815954375769 0.5 1.77245385090552 0.6 1.48919224881282 0.7 1.29805533264756 0.8 1.1642297137253 0.9 1.06862870211932 1 1 1.1 0.951350769866874 1.2 0.918168742399761 1.3 0.897470696306277 1.4 0.887263817503076 1.5 0.886226925452759 1.6 0.893515349287691 1.7 0.908638732853292 1.8 0.931383770980243 1.9 0.961765831907388 2 1
PARI/GP
Built-in
<lang parigp>gamma(x)</lang>
Double-exponential integration
[[+oo],k] means that the function approaches as
<lang parigp>Gamma(x)=intnum(t=0,[+oo,1],t^(x-1)/exp(t))</lang>
Romberg integration
<lang parigp>Gamma(x)=intnumromb(t=0,9,t^(x-1)/exp(t),0)+intnumromb(t=9,max(x,99)^9,t^(x-1)/exp(t),2)</lang>
Stirling approximation
<lang parigp>Stirling(x)=x--;sqrt(2*Pi*x)*(x/exp(1))^x</lang>
Perl
<lang perl>use strict; use warnings; use constant pi => 4*atan2(1, 1); use constant e => exp(1);
- Normally would be: use Math::MPFR
- but this will use it if it's installed and ignore otherwise
my $have_MPFR = eval { require Math::MPFR; Math::MPFR->import(); 1; };
sub Gamma {
my $z = shift;
my $method = shift // 'lanczos';
if ($method eq 'lanczos') {
use constant g => 9;
$z < .5 ? pi / sin(pi * $z) / Gamma(1 - $z, $method) :
sqrt(2* pi) *
($z + g - .5)**($z - .5) *
exp(-($z + g - .5)) *
do {
my @coeff = qw{
1.000000000000000174663
5716.400188274341379136
-14815.30426768413909044
14291.49277657478554025
-6348.160217641458813289
1301.608286058321874105
-108.1767053514369634679
2.605696505611755827729
-0.7423452510201416151527e-2
0.5384136432509564062961e-7
-0.4023533141268236372067e-8
};
my ($sum, $i) = (shift(@coeff), 0);
$sum += $_ / ($z + $i++) for @coeff;
$sum;
}
} elsif ($method eq 'taylor') {
$z < .5 ? Gamma($z+1, $method)/$z :
$z > 1.5 ? ($z-1)*Gamma($z-1, $method) :
do { my $s = 0; ($s *= $z-1) += $_ for qw{ 0.00000000000000000002 -0.00000000000000000023 0.00000000000000000141 0.00000000000000000119 -0.00000000000000011813 0.00000000000000122678 -0.00000000000000534812 -0.00000000000002058326 0.00000000000051003703 -0.00000000000369680562 0.00000000000778226344 0.00000000010434267117 -0.00000000118127457049 0.00000000500200764447 0.00000000611609510448 -0.00000020563384169776 0.00000113302723198170 -0.00000125049348214267 -0.00002013485478078824 0.00012805028238811619 -0.00021524167411495097 -0.00116516759185906511 0.00721894324666309954 -0.00962197152787697356 -0.04219773455554433675 0.16653861138229148950 -0.04200263503409523553 -0.65587807152025388108 0.57721566490153286061 1.00000000000000000000 }; 1/$s; }
} elsif ($method eq 'stirling') {
no warnings qw(recursion);
$z < 100 ? Gamma($z + 1, $method)/$z :
sqrt(2*pi*$z)*($z/e + 1/(12*e*$z))**$z / $z;
} elsif ($method eq 'MPFR') {
my $result = Math::MPFR->new();
Math::MPFR::Rmpfr_gamma($result, Math::MPFR->new($z), 0);
$result;
} else { die "unknown method '$method'" }
}
for my $method (qw(MPFR lanczos taylor stirling)) {
next if $method eq 'MPFR' && !$have_MPFR;
printf "%10s: ", $method;
print join(' ', map { sprintf "%.12f", Gamma($_/3, $method) } 1 .. 10);
print "\n";
}</lang>
- Output:
MPFR: 2.678938534708 1.354117939426 1.000000000000 0.892979511569 0.902745292951 1.000000000000 1.190639348759 1.504575488252 2.000000000000 2.778158480438
lanczos: 2.678938534708 1.354117939426 1.000000000000 0.892979511569 0.902745292951 1.000000000000 1.190639348759 1.504575488252 2.000000000000 2.778158480438
taylor: 2.678938534708 1.354117939426 1.000000000000 0.892979511569 0.902745292951 1.000000000000 1.190639348759 1.504575488252 2.000000000000 2.778158480438
stirling: 2.678938532866 1.354117938504 0.999999999306 0.892979510955 0.902745292336 0.999999999306 1.190639347940 1.504575487227 1.999999998611 2.778158478527
Phix
<lang Phix>sequence c = repeat(0,12)
function gamma(atom z)
atom accm = c[1]
if accm=0 then
accm = sqrt(2*PI)
c[1] = accm
atom k1_factrl = 1 -- (k - 1)!*(-1)^k with 0!==1
for k=2 to 12 do
c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl
k1_factrl *= -(k-1)
end for
end if
for k=2 to 12 do
accm += c[k]/(z+k-1)
end for
accm *= exp(-(z+12))*power(z+12,z+0.5) -- Gamma(z+1)
return accm/z
end function
procedure sq(atom x, atom mul) atom p = x*mul
printf(1,"%18.16g,%18.15g\n",{x,p*p})
end procedure
procedure si(atom x)
printf(1,"%18.15f\n",{x})
end procedure
sq(gamma(-3/2),3/4) sq(gamma(-1/2),-1/2) sq(gamma(1/2),1) si(gamma(1)) sq(gamma(3/2),2) si(gamma(2)) sq(gamma(5/2),4/3) si(gamma(3)) sq(gamma(7/2),8/15) si(gamma(4))</lang>
- Output:
2.363271801207354, 3.14159265358979 -3.544907701811032, 3.14159265358979 1.772453850905515, 3.14159265358979 1.000000000000001 0.8862269254527643, 3.14159265358984 1.000000000000010 1.329340388179146, 3.14159265358984 2.000000000000024 3.323350970447942, 3.14159265358998 6.000000000000175
mpfr version
Above translated to mpfr, with higher accuracy and more iterations as per REXX, and compared against the builtin.
<lang Phix>include mpfr.e mpfr_set_default_prec(-87) -- 87 decimal places.
