In this case a logical and operation should be

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Unformatted text preview: alf of the multiply operations), this negative bit must be handled. One strategy to overcome this is to use a bit mask to mask o the most signi cant bit. In this case, a logical AND operation should be performed on the random, negative integer and the bit mask, imask=z'7fffffff', which is a leading 0 followed by 31 ones (i.e. 011111111111111111111111111111112 ). This has the e ect of the zeroing out the sign bit, forcing the number to be positive. Viz., Remarks Table 3: Random Sequence of Example 3 - LCG(5 0 37 1) \Random" \Random" \Random" n Integer - Xn Binary - Xn Real - Xn 0 1 000001 0.0270 1 5 000101 0.1351 2 25 011001 0.6757 3 14 001110 0.3784 4 33 100001 0.8919 5 17 010001 0.4595 6 11 001011 0.2973 7 18 010010 0.4865 8 16 010000 0.4324 9 6 000110 0.1622 10 30 011110 0.8108 11 2 000010 0.0541 12 10 001010 0.2703 13 13 001101 0.3514 14 28 011100 0.7568 15 29 011101 0.7838 16 34 100010 0.9189 17 22 010110 0.5946 18 36 100100 0.9730 19 32 100000 0.8649 20 12 001100 0.3243 21 23 010111 0.6216 22 4 000100 0.1081 23 20 010100 0.5405 24 26 011010 0.7027 25 19 010011 0.5135 26 21 010101 0.5676 27 31 011111 0.8378 28 7 000111 0.1892 29 35 100011 0.9459 30 27 011011 0.7297 31 24 011000 0.6486 32 9 001001 0.2432 17 18 1 0 1 1 0 1 1 0 0 1 Xn = 730 mod IAND 0 0 0 0 1 0 1 1 1 1 MASK = 31 (31 = 25 - 1) 1 1 0 0 1 Xn = 25 = 0 0 0 0 0 Figure 5: Modulus Operation with m = 25 using IAND xn = iand(xn, imask) However, masking works only when m is a power of two. Otherwise, it destroys the sequence of numbers generated, and the theory no longer applies. In the general case, unless the number resulting from the multiplication aXn ts in the register, special coding must be done. An approach to this is to perform remaindering, by decomposing the multiply operation into steps in which intermediate results are contained in 32 bits. This is the strategy used by Park and Miller Park and Miller, 1988]. Note that, in the language C, one can avoid the issue of the sign bit by using an unsigned integer. 3.3 Using Logical Masks to Perform Modulo 2M Operations Generators where the modulus operation must be performed (requiring an integer division) are more costly to implement than are those with moduli of 2M , where the integer division and remaindering can be accomplished much more e ciently. We illustrate this as follows. With a divisor of 2M , after the multiplication of aXn , the next seed is obtained simply by performing a logical AND of Xn with a mask of (M ; 1) ones, right-justi ed (see Figure 5). This is an extremely e cient operation on binary computers. Many computer languages have a bitwise AND intrinsic function. Use of the AND operation also avoids the problem with negative integers, discussed above. A FORTRAN implementation would look like: xn = iand(xn, mask) 3.4 Summary of LCG Properties Following Anderson Anderson, 1990], we summarize the salient features and the recommendations for three widely used types of linear, congruential random number generators. Remarks 19 Multiplicative, congruential generators are adequate to good for many applications. They are not acceptable... for high-dimensional work. They can be very good if speed is a major consideration. Prime moduli are best. However, moduli of the form 2N are faster on binary computers. { Anderson (1990) m = 2M c > 0 : The full period of m = 2M is obtained if and only if a 1 (mod 4), and c is odd (often chosen as 1). Low-order bits are not random! m = 2M c = 0 : The maximum period of this generator is 2M ;2 (one-quarter the modulus), and is obtained if and only if a 3 (mod 8) or a 5 (mod 8) (5 is preferred) and the initial seed is odd. Low-order bits are not random! m = p (prime) c = 0 or c 6= 0 : The maximum period of this generator is p ; 1 and is obtained if and only if a is a primitive element modulo p. Note that there is always one integer which maps only to itself. (In the case of c = 0, X0 = 0 maps to itself.) Low-order bits may or may not be random! Park and Miller Park and Miller, 1988] recommend the following, portable generator: m = p = 231 ; 1 (2 147 483 647), a = 16 807, and c = 0. The code random.f, based upon Park and Miller's algorithm, is supplied with this package. Note that, even when c is non-zero, the maximum period of this generator is still one less than the modulus. Exercise 4 - to establish all cycles in LCGs when m = p, a prime Consider linear, congruential random number generators. If c = 0, it is obvious that X0 = 0 is not a good candidate integer for the initial seed because it maps to itself. In fact, if m = p, a prime, then there is always a number which maps to itself (a constant sequence), even if c 6= 0. Prove this by nding the integer which maps to itself, and which does not appear in the full period sequence of length m ; 1 for the following linear, congruential generators: (a) LCG(5 0 37 X0 ) (b) LCG(5 1 37 X0 ) (c) LCG(5 5 37 X0 ) Exercise 5 - exploring all possible sequences in simple LCGs For the linear, congruential genera...
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