hp41programs

Alea Random Number Generators for the HP-41
 

Overview
 

-Several pseudo random number generators are listed on this web page:
-"RNG1" "RNG2" "RNG3" work on every HP-41.
-"RNG4" requires a Time-Module.
-Finally, the last program is an attempt to play ( win? ) the lottery...
 

Program#1
 

-A well known RNG is given by the formula:  xn+1 = FRC ( 9821 xn + 0.211327 ) which provides 1 million random numbers.
-The following program gives 1,000,000,000  random numbers r  ( 0 <= r < 1 ). The formula  xn+1 = FRC ( 98 xn + 0.236067977 ) is used.
-The coefficient 98 = 43,046,721 may be replaced by  a   where  a = 1 ( mod 20 )
                     and  0.236067977  may be replaced by  b  where   b*109 is not divisible by 2 or 5.
 
 

01  LBL "RNG1"
02  9
03  ENTER^
04  ENTER^
05  R^
06  *
07  FRC            
08  *
09  FRC
10  *
11  FRC
12  *
13  FRC            
14  *
15  FRC
16  *
17  FRC
18  *
19  FRC            
20  *
21  FRC
22  5
23  SQRT
24  +
25  FRC            
26  END
 

 
( 35 bytes / SIZE 001 )
 
 

      STACK        INPUTS      OUTPUTS
           X             xn           xn+1

 
Example:

  0.2   XEQ "RNG1"  yields  0.436067977
                                   R/S    0.779021394   ... etc ...
 

Program#2
 

-"RNG2" provides 9,999,999,996 random numbers with the formula:  xn+1 = ( 1059 xn ) MOD p   where p = 9,999,999,967 is the greatest prime < 1010
   xn are integers between 0 and p ( exclusive ) which are then divided by p to be reduced to a number between 0 and 1.
-This routine works well because the MOD function gives exact results even when the operands are greater than 1010.
-Actually, the exponent 59 may be replaced by any integer m provided m is relatively prime to p-1 = 2*3*11*457*331543
  but I don't know what is the best choice.
-Unlike "RNG1" and other routines based upon the same type of formulae, the least significant digits don't go through any cycle of ten, one hundred and so on.
-Register R00 is used to store the different  xn integers.
 
 

 01  LBL "RNG2"
 02  RCL 00
 03   E59
 04  *
 05  10
 06  10^X
 07  33
 08  -
 09  MOD
 10  STO 00
 11  LASTX
 12  /
 13  END

 
( 26 bytes / SIZE 001 )
 
 

      STACK        INPUTS      OUTPUTS
           X             /       0 < r < 1

 
Example:

    1  STO 00
        XEQ "RNG2" gives  0.3129146797   ( and R00 = 3129146787 = 1059 ( mod p ) )
                R/S                   0.6904570204    ( R00 = 6904570181 ) ... etc ...
 

-Actually if p is a prime,  ( Z/pZ-{0} ; * ) is a group
  and if a is an integer,  the number of distinct elements in the subset   { 1 ; a ; a2 ; ....... ; ak ; .... } ( mod p )   divides  p-1
-If  p-1 is the smallest positive integer q such that aq = 1 ( mod p ) , then the sequence   a ; a2 ; ....... ; ak ; .... ; ap-1 ( mod p )
  is a permutation of  1 ; 2 ; ...... ; p-1
-In particular, if  p = 2p' + 1  where p' is also a prime, and if ap' is not equal to 1 ( mod p )  then  a  satisfies the required property.
-For instance,  p = 7,841,296,787 = 2*3,920,648,393 + 1              7,841,296,787 and 3,920,648,393 are primes and  -1024 = 4,851,307,369  ( mod p )
  satisfies  (-1024)p' = -1  therefore the routine:

  E24
  *
  CHS
  7841296787
  MOD                  gives  7,841,296,786  random integers

-These ideas may be used to create your own RNG.
 

Program#3
 

-The following algorithm is given by Clifford Pickover in "Keys to Infinity" ( John Wiley & Sons )  ISBN 0-471-11857-5
 
 

 01  LBL "RNG3"
 02  LN
 03   E2
 04  *
 05  1
 06  MOD
 07  END

 
( 17 bytes / SIZE 001 )
 
 

      STACK        INPUTS      OUTPUTS
           X             xn           xn+1

 
Example:

    0.1   XEQ "RNG3" produces  0.74149070
                   R/S                          0.09073404  ... etc ...
 

Program#4
 
 
 

 01  LBL "RNG4"
 02  DATE
 03  TIME
 04  +
 05   E49
 06  *
 07  PI
 08  MOD
 09  LN1+X
 10  R-D
 11  FRC
 12  END

 
( 25 bytes / SIZE 000 )
 
 

      STACK        INPUTS      OUTPUTS
           X             /       0 < r < 1

 
-I cannot give any example since the result depends on the instant you press R/S
 

Winning the Lottery?
 

-If you need, for instance, 7 integers between 1 and 49 , you can use 7 random numbers between 0 and 1,
  multiply them by 49 , add 1 and take the integer part of the result.
-The small routine hereafter is another possibility, if you accept to calculate the integer part in your mind:
 
 
 

 01  LBL "$$$"
 02  R-D
 03  49
 04  MOD
 05  1
 06  +
 07  END

 
( 16 bytes / SIZE 000 )
 
 

      STACK        INPUTS      OUTPUTS
           X             r      0 < N < 50

 
Example:

   41  XEQ "$$$"  yields   47.1270
              R/S                      6.1759
              R/S                    11.8535
              R/S                    43.1567
              R/S                    23.6963
              R/S                    35.6962
              R/S                    37.2418         suggesting     47-06-11-43-23-35-37
 

Notes:

1-I'm not a statistician and I can't assure all these RNGs would stand up to sophisticated tests, but one may use his imagination in devising variations.
2-If you win one million dollars thanks to one of these programs, I accept to share the jackpot...