Unformatted text preview: a vector length of 64, for example, you will need a vector
of (a a2 a3 : : : a64). Race your vectorized version of ranf() vs. Cray's ranf(), and see 28
Initial Seed
x Left Sequence ● XL,n+1 = aL XL,n + cL L(x) L2(x) ● ● ● L3(x) XR,n+1 = aR XR,n+ cR ● ● ● Right Sequences ● LR(x) RL(x) ● RL2(x) LRL(x) ● ● R2L(x) R(x) L2R(x) ● ● ● R2(x) ● RLR(x) LR2(x) R3(x) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Figure 7: Tree Generated from OLeftO and ORightO Generator
how close you can come to their execution time. To do this, you will have to generate long
vectors of random numbers, and use them elsewhere in the code. Use second() to obtain
elapsed CPU time. Exercise 9: Leapahead in an LCG For the LCG, Xn+1 = aXn + c, where c 6= 0, determine ak and ck so that Xn+k = ak Xn + ck ,
^
^
^
^
resulting in a leapahead of k places on the original LCG cycle. 6 Lagged Fibonacci Generators
Lagged Fibonacci pseudorandom number generators have become increasingly popular in
recent years. These generators are so named because of their similarity to the familiar Remarks 29 Fibonacci sequence:
1 1 2 3 5 8 13 : : : de ned by: Xn = Xn;1 + Xn;2
where the rst two values, X0 and X1, must be supplied. This formula is generalized to give
a family of pseudorandom number generators of the form: Xn = Xn;` + Xn;k (mod m) where ` > k > 0
and where, instead of two initial values, ` initial values, X0, ... , X`;1 , are needed in order
to compute the next sequence element. In this expression the \lags" are k and `, so that
the current value of X is determined by the value of X k places ago and ` places ago. In
addition, for most applications of interest m is a power of two. That is, m = 2M . (This
type of pseudorandom number generator, along with several others, has been extensively
tested for randomness properties by Marsaglia Marsaglia, 1985] and has been given high
marks. The only de ciency found was related to what he calls the Birthday Spacings test,
for low values of ` and k. The interested reader is referred to Marsaglia, 1985.)
With proper choice of k, `, and the rst ` values of X , the period, P , of this generator
is equal to (2` ; 1)
2(M ;1). Proper choice of ` and k here means that the trinomial
x` + xk + 1 is primitive over the integers mod 2. The only condition on the rst ` values is
that at least one of them must be odd. Using the notation LFG(` k M ) to indicate the lags
and the power of two modulus, examples of two commonly used versions of these generators
are:
1) LFG(17 5 31): ` = 17, k = 5, M = 31 ) P 247
2) LFG(55 24 31): ` = 55, k = 24, M = 31 ) P 285
Obviously, the value of the modulus, m, does not by itself limit the period of the generator,
as it does in the case of an LCG. Note also that lagged Fibonacci pseudorandom number
generation is computationally simple: an integer add, a logical AND (to accomplish the mod
2M operation), and the decrementing of two array pointers are the only operations required
to produce a new random number X . Furthermore, with a large enough value of k, limited
vectorization can be achieved. The major drawback in the case of this type of generator is
the fact that ` words of memory must be kept current. An LCG requires only one: the last
value of X generated.
We now look at some of the theory of these generators, with a view toward ensuring their
proper use. Conceptually, a Fibonacci generator acts the same as a linear shift register, and
if we set M = 1 so that m = 21 , then we have a binary linear shift register. Figure 8 is
a diagram of a particular binary linear shift register, where ` = 10, k = 7, and every Xj is
either a 1 or a 0. (We will use this choice of ` and k in several examples in order to illustrate
in a nontrivial way some of the properties of this type of generator.) The arrows are there
to depict the motion of the shift register as it makes the transition from one state to the
next. This is called advancing the register. Two such advances are shown in Figure 9. 30
Addition mod 2 X(n10) X(n9) bit 9 bit 8 X(n8) X(n7) bit 7 bit 6 X(n6) X(n5) bit 5 bit 4 X(n4) X(n3) X(n2) X(n1) bit 3 bit 2 bit 1 bit 0 Figure 8: State Transition Diagram for Binary Shift Register
If we set m equal to a higher power of two, say m = 24, i.e., M = 4, then the singlebit
values of the X 's in Figure 8 will instead be represented by M bits each. Figure 10 is a
diagram of the mod16 linear shift register associated with the equation Xn = Xn;10 + Xn;7 (mod 24)
In looking at this more general Fibonacci generator, LFG(10 7 4), two things are worth
noting:
1. If we look for a moment at only the least signi cant bits of the elements in this register,
that is, only those bits in the bottom row, then we see that the behavior of this row is
una ected by bits in the higher rows. The contents of the bottom row will therefore
be indistinguishable from those of the binary shift register in Figure 8.
This is no surprise  when we add two numbers, the answer in the one's place does
not depend in any way on the dig...
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 Fall '14
 Randomness, random number, Pseudorandom number generator, congruential generators, LCGs

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