sequence c = mpfr_inits(40)
function gamma(atom z)
mpfr accm = c[1]
if mpfr_cmp_si(accm,0)=0 then
-- c[1] := sqrt(2*PI)
mpfr_const_pi(accm)
mpfr_mul_si(accm,accm,2)
mpfr_sqrt(accm,accm)
-- k1_factrl = (k - 1)!*(-1)^k with 0!==1
mpfr k1_factrl = mpfr_init(1),
tmk = mpfr_init(),
p = mpfr_init()
for k=2 to length(c) do
-- c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl
mpfr_set_si(tmk,length(c)+1-k)
mpfr_exp(c[k],tmk)
mpfr_set_d(p,k-1.5)
mpfr_pow(p,tmk,p)
mpfr_div(p,p,k1_factrl)
mpfr_mul(c[k],c[k],p)
-- k1_factrl *= -(k-1)
mpfr_mul_si(k1_factrl,k1_factrl,-(k-1))
end for
end if
accm = mpfr_init_set(accm)
for k=2 to length(c) do
-- accm += c[k]/(z+k-1)
mpfr ck = mpfr_init_set(c[k]),
zk = mpfr_init(z+k-1)
mpfr_div(ck,ck,zk)
mpfr_add(accm,accm,ck)
end for
atom zc = z+length(c)
-- accm *= exp(-zc)*power(zc,z+0.5) -- Gamma(z+1)
mpfr ez = mpfr_init(-zc),
p = mpfr_init(zc),
zh = mpfr_init(z+0.5)
mpfr_exp(ez,ez)
mpfr_pow(p,p,zh)
mpfr_mul(accm,accm,ez)
mpfr_mul(accm,accm,p)
-- return accm/z
mpfr_set_d(ez,z)
mpfr_div(accm,accm,ez)
return accm
end function
function gamma2(atom z)
mpfr r = mpfr_init(z) mpfr_gamma(r,r) return r
end function
constant FMT = "%43.40Rf"
procedure sq(mpfr x, integer n, d=1)
mpfr p = mpfr_init_set(x)
mpfr_mul_si(p,p,n)
mpfr_div_si(p,p,d)
mpfr_mul(p,p,p)
string xs = mpfr_sprintf(FMT,x),
ps = mpfr_sprintf(FMT,p)
printf(1,"%s,%s\n",{xs,ps})
end procedure
procedure si(mpfr x)
string xs = mpfr_sprintf(FMT,x) printf(1,"%s\n",trim_tail(xs,".0"))
end procedure
sq(gamma(-3/2),3,4) sq(gamma(-1/2),-1,2) sq(gamma(1/2),1) si(gamma(1)) sq(gamma(3/2),2) si(gamma(2)) sq(gamma(5/2),4,3) si(gamma(3)) sq(gamma(7/2),8,15) si(gamma(4)) puts(1,"mpfr_gamma():\n") sq(gamma2(-3/2),3,4) sq(gamma2(-1/2),-1,2) sq(gamma2(1/2),1) si(gamma2(1)) sq(gamma2(3/2),2) si(gamma2(2)) sq(gamma2(5/2),4,3) si(gamma2(3)) sq(gamma2(7/2),8,15) si(gamma2(4))</lang>
- Output:
2.3632718012073547030642233111215269103967, 3.1415926535897932384626433832795028841972 -3.5449077018110320545963349666822903655951, 3.1415926535897932384626433832795028841972 1.7724538509055160272981674833411451827975, 3.1415926535897932384626433832795028841972 1 0.8862269254527580136490837416705725913988, 3.1415926535897932384626433832795028841972 1 1.3293403881791370204736256125058588870982, 3.1415926535897932384626433832795028841972 2 3.3233509704478425511840640312646472177454, 3.1415926535897932384626433832795028841972 6 mpfr_gamma(): 2.3632718012073547030642233111215269103967, 3.1415926535897932384626433832795028841972 -3.5449077018110320545963349666822903655951, 3.1415926535897932384626433832795028841972 1.7724538509055160272981674833411451827975, 3.1415926535897932384626433832795028841972 1 0.8862269254527580136490837416705725913988, 3.1415926535897932384626433832795028841972 1 1.3293403881791370204736256125058588870982, 3.1415926535897932384626433832795028841972 2 3.3233509704478425511840640312646472177454, 3.1415926535897932384626433832795028841972 6
Phixmonti
<lang Phixmonti>0.577215664901 var gamma -0.65587807152056 var coeff -0.042002635033944 var quad 0.16653861138228 var qui -0.042197734555571 var theSet
def recigamma var z /# n -- n #/
z 6 power theSet * z 5 power qui * z 4 power quad * z 3 power coeff * z 2 power gamma * z + + + + +
enddef
/# without var def recigamma
dup 6 power theSet * swap dup 5 power qui * swap dup 4 power quad * swap dup 3 power coeff * swap dup 2 power gamma * swap + + + + +
enddef
- /
def gammafunc /# n -- n #/
dup 1 == if
else
dup abs 0.5 <= if
recigamma 1 swap /
else
dup 1 - gammafunc swap 1 - *
endif
endif
enddef
0.1 2.1 .1 3 tolist for
dup print " = " print gammafunc print nl
endfor</lang>
PicoLisp
<lang PicoLisp>(scl 28)
(de *A
~(flip
(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 ) ) )
(de gamma (X)
(let (Y (- X 1.0) Sum (car *A))
(for A (cdr *A)
(setq Sum (+ A (*/ Sum Y 1.0))) )
(*/ 1.0 1.0 Sum) ) )</lang>
- Output:
: (for I (range 1 10) (prinl (round (gamma (*/ I 1.0 3)) 14)) ) 2.67893853470775 1.35411793942640 1.00000000000000 0.89297951156925 0.90274529295093 1.00000000000000 1.19063934875900 1.50457548825154 1.99999999999397 2.77815847933858
PL/I
<lang PL/I>/* From Rosetta Fortran */ test: procedure options (main);
declare i fixed binary;
on underflow ;
put skip list ('Lanczos', 'Builtin' );
do i = 1 to 10;
put skip list (lanczos_gamma(i/3.0q0), gamma(i/3.0q0) );
end;
lanczos_gamma: procedure (a) returns (float (18)) recursive;
declare a float (18); declare pi float (18) value (3.14159265358979324E0); declare cg fixed binary initial ( 7 );
/* these precomputed values are taken by the sample code in Wikipedia, */
/* and the sample itself takes them from the GNU Scientific Library */
declare p(0:8) float (18) static initial
( 0.99999999999980993e0, 676.5203681218851e0, -1259.1392167224028e0,
771.32342877765313e0, -176.61502916214059e0, 12.507343278686905e0,
-0.13857109526572012e0, 9.9843695780195716e-6, 1.5056327351493116e-7 );
declare ( t, w, x ) float (18); declare i fixed binary;
x = a;
if x < 0.5 then
return ( pi / ( sin(pi*x) * lanczos_gamma(1.0-x) ) );
else
do;
x = x - 1.0;
t = p(0);
do i = 1 to cg+2;
t = t + p(i)/(x+i);
end;
w = x + float(cg) + 0.5;
return ( sqrt(2*pi) * w**(x+0.5) * exp(-w) * t );
end;
end lanczos_gamma;
end test;</lang>
- Output:
Lanczos Builtin 2.67893853470774706E+0000 2.678938534707747630E+0000 1.35411793942640071E+0000 1.354117939426400420E+0000 1.00000000000000021E+0000 1.000000000000000000E+0000 8.92979511569249470E-0001 8.929795115692492110E-0001 9.02745292950933961E-0001 9.027452929509336110E-0001 1.00000000000000048E+0000 1.000000000000000000E+0000 1.19063934875899964E+0000 1.190639348758998950E+0000 1.50457548825155704E+0000 1.504575488251556020E+0000 2.00000000000000154E+0000 2.000000000000000000E+0000 2.77815848043766660E+0000 2.778158480437664210E+0000
PowerShell
I would download the Math.NET Numerics dll(s). Documentation and download at: http://cyber-defense.sans.org/blog/2015/06/27/powershell-for-math-net-numerics/comment-page-1/ <lang PowerShell> Add-Type -Path "C:\Program Files (x86)\Math\MathNet.Numerics.3.12.0\lib\net40\MathNet.Numerics.dll"
1..20 | ForEach-Object {[MathNet.Numerics.SpecialFunctions]::Gamma($_ / 10)} </lang>
- Output:
9.51350769866874 4.5908437119988 2.99156898768759 2.21815954375769 1.77245385090552 1.48919224881282 1.29805533264756 1.1642297137253 1.06862870211932 1 0.951350769866874 0.918168742399759 0.897470696306277 0.887263817503075 0.88622692545276 0.89351534928769 0.908638732853289 0.931383770980245 0.961765831907388 1
PureBasic
Below is PureBasic code for:
- Complete Gamma function
- Natural Logarithm of the Complete Gamma function
- Factorial function
<lang PureBasic>Procedure.d Gamma(x.d) ; AKJ 01-May-10
- Complete Gamma function for x>0 and x<2 (approx)
- Extended outside this range via
- Gamma(x+1) = x*Gamma(x)
- Based on http://rosettacode.org/wiki/Gamma_function [Ada]
