Unformatted text preview: scale" or granularity of our problem  the RNG is incapable of producing random
numbers which ll our attribute space (here, our 8 bins). There are two separate
issues to consider here. The least troublesome is the nite precision existing in all
computers, which results in a roundo error to the precision with which integers can
be represented, or with which the oating point divide is accomplished. More serious is
the second issue: the interaction of the sequence of random numbers produced by our
generator with our application. This interaction is particularly troublesome when our
application requires ntuples of random numbers, instead of just one random number
at a time. Later, we shall illustrate this e ect by displaying plots of individual points
generated using various RNGs. In some applications, several random numbers are
needed at a time. For example, when tracing particles among surfaces, three random
numbers are required to obtain outgoing direction (one for outgoing azimuthal angle,
and one for outgoing polar angle) and energy (usually, only one is required to obtain Remarks 15 Table 2: Random Sequence of Example 2  LCG(5 0 16 1)
\Random" \Random" \Random"
n
Integer  Xn Binary  Xn Real  Xn
0
1
0001
0.0625
1
5
0101
0.3125
2
9
1001
0.5625
3
13
1101
0.8125
Average
0.4375
Variance
0.0781
the energy). Example 3 LCG (5, 0, 37, 1) We obtain the sequence
1,5,25,14,33,17,11,18,16,6,30,2,10,13,28,29,34,22,
36,32,12,23,4,20,26,19,21,31,7,35,27,24,9,8,3,15, : : :
Table 3 provides the sequence throughout the period. Here, because we use a prime number
as the divisor for the modulus operation and c = 0, we obtain a period one less than modulus
37 (0 is not possible, as it maps to itself, so we obtain a period of 36). Indeed, when m = p,
a prime, the maximum period, Pmax , is m ; 1, even if c 6= 0. Thus, for linear, congruential
generators with a prime modulus, using a nonzero c does not increase the period.
Here, the low order bits, while not exhibiting a discernible pattern, do not appear as
\random" as one might expect. Indeed, as is shown in Altman Altman, 1988], the bitwise
randomnes properties of LCGs should be considered on a case by case basis. He provides
examples of LCGs with prime moduli that fail bitwise testing, but points out, for example,
that LCG(13445 0 231 ; 1 X0 ) does pass the bitwise randomness test. Exercise 3  to illustrate the generation of random numbers using
LCG(13445, 0, 65536, 16811) and to partition the random real
numbers into bins Obtain and edit the linear congruential program ranlc.f. Supply the additional code to
compute the average and the variance. For sample sizes of 10, 100, 1,000, and 10,000, run
the code and plot the errors in the average and variance versus sample size. Discuss how
the error in these quantities varies with the sample size. Use the following parameters:
a = 16807, c = 0, and m = 216(= 65536). 16 3.2 Initial Seed Now, we address establishing the initial seed. When debugging, it is important to implement
the algorithm to reproduce the same stream of random numbers on successive runs. If the
run is a debug run (noted, perhaps, by a parameter in the input le), the seed should be set
to a constant initial value, such as a large prime number (it should be odd, as this will satisfy
period conditions for any modulus). Otherwise, the initial seed should be set to a \random"
odd value. Anderson 1990] recommends setting the initial seed X0 to the following integer: X0 = iyr + 100 (imonth ; 1 + 12 (iday ; 1 + 31 (ihour + 24 (imin + 60 isec))))
where the variables on the righthand side are the integer values of the date and time. Note
that the year is 2 digits long, i.e. the domain of iyr is 0 99]. However, we have found it
preferable to introduce the maximum variation in the seed into the least signi cant bits by
using the second of this century, rather than the most signi cant bits. Ergo, we prefer the
following:
X0 = isec + 60 (imin + 60 (ihr + 24 (iday ; 1 + 31 (imon ; 1 + 12 iyr))))
and, to ensure X0 is odd (in FORTRAN):
x0 = ior(x0 1) Arithmetical Considerations
Over ow and Negative Integers Consider performing the operation aXn + c (mod m). A large value of a is desirable to
provide su cient randomness. A large value of m is also desired, so that the period is
kept long. For example, on 32 bit computers, a and m are often 31 bits long, as the most
signi cant bit (the 32nd bit) is generally used to indicate sign. When two 31 bit integers
are multiplied together, a possibly 62 bit integer results. Thus, an over ow almost always
occurs. Fortunately, oating point multipliers (and software emulations thereof) are designed
to throw away the most signi cant bits, and retain the least signi cant 32 bits.
However, if the result of the multiplication is so as to have the most signi cant bit (the
32nd bit) set, then the computer may treat this as a negative integer, which is incompatible
with the algorithm above. If this happens (it occurs during oneh...
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 Fall '14
 Randomness, random number, Pseudorandom number generator, congruential generators, LCGs

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