Two modulo-9 LCGs show how different parameters lead to different cycle lengths. Numerical recipes in fortran 90 volume 1 pdf row shows the state evolving until it repeats.
0, and a seed of 1, which produces a cycle of length 6. The second row is the same generator with a seed of 3, which produces a cycle of length 2. 9 with any seed in . The method represents one of the oldest and best-known pseudorandom number generator algorithms.
0, the method is called a mixed congruential generator. A benefit of LCGs is that with appropriate choice of parameters, the period is known and long. Although not the only criterion, too short a period is a fatal flaw in a pseudorandom number generator. While LCGs are capable of producing pseudorandom numbers which can pass formal tests for randomness, this is extremely sensitive to the choice of the parameters m and a. Historically, poor choices for a have led to ineffective implementations of LCGs. A particularly illustrative example of this is RANDU, which was widely used in the early 1970s and led to many results which are currently being questioned because of the use of this poor LCG. This is the original Lehmer RNG formulation.
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1 if the multiplier a is chosen to be a primitive element of the integers modulo m. One disadvantage of a prime modulus is that the modular reduction requires a double-width product and an explicit reduction step. If a double-width product is unavailable, and the multiplier is chosen carefully, Schrage’s method may be used. Thus, the difference will lie in the range , and can be reduced to with a single conditional add.
If a prime just less than a power of 2 is used, sometimes the missing values are simply ignored. 264, produces a particularly efficient LCG, because this allows the modulus operation to be computed by simply truncating the binary representation. In fact, the most significant bits are usually not computed at all. The initial state X0 must be odd, and the low three bits of X alternate between are not useful. A more serious issue with the use of a power-of-two modulus is that the low bits have a shorter period than the high bits.
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