Protected Dim A.d(28) A(0) = 1.0 A(1) = 0.5772156649015328606 A(2) =-0.6558780715202538811 A(3) =-0.0420026350340952355 A(4) = 0.1665386113822914895 A(5) =-0.0421977345555443368 ; was ...33675 A(6) =-0.0096219715278769736 A(7) = 0.0072189432466630995 A(8) =-0.0011651675918590651 A(9) =-0.0002152416741149510 A(10) = 0.0001280502823881162 A(11) =-0.0000201348547807882 A(12) =-0.0000012504934821427 A(13) = 0.0000011330272319817 A(14) =-0.0000002056338416978 A(15) = 0.0000000061160951045 A(16) = 0.0000000050020076445 A(17) =-0.0000000011812745705 A(18) = 0.0000000001043426712 A(19) = 0.0000000000077822634 A(20) =-0.0000000000036968056 A(21) = 0.0000000000005100370 A(22) =-0.0000000000000205833 A(23) =-0.0000000000000053481 A(24) = 0.0000000000000012268 A(25) =-0.0000000000000001181 A(26) = 0.0000000000000000012 A(27) = 0.0000000000000000014 A(28) =-0.0000000000000000002
- A(29) = 0.00000000000000000002
Protected y.d, Prod.d, Sum.d, N If x<=0.0: ProcedureReturn 0.0: EndIf ; Error y = x-1.0: Prod = 1.0 While y>=1.0
Prod*y: y-1.0 ; Recurse using Gamma(x+1) = x*Gamma(x)
Wend Sum= A(28) For N = 27 To 0 Step -1
Sum*y+A(N)
Next N If Sum=0.0: ProcedureReturn Infinity(): EndIf ProcedureReturn Prod / Sum EndProcedure
Procedure.d GammLn(x.d) ; AKJ 01-May-10
- Returns Ln(Gamma()) or 0 on error
- Based on Numerical Recipes gamma.h
Protected j, tmp.d, y.d, ser.d Protected Dim cof.d(13) cof(0) = 57.1562356658629235 cof(1) = -59.5979603554754912 cof(2) = 14.1360979747417471 cof(3) = -0.491913816097620199 cof(4) = 0.339946499848118887e-4 cof(5) = 0.465236289270485756e-4 cof(6) = -0.983744753048795646e-4 cof(7) = 0.158088703224912494e-3 cof(8) = -0.210264441724104883e-3 cof(9) = 0.217439618115212643e-3 cof(10) = -0.164318106536763890e-3 cof(11) = 0.844182239838527433e-4 cof(12) = -0.261908384015814087e-4 cof(13) = 0.368991826595316234e-5 If x<=0: ProcedureReturn 0: EndIf ; Bad argument y = x tmp = x+5.2421875 tmp = (x+0.5)*Log(tmp)-tmp ser = 0.999999999999997092 For j=0 To 13
y + 1: ser + cof(j)/y
Next j ProcedureReturn tmp+Log(2.5066282746310005*ser/x) EndProcedure
Procedure Factorial(x) ; AKJ 01-May-10
ProcedureReturn Gamma(x+1)
EndProcedure</lang>
- Examples
<lang PureBasic>Debug "Gamma()" For i = 12 To 15
Debug StrD(i/3.0, 3)+" "+StrD(Gamma(i/3.0))
Next i Debug "" Debug "Ln(Gamma(5.0)) = "+StrD(GammLn(5.0), 16) ; Ln(24) Debug "" Debug "Factorial 6 = "+StrD(Factorial(6), 0) ; 72</lang>
- Output:
[Debug] Gamma(): [Debug] 4.000 6.0000000000 [Debug] 4.333 9.2605282681 [Debug] 4.667 14.7114047740 [Debug] 5.000 24.0000000000 [Debug] [Debug] Ln(Gamma(5.0)) = 3.1780538303479458 [Debug] [Debug] Factorial 6 = 720
Python
Procedural
<lang python>_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
)
def gamma (x):
y = float(x) - 1.0;
sm = _a[-1];
for an in _a[-2::-1]:
sm = sm * y + an;
return 1.0 / sm;
if __name__ == '__main__':
for i in range(1,11):
print " %20.14e" % gamma(i/3.0)
</lang>
- Output:
2.67893853470775e+00 1.35411793942640e+00 1.00000000000000e+00 8.92979511569249e-01 9.02745292950934e-01 1.00000000000000e+00 1.19063934875900e+00 1.50457548825154e+00 1.99999999999397e+00 2.77815847933857e+00
Functional
In terms of fold/reduce:
<lang python>Gamma function
from functools import reduce
- gamma_ :: [Float] -> Float -> Float
def gamma_(tbl):
Gamma function.
def go(x):
y = float(x) - 1.0
return 1.0 / reduce(
lambda a, x: a * y + x,
tbl[-2::-1],
tbl[-1]
)
return lambda x: go(x)
- TBL :: [Float]
TBL = [
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
]
- TEST ----------------------------------------------------
- main :: IO()
def main():
Gamma function over a range of values.
gamma = gamma_(TBL)
print(
fTable(' i -> gamma(i/3):\n')(repr)(lambda x: "%0.7e" % x)(
lambda x: gamma(x / 3.0)
)(enumFromTo(1)(10))
)
- GENERIC -------------------------------------------------
- enumFromTo :: (Int, Int) -> [Int]
def enumFromTo(m):
Integer enumeration from m to n. return lambda n: list(range(m, 1 + n))
- FORMATTING -------------------------------------------------
- fTable :: String -> (a -> String) ->
- (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
def go(xShow, fxShow, f, xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
return s + '\n' + '\n'.join(map(
lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
xs, ys
))
return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
i -> gamma(i/3): 1 -> 2.6789385e+00 2 -> 1.3541179e+00 3 -> 1.0000000e+00 4 -> 8.9297951e-01 5 -> 9.0274529e-01 6 -> 1.0000000e+00 7 -> 1.1906393e+00 8 -> 1.5045755e+00 9 -> 2.0000000e+00 10 -> 2.7781585e+00
R
Lanczos' approximation is loosely converted from the Octave code.
<lang r>stirling <- function(z) sqrt(2*pi/z) * (exp(-1)*z)^z
nemes <- function(z) sqrt(2*pi/z) * (exp(-1)*(z + (12*z - (10*z)^-1)^-1))^z
lanczos <- function(z) {
if(length(z) > 1)
{
sapply(z, lanczos)
} else
{
g <- 7
p <- c(0.99999999999980993, 676.5203681218851, -1259.1392167224028,
771.32342877765313, -176.61502916214059, 12.507343278686905,
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7)
z <- as.complex(z)
if(Re(z) < 0.5)
{
pi / (sin(pi*z) * lanczos(1-z))
} else
{
z <- z - 1
x <- p[1] + sum(p[-1]/seq.int(z+1, z+g+1))
tt <- z + g + 0.5
sqrt(2*pi) * tt^(z+0.5) * exp(-tt) * x
}
}
}
spouge <- function(z, a=49) {
if(length(z) > 1)
{
sapply(z, spouge)
} else
{
z <- z-1
k <- seq.int(1, a-1)
ck <- rep(c(1, -1), len=a-1) / factorial(k-1) * (a-k)^(k-0.5) * exp(a-k)
(z + a)^(z+0.5) * exp(-z - a) * (sqrt(2*pi) + sum(ck/(z+k)))
}
}
- Checks
z <- (1:10)/3 all.equal(gamma(z), stirling(z)) # Mean relative difference: 0.07181942 all.equal(gamma(z), nemes(z)) # Mean relative difference: 0.003460549 all.equal(as.complex(gamma(z)), lanczos(z)) # TRUE all.equal(gamma(z), spouge(z)) # TRUE data.frame(z=z, stirling=stirling(z), nemes=nemes(z), lanczos=lanczos(z), spouge=spouge(z), builtin=gamma(z))</lang>
- Output:
z stirling nemes lanczos spouge builtin 1 0.3333333 2.1569760 2.6290752 2.6789385+0i 2.6789385 2.6789385 2 0.6666667 1.2028507 1.3515736 1.3541179+0i 1.3541179 1.3541179 3 1.0000000 0.9221370 0.9996275 1.0000000+0i 1.0000000 1.0000000 4 1.3333333 0.8397427 0.8928835 0.8929795+0i 0.8929795 0.8929795 5 1.6666667 0.8591902 0.9027098 0.9027453+0i 0.9027453 0.9027453 6 2.0000000 0.9595022 0.9999831 1.0000000+0i 1.0000000 1.0000000 7 2.3333333 1.1491064 1.1906296 1.1906393+0i 1.1906393 1.1906393 8 2.6666667 1.4584904 1.5045690 1.5045755+0i 1.5045755 1.5045755 9 3.0000000 1.9454032 1.9999951 2.0000000+0i 2.0000000 2.0000000 10 3.3333333 2.7097638 2.7781544 2.7781585+0i 2.7781585 2.7781585
Racket
<lang Racket>#lang racket (define (gamma number)
(if (> 1/2 number)
(/ pi (* (sin (* pi number))
(gamma (- 1.0 number))))
(let ((n (sub1 number))
(c '(0.99999999999980993 676.5203681218851 -1259.1392167224028
771.32342877765313 -176.61502916214059 12.507343278686905
-0.13857109526572012 9.9843695780195716e-6 1.5056327351493116e-7)))
(* (sqrt (* pi 2))
(expt (+ n 7 0.5) (+ n 0.5))
(exp (- (+ n 7 0.5)))
(+ (car c)
(apply +
(for/list ((i (in-range (length (cdr c)))) (x (in-list (cdr c))))
(/ x (+ 1 n i)))))))))
(map gamma '(0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0))
- ->
- '(9.513507698668736
- 4.590843711998802
- 2.9915689876875904
- 2.218159543757687
- 1.7724538509055159
- 1.489192248812818
- 1.2980553326475577
- 1.1642297137253037
- 1.068628702119319
- 1.0)</lang>
Raku
(formerly Perl 6) <lang perl6>sub Γ(\z) {
constant g = 9;
z < .5 ?? pi/ sin(pi * z) / Γ(1 - z) !!
