Saturday, January 14, 2023

Swiss Micros DM41X: Infinite Integrals by Gaussian Quadrature

Swiss Micros DM41X:   Infinite Integrals by Gaussian Quadrature



Introduction


The program INFGAUS calculates the integral:


∫ e^-x * f(x) dx from x = a to x = ∞


f(x) is the subroutine FX.   The subroutine starts with the x value on the stack and ends with the RTN command.


The HP 45 algorithm, which is incorporated into the program INFGAUS estimates the integral by the sum


e^-a * Σ(( w_i * f(z_i + a)) for i=1 to 3)



where 


w_1 = 0.71109390099

z_1 = 0.4157745568


w_2 = 0.2785177336

z_2 = 2.29428036


w_3 = 0.0103892565

z_3 = 6.289945083




DM41X Program:  INFGAUS


The code can work is for the entire HP 41C/DM41 family.  The calculator is set to Fix 2 mode.  Registers 01 through 08 are needed.  


01  LBL^T INFGAUS

02  .4157745568

03  STO 01

04  2.29428036

05  STO 02

06  6.289945083

07  STO 03

08  .7110930099

09  STO 04

10  .278517736

11  STO 05

12  .0103892565

13  STO 06

14  FIX 2

15  ^T A?

16  PROMPT

17  STO 07

18  RCL 01

19  +

20  XEQ^T FX

21  RCL 04

22  *

23  STO 08

24  RCL 07

25  RCL 02

26  +

27  XEQ^T FX

28  RCL 05

29  *

30  ST+ 08

31  RCL 03

32  RCL 07

33  +

34  XEQ^T FX

35  RCL 06

36  *

37  RCL 08

38  +

39  RCL 07

40  CHS

41  E↑X

42  *

43  STO 08

44  END


Examples


Example 1


∫ e^-x * x^3.99 dx from x = 0 to ∞   (calculate Γ(4.99))


LBL^T FX

3.99

Y↑X

RTN


A? 0

Result:  23.64



Example 2


∫ e^-x * x^2 ÷ (x - 1)  dx from x = 2 to ∞   


LBL^T FX

X↑2

LASTx

1

-

/

RTN


A?  2

Result:  0.62



Example 3


∫ (e^-x)^2 dx from x = 0 to ∞   

= ∫ (e^-x) * (e^-x) dx from x = 0 to ∞   



LBL^T FX

CHS

E↑X

RTN


A?  0

Result:  0.50  (it turns out 0.5 is the exact answer)




Source


HP-45 Applications Handbook  Hewlett Packard Company.  1974.



Remember, my regular posting schedule for January and February 2023 is on Saturdays only. 


Eddie 


All original content copyright, © 2011-2023.  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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