Saturday, December 10, 2022

HP 20S: Gamma Function Approximation (Stirling's Formula)

HP 20S:  Gamma Function Approximation (Stirling's Formula)



Introduction


The gamma function uses the approximation sequence:


Let t = x + 9


Then calculate:


Let G = Γ(t) ≈ 

exp( ln √(2 × π ÷ t) + t × ln t - t + (12 × t)^-1 - (360 × t^3)^-1 + (1260 × t^5)^-1 )


Note:

(360 × t^3)^-1  = (12 × t)^-1 × (30 × t^2)^-1

(1260 × t^5)^-1 = (360 × t^3)^-1 × (3.5 × t^2)^-1


While x > t:  

G = G ÷ x

x = x + 1

End Loop


Display G as the final answer 


The approximation polynomial is used for higher values because for the approximation is more accurate for higher values.  



HP 20S:  Gamma Approximation

(63 steps)


Key Code:  { Key }


61, 41, b :  { LBL B }

21, 1 :  { STO 1 }

75 :   { + }

9 :  { 9 }

74 : { = }

21, 2 : { STO 2 }

32 : { +/- }

21, 0 : { STO 0 }

32 : { +/- } 

55 : { × }

13 : { LN }

74 : { = }

21, 75, 0 : { STO+ 0 }

2 :  { 2 }

55 : { × }

61, 22 : { π }

45 : { ÷ }

22, 2 : { RCL 2 }

74 : { = }

11 : { √ }

13 : { LN }

21, 75, 0 : { STO+ 0 }

1 : { 1 }

2 : { 2 }

55 : { × }

22, 2 : { RCL 2 }

74 : { = }

15 : { 1/x }

21, 75, 0 : { STO+ 0 }

45 : { ÷ }

3 : { 3 }

0 : { 0 }

45 : { ÷ }

22, 2 : { RCL 2 }

51, 11 :  { x^2 }

74 : { = }

21, 65, 0 : { STO- 0 }

45 : { ÷ }

3 : { 3 }

73 : { . }

5 : { 5 }

45 : { ÷ }

22, 2 : { RCL 2 }

51, 11 : { x^2 }

74 : { = }

21, 75, 0 : { STO+ 0 }

22, 0 : { RCL 0 }

12 : { e^x }

21, 0 : { STO 0 }

61, 41, 0 : { LBL 0 }

22, 1 : { RCL 1 }

21, 45, 0 : { STO÷ 0 }

1 : { 1 }

21, 75, 1 : { STO+ 1 }

22, 2 : { RCL 2 }

31 : { INPUT }

22, 1 : { RCL 1 }

61, 42 : { x≤y? }

51, 41, 1 : { GTO 1 }

51, 41, 0 : { GTO 0 }

61, 41, 1 : { LBL 1 }

22, 0 : { RCL 0 }

61, 26 : { RTN }



Examples


Γ(0.5) returns 1.77245385109


Γ(4.4) returns 10.1361018514


Calculate the gamma function, press [ XEQ ] B. 


This program is based on the approximation code of the HP 25.  



Source:


Davidson, Jim, and John Vlissides.  "HP-25 Program-Gamma Function"  ENTER: 65 NOTES  Vol. 3 No. 10  December 1976



All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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