sqrt(2*pi) *
(z + g - 1/2)**(z - 1/2) *
exp(-(z + g - 1/2)) *
[+] <
1.000000000000000174663
5716.400188274341379136
-14815.30426768413909044
14291.49277657478554025
-6348.160217641458813289
1301.608286058321874105
-108.1767053514369634679
2.605696505611755827729
-0.7423452510201416151527e-2
0.5384136432509564062961e-7
-0.4023533141268236372067e-8
> Z* 1, |map 1/(z + *), 0..*
}
say Γ($_) for 1/3, 2/3 ... 10/3;</lang>
- Output:
2.67893853470775 1.3541179394264 1 0.892979511569248 0.902745292950934 1 1.190639348759 1.50457548825155 2 2.77815848043766
REXX
Taylor series, 80-digit coefficients
This version uses a Taylor series with 80-digits coefficients with much more accuracy.
As a result, the gamma value for ½ is now 25 decimal digits more accurate than the previous version
(which only used 20 digit coefficients).
Note: The Taylor series isn't much good above values of 6½. <lang rexx>/*REXX program calculates GAMMA using the Taylor series coefficients; ≈80 decimal digits*/
/*The GAMMA function symbol is the Greek capital letter: Γ */
numeric digits 90 /*be able to handle extended precision.*/ parse arg LO HI . /*allow specification of gamma arg/args*/
/* [↓] either show a range or a ··· */
do j=word(LO 1, 1) to word(HI LO 9, 1) /* ··· single gamma value.*/
say 'gamma('j") =" gamma(j) /*compute gamma of J and display value.*/
end /*j*/ /* [↑] default LO is one; HI is nine.*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ gamma: procedure; parse arg x; xm=x-1; sum=0
/*coefficients thanks to: Arne Fransén & Staffan Wrigge.*/ #.1 = 1 /* [↓] #.2 is the Euler-Mascheroni constant. */ #.2 = 0.57721566490153286060651209008240243104215933593992359880576723488486772677766467 #.3 = -0.65587807152025388107701951514539048127976638047858434729236244568387083835372210 #.4 = -0.04200263503409523552900393487542981871139450040110609352206581297618009687597599 #.5 = 0.16653861138229148950170079510210523571778150224717434057046890317899386605647425 #.6 = -0.04219773455554433674820830128918739130165268418982248637691887327545901118558900 #.7 = -0.00962197152787697356211492167234819897536294225211300210513886262731167351446074 #.8 = 0.00721894324666309954239501034044657270990480088023831800109478117362259497415854 #.9 = -0.00116516759185906511211397108401838866680933379538405744340750527562002584816653
- .10 = -0.00021524167411495097281572996305364780647824192337833875035026748908563946371678
- .11 = 0.00012805028238811618615319862632816432339489209969367721490054583804120355204347
- .12 = -0.00002013485478078823865568939142102181838229483329797911526116267090822918618897
- .13 = -0.00000125049348214267065734535947383309224232265562115395981534992315749121245561
- .14 = 0.00000113302723198169588237412962033074494332400483862107565429550539546040842730
- .15 = -0.00000020563384169776071034501541300205728365125790262933794534683172533245680371
- .16 = 0.00000000611609510448141581786249868285534286727586571971232086732402927723507435
- .17 = 0.00000000500200764446922293005566504805999130304461274249448171895337887737472132
- .18 = -0.00000000118127457048702014458812656543650557773875950493258759096189263169643391
- .19 = 0.00000000010434267116911005104915403323122501914007098231258121210871073927347588
- .20 = 0.00000000000778226343990507125404993731136077722606808618139293881943550732692987
- .21 = -0.00000000000369680561864220570818781587808576623657096345136099513648454655443000
- .22 = 0.00000000000051003702874544759790154813228632318027268860697076321173501048565735
- .23 = -0.00000000000002058326053566506783222429544855237419746091080810147188058196444349
- .24 = -0.00000000000000534812253942301798237001731872793994898971547812068211168095493211
- .25 = 0.00000000000000122677862823826079015889384662242242816545575045632136601135999606
- .26 = -0.00000000000000011812593016974587695137645868422978312115572918048478798375081233
- .27 = 0.00000000000000000118669225475160033257977724292867407108849407966482711074006109
- .28 = 0.00000000000000000141238065531803178155580394756670903708635075033452562564122263
- .29 = -0.00000000000000000022987456844353702065924785806336992602845059314190367014889830
- .30 = 0.00000000000000000001714406321927337433383963370267257066812656062517433174649858
- .31 = 0.00000000000000000000013373517304936931148647813951222680228750594717618947898583
- .32 = -0.00000000000000000000020542335517666727893250253513557337960820379352387364127301
- .33 = 0.00000000000000000000002736030048607999844831509904330982014865311695836363370165
- .34 = -0.00000000000000000000000173235644591051663905742845156477979906974910879499841377
- .35 = -0.00000000000000000000000002360619024499287287343450735427531007926413552145370486
- .36 = 0.00000000000000000000000001864982941717294430718413161878666898945868429073668232
- .37 = -0.00000000000000000000000000221809562420719720439971691362686037973177950067567580
- .38 = 0.00000000000000000000000000012977819749479936688244144863305941656194998646391332
- .39 = 0.00000000000000000000000000000118069747496652840622274541550997151855968463784158
- .40 = -0.00000000000000000000000000000112458434927708809029365467426143951211941179558301
- .41 = 0.00000000000000000000000000000012770851751408662039902066777511246477487720656005
- .42 = -0.00000000000000000000000000000000739145116961514082346128933010855282371056899245
- .43 = 0.00000000000000000000000000000000001134750257554215760954165259469306393008612196
- .44 = 0.00000000000000000000000000000000004639134641058722029944804907952228463057968680
- .45 = -0.00000000000000000000000000000000000534733681843919887507741819670989332090488591
- .46 = 0.00000000000000000000000000000000000032079959236133526228612372790827943910901464
- .47 = -0.00000000000000000000000000000000000000444582973655075688210159035212464363740144
- .48 = -0.00000000000000000000000000000000000000131117451888198871290105849438992219023663
- .49 = 0.00000000000000000000000000000000000000016470333525438138868182593279063941453996
- .50 = -0.00000000000000000000000000000000000000001056233178503581218600561071538285049997
- .51 = 0.00000000000000000000000000000000000000000026784429826430494783549630718908519485
- .52 = 0.00000000000000000000000000000000000000000002424715494851782689673032938370921241
- =52; do k=# by -1 for #
sum=sum*xm + #.k
end /*k*/
return 1/sum</lang>
- output when using the input of: 0.5
gamma(0.5) = 1.77245385090551602729816748334114518279754945612238712821380509003635917689651032047826593
Note that: Γ(½) = √ =
1.77245 38509 05516 02729 81674 83341 14518 27975 49456 12238 71282 13807 78985 29112 84591 03218 13749 50656 73854 46654 16226 82362 +
to 110 digits past the decimal point, the vinculum (overbar) marks the difference digit from the computed value (by this REXX program) of gamma(½).
Spouge's approximation, using 87 digit coefficients
This REXX version is a translation of Phix but with more (decimal digits) precision and more steps.
Many of the "normal" high-level mathematical functions aren't available in REXX, so some of them (RYO) are included here. <lang rexx>/*REXX program calculates the gamma function using Spouge's approximation with 87 digits*/ e=2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138 numeric digits length(e) - length(.) /*use the number of decimal digits in E*/ c.= 0
- = 40 /*#: the number of steps in GAMMA func*/
call sq gamma(-3/2), 3/4
call sq gamma(-1/2), -1/2
call sq gamma( 1/2), 1
call si gamma( 1 )
call sq gamma( 3/2), 2
call si gamma( 2 )
call sq gamma( 5/2), 4/3
call si gamma( 3 )
call sq gamma( 7/2), 8/15
call si gamma( 4 )
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ gamma: procedure expose c. e #; parse arg z; #p= # + 1
accm = c.1
if accm==0 then do; accm= sqrt( 2*pi() )
c.1 = accm
kfact = 1
do k=2 to #
c.k= exp(#p-k) * pow(#p-k, k-1.5) / kfact
kfact = kfact * -(k-1)
end /*k*/
end
do j=2 to #; accm = accm + c.j / (z+j-1)
end /*k*/
return (accm * exp(-(z+#)) * pow(z+#, z+0.5) ) / z
/*──────────────────────────────────────────────────────────────────────────────────────*/ pi: return 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348 fmt: parse arg n,p,a; _= format(n,p,a); L= length(_); return left( strip0(_), L) isInt: return datatype(arg(1), 'W') /*is the argument an integer? */ sq: procedure expose #; parse arg x,mu; say fmt(x,9,#) fmt((x*mu)**2,9,#); return si: procedure expose #; parse arg x; say fmt(x,9,#); return strip0: procedure; arg _; if pos(., _)\==0 then _= strip(strip(_,'T',0),'T',.); return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ exp: procedure expose e; arg x; ix= x%1; if abs(x-ix)>.5 then ix=ix+sign(x); x= x-ix; z=1
_=1; w=1; do j=1; _= _*x/j; z= (z+_)/1; if z==w then leave; w=z
end /*j*/; if z\==0 then z= e**ix * z; return z
/*──────────────────────────────────────────────────────────────────────────────────────*/ ln: procedure; parse arg x; call e; ig= x>1.5; is= 1-2*(ig\==1); ii= 0; xx= x
do while ig & xx>1.5 | \ig & xx<.5; _=e
do k=-1; iz=xx*_**-is; if k>=0&(ig&iz<1|\ig&iz>.5) then leave; _=_*_; izz=iz; end
xx= izz; ii= ii+is*2**k; end /*while*/; x= x*e**-ii-1; z=0; _= -1; p=z
do k=1; _=-_*x; z=z+_/k; if z=p then leave; p=z; end; /*k*/; return z+ii
/*──────────────────────────────────────────────────────────────────────────────────────*/ pow: procedure; parse arg x,y; if y=0 then return 1; if x=0 then return 0
if isInt(y) then return x**y; if isInt(1/y) then return root(x, 1/y)
if abs(y//1)=.5 then return sqrt(x)**sign(y)*x**(y%1); return exp( y*ln(x) )
/*──────────────────────────────────────────────────────────────────────────────────────*/ root: procedure; parse arg x 1 ox,y 1 oy; if x=0 | y=1 then return x/1
if \isInt(y) then return $pow(x, 1/y)
if y==2 then return sqrt(x); if y==-2 then return 1/sqrt(x); return rooti(x,y)/1
/*──────────────────────────────────────────────────────────────────────────────────────*/ rooti: x=abs(x); y=abs(y); a= digits() + 5; m= y-1; d= 5
parse value format(x,2,1,,0) 'E0' with ? 'E' _ .; g= (?/y'E'_ % y) + (x>1)
do until d==a; d=min(d+d, a); numeric digits d; o=0
do until o=g; o=g; g= format((m*g**y+x)/y/g**m,,d-2); end; end
_= g*sign(ox); if oy<0 then _= 1/_; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric digits; h=d+6
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .; g=g *.5'e'_ %2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return g/1</lang>
- output when using the default input:
2.3632718012073547030642233111215269103967 3.1415926535897932384626433832795028841972
-3.5449077018110320545963349666822903655951 3.1415926535897932384626433832795028841972
1.7724538509055160272981674833411451827975 3.1415926535897932384626433832795028841972
1
0.8862269254527580136490837416705725913988 3.1415926535897932384626433832795028841972
1
1.3293403881791370204736256125058588870982 3.1415926535897932384626433832795028841972
2
3.3233509704478425511840640312646472177454 3.1415926535897932384626433832795028841972
6
Ring
<lang ring> 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
</lang>
RLaB
RLaB through GSL has the following functions related to the Gamma function, namely, Gamma, GammaRegularizedC, LogGamma, RecGamma, and Pochhammer, where
- , the Gamma function;
- , the regularized Gamma function which is also known as the normalized incomplete Gamma function;
- , which the GSL calls the complementary normalized Gamma function;
- ;
- ;
- .
Ruby
Taylor series
<lang ruby>$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 ]
def gamma(x)
y = Float(x) - 1
1.0 / $a.reverse.inject {|sum, an| sum * y + an}
end
(1..10).each {|i| puts format("%.14e", gamma(i/3.0))}</lang>
- Output:
2.67893853470775e+00 1.35411793942640e+00 1.00000000000000e+00 8.92979511569249e-01 9.02745292950934e-01 1.00000000000000e+00 1.19063934875900e+00 1.50457548825154e+00 1.99999999999397e+00 2.77815847933857e+00
Built in
<lang ruby>(1..10).each{|i| puts Math.gamma(i/3.0)}</lang>
- Output:
2.678938534707748 1.3541179394264005 1.0 0.8929795115692493 0.9027452929509336 1.0 1.190639348758999 1.5045754882515558 2.0 2.7781584804376647
Scala
<lang scala>import java.util.Locale._
object Gamma {
def stGamma(x:Double):Double=math.sqrt(2*math.Pi/x)*math.pow((x/math.E), x)
def laGamma(x:Double):Double={
val p=Seq(676.5203681218851, -1259.1392167224028, 771.32342877765313,
-176.61502916214059, 12.507343278686905, -0.13857109526572012,
9.9843695780195716e-6, 1.5056327351493116e-7)
if(x < 0.5) {
math.Pi/(math.sin(math.Pi*x)*laGamma(1-x))
} else {
val x2=x-1
val t=x2+7+0.5
val a=p.zipWithIndex.foldLeft(0.99999999999980993)((r,v) => r+v._1/(x2+v._2+1))
math.sqrt(2*math.Pi)*math.pow(t, x2+0.5)*math.exp(-t)*a
}
}
def main(args: Array[String]): Unit = {
println("Gamma Stirling Lanczos")
for(x <- 0.1 to 2.0 by 0.1)
println("%.1f -> %.16f %.16f".formatLocal(ENGLISH, x, stGamma(x), laGamma(x)))
}
}</lang>
- Output:
Gamma Stirling Lanczos 0.1 -> 5.6971871489771690 9.5135076986687340 0.2 -> 3.3259984240223925 4.5908437119988030 0.3 -> 2.3625300362696198 2.9915689876875904 0.4 -> 1.8414763359362354 2.2181595437576870 0.5 -> 1.5203469010662807 1.7724538509055159 0.6 -> 1.3071588574483560 1.4891922488128180 0.7 -> 1.1590532921139200 1.2980553326475577 0.8 -> 1.0533709684256085 1.1642297137253035 0.9 -> 0.9770615078776956 1.0686287021193193 1.0 -> 0.9221370088957892 1.0000000000000002 1.1 -> 0.8834899531687038 0.9513507698668728 1.2 -> 0.8577553353965909 0.9181687423997607 1.3 -> 0.8426782594483921 0.8974706963062777 1.4 -> 0.8367445486370817 0.8872638175030760 1.5 -> 0.8389565525264964 0.8862269254527583 1.6 -> 0.8486932421525738 0.8935153492876904 1.7 -> 0.8656214717938840 0.9086387328532912 1.8 -> 0.8896396352879945 0.9313837709802430 1.9 -> 0.9208427218942294 0.9617658319073875 2.0 -> 0.9595021757444918 1.0000000000000010
Scheme
for Lanczos and Stirling
for Taylor
<lang scheme> (import (scheme base)
(scheme inexact)
(scheme write))
(define PI 3.14159265358979323846264338327950) (define e 2.7182818284590452353602875)
(define gamma-lanczos
(let ((p '(676.5203681218851 -1259.1392167224028 771.32342877765313
-176.61502916214059 12.507343278686905 -0.13857109526572012
9.9843695780195716e-6 1.5056327351493116e-7)))
(lambda (x)
(if (< x 0.5)
(/ PI (* (sin (* PI x)) (gamma-lanczos (- 1 x))))
(let* ((x2 (- x 1))
(t (+ x2 7 0.5))
(a (do ((ps p (cdr ps))
(idx 0 (+ 1 idx))
(res 0.99999999999980993 (+ res
(/ (car ps)
(+ x2 idx 1)))))
((null? ps) res))))
(* (sqrt (* 2 PI)) (expt t (+ x2 0.5)) (exp (- t)) a))))))
(define (gamma-stirling x)
(* (sqrt (* 2 (/ PI x))) (expt (/ x e) x)))
(define gamma-taylor
(let ((a (reverse
'(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))))
(lambda (x)
(let ((y (- x 1)))
(do ((as a (cdr as))
(res 0 (+ (car as) (* res y))))
((null? as) (/ 1 res)))))))
(do ((i 0.1 (+ i 0.1)))
((> i 2.01) )
(display (string-append "Gamma ("
(number->string i)
"): "
"\n --- Lanczos : "
(number->string (gamma-lanczos i))
"\n --- Stirling: "
(number->string (gamma-stirling i))
"\n --- Taylor : "
(number->string (gamma-taylor i))
"\n")))
</lang>
- Output:
Gamma (0.1): --- Lanczos : 9.513507698668736 --- Stirling: 5.69718714897717 --- Taylor : 9.513507698668734 Gamma (0.2): --- Lanczos : 4.590843711998803 --- Stirling: 3.3259984240223925 --- Taylor : 4.5908437119988035 Gamma (0.30000000000000004): --- Lanczos : 2.9915689876875904 --- Stirling: 2.3625300362696198 --- Taylor : 2.991568987687591 Gamma (0.4): --- Lanczos : 2.218159543757687 --- Stirling: 1.8414763359362354 --- Taylor : 2.2181595437576886 Gamma (0.5): --- Lanczos : 1.7724538509055159 --- Stirling: 1.5203469010662807 --- Taylor : 1.772453850905516 Gamma (0.6): --- Lanczos : 1.489192248812818 --- Stirling: 1.307158857448356 --- Taylor : 1.489192248812817 Gamma (0.7): --- Lanczos : 1.2980553326475577 --- Stirling: 1.15905329211392 --- Taylor : 1.298055332647558 Gamma (0.7999999999999999): --- Lanczos : 1.1642297137253035 --- Stirling: 1.0533709684256085 --- Taylor : 1.1642297137253033 Gamma (0.8999999999999999): --- Lanczos : 1.0686287021193193 --- Stirling: 0.9770615078776956 --- Taylor : 1.0686287021193195 Gamma (0.9999999999999999): --- Lanczos : 1.0000000000000002 --- Stirling: 0.9221370088957892 --- Taylor : 1.0000000000000002 Gamma (1.0999999999999999): --- Lanczos : 0.9513507698668728 --- Stirling: 0.8834899531687039 --- Taylor : 0.9513507698668733 Gamma (1.2): --- Lanczos : 0.9181687423997607 --- Stirling: 0.8577553353965909 --- Taylor : 0.9181687423997608 Gamma (1.3): --- Lanczos : 0.8974706963062777 --- Stirling: 0.842678259448392 --- Taylor : 0.8974706963062773 Gamma (1.4000000000000001): --- Lanczos : 0.8872638175030759 --- Stirling: 0.8367445486370818 --- Taylor : 0.8872638175030753 Gamma (1.5000000000000002): --- Lanczos : 0.8862269254527583 --- Stirling: 0.8389565525264964 --- Taylor : 0.886226925452758 Gamma (1.6000000000000003): --- Lanczos : 0.8935153492876904 --- Stirling: 0.8486932421525738 --- Taylor : 0.8935153492876904 Gamma (1.7000000000000004): --- Lanczos : 0.9086387328532912 --- Stirling: 0.865621471793884 --- Taylor : 0.9086387328532904 Gamma (1.8000000000000005): --- Lanczos : 0.931383770980243 --- Stirling: 0.8896396352879945 --- Taylor : 0.9313837709802427 Gamma (1.9000000000000006): --- Lanczos : 0.9617658319073875 --- Stirling: 0.9208427218942294 --- Taylor : 0.9617658319073876 Gamma (2.0000000000000004): --- Lanczos : 1.000000000000001 --- Stirling: 0.9595021757444918 --- Taylor : 1.0000000000000002
Scilab
<lang>function x=gammal(z) // Lanczos'
lz=[ 1.000000000190015 ..
76.18009172947146 -86.50532032941677 24.01409824083091 ..
-1.231739572450155 1.208650973866179e-3 -5.395239384953129e-6 ]
if z < 0.5 then
x=%pi/sin(%pi*z)-gammal(1-z)
else
z=z-1.0
b=z+5.5
a=lz(1)
for i=1:6
a=a+(lz(i+1)/(z+i))
end
x=exp((log(sqrt(2*%pi))+log(a)-b)+log(b)*(z+0.5))
end
endfunction
printf("%4s %-9s %-9s\n","x","gamma(x)","gammal(x)") for i=1:30
x=i/10
printf("%4.1f %9f %9f\n",x,gamma(x),gammal(x))
end</lang>
- Output:
x gamma(x) gammal(x) 0.1 9.097779 9.097779 0.2 4.180567 4.180567 0.3 2.585167 2.585167 0.4 1.814074 1.814074 0.5 1.772454 1.772454 0.6 1.489192 1.489192 0.7 1.298055 1.298055 0.8 1.164230 1.164230 0.9 1.068629 1.068629 1.0 1.000000 1.000000 1.1 0.951351 0.951351 1.2 0.918169 0.918169 1.3 0.897471 0.897471 1.4 0.887264 0.887264 1.5 0.886227 0.886227 1.6 0.893515 0.893515 1.7 0.908639 0.908639 1.8 0.931384 0.931384 1.9 0.961766 0.961766 2.0 1.000000 1.000000 2.1 1.046486 1.046486 2.2 1.101802 1.101802 2.3 1.166712 1.166712 2.4 1.242169 1.242169 2.5 1.329340 1.329340 2.6 1.429625 1.429625 2.7 1.544686 1.544686 2.8 1.676491 1.676491 2.9 1.827355 1.827355 3.0 2.000000 2.000000
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "float.s7i";
const func float: gamma (in float: X) is func
result
var float: result is 0.0;
local
const array float: A is [] (
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);
var float: Y is 0.0;
var float: Sum is A[maxIdx(A)];
var integer: N is 0;
begin
Y := X - 1.0;
for N range pred(maxIdx(A)) downto minIdx(A) do
Sum := Sum * Y + A[N];
end for;
result := 1.0 / Sum;
end func;
const proc: main is func
local
var integer: I is 0;
begin
for I range 1 to 10 do
writeln((gamma(flt(I) / 3.0)) digits 15);
end for;
end func;</lang>
- Output:
2.678937911987305 1.354117870330811 1.000000000000000 0.892979443073273 0.902745306491852 1.000000000000000 1.190639257431030 1.504575252532959 1.999999523162842 2.778157949447632
Sidef
<lang ruby>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))
}</lang>
- Output:
2.67893853470775e+00 1.35411793942640e+00 1.00000000000000e+00 8.92979511569249e-01 9.02745292950934e-01 1.00000000000000e+00 1.19063934875900e+00 1.50457548825154e+00 1.99999999999397e+00 2.77815847933858e+00
Lanczos approximation: <lang ruby>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 = 0 const pi = Num.pi
if (z.re < 0.5) {
result = (pi / (sin(pi * z) * gamma(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.re : result
} for i in 1..10 {
say ("%.14e" % gamma(i/3))
}</lang>
- Output:
2.67893853470774e+00 1.35411793942640e+00 1.00000000000000e+00 8.92979511569252e-01 9.02745292950931e-01 1.00000000000000e+00 1.19063934875900e+00 1.50457548825155e+00 2.00000000000000e+00 2.77815848043767e+00
A simpler implementation: <lang ruby>define ℯ = Num.e define τ = Num.tau func Γ(t) {
t < 20 ? (__FUNC__(t + 1) / t)
: (sqrt(τ*t) * pow(t/ℯ + 1/(12*ℯ*t), t) / t)
} for i in (1..10) {
say ("%.14e" % Γ(i/3))
}</lang>
- Output:
2.67893831294932e+00 1.35411783267848e+00 9.99999913007168e-01 8.92979437649773e-01 9.02745221785653e-01 9.99999913007168e-01 1.19063925019970e+00 1.50457536964275e+00 1.99999982601434e+00 2.77815825046596e+00
Stata
This implementation uses the Taylor expansion of 1/gamma(1+x). The coefficients were computed with Python and mpmath (see below). The results are compared to Mata's gamma function for each real between 1/100 and 100, by steps of 1/100.
<lang stata>mata _gamma_coef = 1.0,
5.772156649015328606065121e-1,
-6.558780715202538810770195e-1, -4.200263503409523552900393e-2,
1.665386113822914895017008e-1,
-4.219773455554433674820830e-2, -9.621971527876973562114922e-3,
7.218943246663099542395010e-3,
-1.165167591859065112113971e-3, -2.152416741149509728157300e-4,
1.280502823881161861531986e-4,
-2.013485478078823865568939e-5, -1.250493482142670657345359e-6,
1.133027231981695882374130e-6,
-2.056338416977607103450154e-7,
6.116095104481415817862499e-9, 5.002007644469222930055665e-9,
-1.181274570487020144588127e-9,
1.04342671169110051049154e-10, 7.782263439905071254049937e-12,
-3.696805618642205708187816e-12,
5.100370287454475979015481e-13,
-2.05832605356650678322243e-14, -5.348122539423017982370017e-15,
1.226778628238260790158894e-15,
-1.181259301697458769513765e-16,
1.186692254751600332579777e-18, 1.412380655318031781555804e-18,
-2.298745684435370206592479e-19,
1.714406321927337433383963e-20
function gamma_(x_) {
external _gamma_coef x = x_ y = 1 while (x<0.5) y = y/x++ while (x>1.5) y = --x*y z = _gamma_coef[30] x-- for (i=29; i>=1; i--) z = z*x+_gamma_coef[i] return(y/z)
}
function map(f,a) {
n = rows(a)
p = cols(a)
b = J(n,p,.)
for (i=1; i<=n; i++) {
for (j=1; j<=p; j++) {
b[i,j] = (*f)(a[i,j])
}
}
return(b)
}
x=(1::1000)/100 u=map(&gamma(),x) v=map(&gamma_(),x) max(abs((v-u):/u)) end</lang>
Output
9.80341e-15
Here follows the Python program to compute coefficients.
<lang python>from mpmath import mp
mp.dps = 50
def gamma_coef(n):
a = [mp.mpf(1), mp.mpf(mp.euler)]
for k in range(3, n + 1):
s = sum((-1)**j * mp.zeta(j) * a[k - j - 1] for j in range(2, k))
a.append((s - a[1] * a[k - 2]) / (1 - k * a[0]))
return a
def horner(a, x):
y = 0
for s in reversed(a):
y = y * x + s
return y
gc = gamma_coef(30)
def gamma_approx(x):
y = mp.mpf(1)
while x < 0.5:
y /= x
x += 1
while x > 1.5:
x -= 1
y *= x
return y / horner(gc, x - 1)
for x in gc:
print(mp.nstr(x, 25))</lang>
Tcl
<lang tcl>package require math package require math::calculus
- gamma(1) and gamma(1.5)
set f 1.0 set f2 [expr {sqrt(acos(-1.))/2.}]
for {set x 1.0} {$x <= 10.0} {set x [expr {$x + 0.5}]} {
# method 1 - numerical integration, Romberg's method, special # case for an improper integral
set g1 [math::calculus::romberg \
[list apply {{x t} {expr {$t ** ($x-1) * exp(-$t)}}} $x] \
0 1 -relerror 1e-8]
set g2 [math::calculus::romberg_infinity \
[list apply {{x t} {expr {$t ** ($x-1) * exp(-$t)}}} $x] \
1 Inf -relerror 1e-8]
set gamma [expr {[lindex $g1 0] + [lindex $g2 0]}]
# method 2 - library function
set libgamma [expr {exp([math::ln_Gamma $x])}]
# method 3 - special forms for integer and half-integer arguments
if {$x == entier($x)} {
puts [format {%4.1f %13.6f %13.6f %13.6f} $x $gamma $libgamma $f]
set f [expr $f * $x]
} else {
puts [format {%4.1f %13.6f %13.6f %13.6f} $x $gamma $libgamma $f2]
set f2 [expr $f2 * $x]
}
}</lang>
- Output:
1.0 1.000000 1.000000 1.000000 1.5 0.886228 0.886227 0.886227 2.0 1.000000 1.000000 1.000000 2.5 1.329340 1.329340 1.329340 3.0 2.000000 2.000000 2.000000 3.5 3.323351 3.323351 3.323351 4.0 6.000000 6.000000 6.000000 4.5 11.631731 11.631728 11.631728 5.0 24.000009 24.000000 24.000000 5.5 52.342778 52.342778 52.342778 6.0 120.000000 120.000000 120.000000 6.5 287.885278 287.885278 287.885278 7.0 720.000001 720.000000 720.000000 7.5 1871.254311 1871.254305 1871.254306 8.0 5040.000032 5039.999999 5040.000000 8.5 14034.298267 14034.407291 14034.407293 9.0 40320.000705 40319.999992 40320.000000 9.5 119292.464880 119292.461971 119292.461995 10.0 362880.010950 362879.999927 362880.000000
TI-83 BASIC
There is an hidden Gamma function in TI-83. Factorial (!) is implemented in increments of 0.5 :
.5! -> .8862269255
As far as Gamma(n)=(n-1)! , we have everything needed.
Stirling's approximation
<lang ti83b>for(I,1,10) I/2→X X^(X-1/2)e^(-X)√(2π)→Y Disp X,(X-1)!,Y Pause End</lang>
- Output:
The output display for x=0.5 to 5 by 0.5 : x, (x-1)!, Y(x) . Y(x) is Stirling's approximation of Gamma.
0.5
1.772453851
1.520346901
1
1
.9221370089
1.5
.8862269255
.8389565525
2
1
.9595021757
2.5
1.329340388
1.285978615
3
2
1.945403197
3.5
3.32335097
3.245363748
4
6
5.876543783
4.5
11.6317284
11.41865156
5
24
23.60383359
Lanczos' approximation
<lang ti83b>for(I,1,10) I/2→X e^(ln((1.0 +76.18009173/(X+1) -86.50532033/(X+2) +24.01409824/(X+3) -1.231739572/(X+4) +1.208650974E-3/(X+5) -5.395239385E-6/(X+6) )√(2π)/X) +(X+.5)ln(X+5.5)-X-5.5)->Y Disp X,(X-1)!,Y Pause End</lang>
- Output:
The output display for x=0.5 to 5 by 0.5 : x, (x-1)!, Y(x) . Y(x) is Lanczos's approximation of Gamma.
0.5
1.772453851
1.772453851
1
1
1
1.5
.8862269255
.8862269254
2
1
1
2.5
1.329340388
1.329340388
3
2
2
3.5
3.32335097
3.32335097
4
6
6
4.5
11.6317284
11.6317284
5
24
24
Visual FoxPro
Translation of BBC Basic but with OOP extensions. Also some ideas from Numerical Methods (Press et al). <lang vfp> 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 </lang>
- Output:
zval gamval 0.1 9.513507698669704 0.2 4.590843712000122 0.3 2.991568987689402 0.4 2.218159543760185 0.5 1.772453850902053 0.6 1.489192248811141 0.7 1.298055332646772 0.8 1.164229713724969 0.9 1.068628702119210 1.0 1.000000000000000 1.1 0.951350769866919 1.2 0.918168742399821 1.3 0.897470696306335 1.4 0.887263817503125 1.5 0.886226925452796 1.6 0.893515349287718 1.7 0.908638732853309 1.8 0.931383770980253 1.9 0.961765831907391 2.0 1.000000000000000
Wren
The gamma method in the Math class is based on the Lanczos approximation. <lang ecmascript>import "/fmt" for Fmt import "/math" for Math
var stirling = Fn.new { |x| (2 * Num.pi / x).sqrt * (x / Math.e).pow(x) }
System.print(" x\tStirling\t\tLanczos\n") for (i in 1..20) {
var d = i / 10
System.write("%(Fmt.f(4, d, 2))\t")
System.write("%(Fmt.f(16, stirling.call(d), 14))\t")
System.print("%(Fmt.f(16, Math.gamma(d), 14))")
}</lang>
- Output:
x Stirling Lanczos 0.10 5.69718714897717 9.51350769866875 0.20 3.32599842402239 4.59084371199881 0.30 2.36253003626962 2.99156898768760 0.40 1.84147633593624 2.21815954375769 0.50 1.52034690106628 1.77245385090552 0.60 1.30715885744836 1.48919224881282 0.70 1.15905329211392 1.29805533264756 0.80 1.05337096842561 1.16422971372530 0.90 0.97706150787770 1.06862870211932 1.00 0.92213700889579 1.00000000000000 1.10 0.88348995316870 0.95135076986687 1.20 0.85775533539659 0.91816874239976 1.30 0.84267825944839 0.89747069630628 1.40 0.83674454863708 0.88726381750308 1.50 0.83895655252650 0.88622692545276 1.60 0.84869324215257 0.89351534928769 1.70 0.86562147179388 0.90863873285329 1.80 0.88963963528799 0.93138377098024 1.90 0.92084272189423 0.96176583190739 2.00 0.95950217574449 1.00000000000000
Yabasic
<lang Yabasic>dim c(12)
sub gamma(z)
local accm, k, k1_factrl
accm = c(1)
if accm=0 then
accm = sqrt(2*PI)
c(1) = accm
k1_factrl = 1
for k=2 to 12
c(k) = exp(13-k)*(13-k)^(k-1.5)/k1_factrl
k1_factrl = k1_factrl * -(k-1)
next
end if
for k=2 to 12
accm = accm + c(k)/(z+k-1)
next
accm = accm * exp(-(z+12))*(z+12)^(z+0.5)
return accm/z
end sub
sub si(x)
print x using "%18.13f"
end sub
for i = 0.1 to 2.1 step .1
print i, " = "; : si(gamma(i))
next</lang>
zkl
but without a built in gamma function.
<lang zkl>fcn taylorGamma(x){
var table = T(
0x1p+0, 0x1.2788cfc6fb618f4cp-1,
-0x1.4fcf4026afa2dcecp-1, -0x1.5815e8fa27047c8cp-5,
0x1.5512320b43fbe5dep-3, -0x1.59af103c340927bep-5,
-0x1.3b4af28483e214e4p-7, 0x1.d919c527f60b195ap-8,
-0x1.317112ce3a2a7bd2p-10, -0x1.c364fe6f1563ce9cp-13,
0x1.0c8a78cd9f9d1a78p-13, -0x1.51ce8af47eabdfdcp-16,
-0x1.4fad41fc34fbb2p-20, 0x1.302509dbc0de2c82p-20,
-0x1.b9986666c225d1d4p-23, 0x1.a44b7ba22d628acap-28,
0x1.57bc3fc384333fb2p-28, -0x1.44b4cedca388f7c6p-30,
0x1.cae7675c18606c6p-34, 0x1.11d065bfaf06745ap-37,
-0x1.0423bac8ca3faaa4p-38, 0x1.1f20151323cd0392p-41,
-0x1.72cb88ea5ae6e778p-46, -0x1.815f72a05f16f348p-48,
0x1.6198491a83bccbep-50, -0x1.10613dde57a88bd6p-53,
0x1.5e3fee81de0e9c84p-60, 0x1.a0dc770fb8a499b6p-60,
-0x1.0f635344a29e9f8ep-62, 0x1.43d79a4b90ce8044p-66).reverse();
y := x.toFloat() - 1.0;
sm := table[1,*].reduce('wrap(sm,an){ sm * y + an },table[0]);
return(1.0 / sm);
}</lang> <lang zkl>fcn lanczosGamma(z) { z = z.toFloat();
// Coefficients used by the GNU Scientific Library. // http://en.wikipedia.org/wiki/Lanczos_approximation const g = 7, PI = (0.0).pi; exp := (0.0).e.pow; var table = T( 0.99999_99999_99809_93, 676.52036_81218_851, -1259.13921_67224_028, 771.32342_87776_5313, -176.61502_91621_4059, 12.50734_32786_86905, -0.13857_10952_65720_12, 9.98436_95780_19571_6e-6, 1.50563_27351_49311_6e-7); // Reflection formula. if (z < 0.5) { return(PI / ((PI * z).sin() * lanczosGamma(1.0 - z))); } else { z -= 1; x := table[0]; foreach i in ([1 .. g + 1]){ x += table[i] / (z + i); } t := z + g + 0.5; return((2.0 * PI).sqrt() * t.pow(z + 0.5) * exp(-t) * x); }
}</lang>
- Output:
foreach i in ([1.0 .. 10]) {
x := i / 3.0;
println("%f: %20.19e %20.19e %e".fmt( x,
a:=taylorGamma(x), b:=lanczosGamma(x),(a-b).abs()));
}
0.333333: 2.6789385347077483424e+00 2.6789385347077474542e+00 8.881784e-16 0.666667: 1.3541179394264004632e+00 1.3541179394264002411e+00 2.220446e-16 1.000000: 1.0000000000000000000e+00 1.0000000000000002220e+00 2.220446e-16 1.333333: 8.9297951156924926241e-01 8.9297951156924970650e-01 4.440892e-16 1.666667: 9.0274529295093364212e-01 9.0274529295093353110e-01 1.110223e-16 2.000000: 1.0000000000000000000e+00 1.0000000000000006661e+00 6.661338e-16 2.333333: 1.1906393487589990166e+00 1.1906393487589996827e+00 6.661338e-16 2.666667: 1.5045754882515545159e+00 1.5045754882515582906e+00 3.774758e-15 3.000000: 1.9999999999992210675e+00 2.0000000000000017764e+00 7.807088e-13 3.333333: 2.7781584802531797962e+00 2.7781584804376668885e+00 1.844871e-10
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