Combined Linear Congruential Generator

From Wikipedia, the free encyclopedia
Jump to: navigation, search

A Combined Linear Congruential Generator (CLCG) is a pseudo-random number generator algorithm based on combining two or more linear congruential generators (LCG). A traditional LCG has a period which is inadequate for complex system simulation.[1] By combining two or more LCGs, random numbers with a longer period and better statistical properties can be created.[1] The algorithm is defined as:[2]

where:

— the "modulus" of the first LCG
— the ith input from the jth LCG
— the ith generated random integer

with:

where is a uniformly distributed random number between 0 and 1.

Derivation[edit]

If Wi,1, Wi,2, ..., Wi,k are any independent, discrete, random-variables and one of them is uniformly distributed from 0 to m1 − 2, then Zi is uniformly distributed between 0 and m1 − 2, where:[2]

Let Xi,1, Xi,2, ..., Xi,k be outputs from k LCGs. If Wi,j is defined as Xi,j − 1, then Wi,j will be approximately uniformly distributed from 0 to mj − 1.[2] The coefficient "(−1)j−1" implicitly performs the subtraction of one from Xi,j.[1]

Properties[edit]

The CLCG provides an efficient way to calculate pseudo-random numbers. The LCG algorithm is computationally inexpensive to use.[3] The results of multiple LCG algorithms are combined through the CLCG algorithm to create pseudo-random numbers with a longer period than is achievable with the LCG method by itself.[3]

The period of a CLCG is dependent on the seed value used to initiate the algorithm. The maximum period of a CLCG is defined by the function:[1]

Example[edit]

The following is an example algorithm designed for use in 32 bit computers:[2]

LCGs are used with the following properties:

The CLCG algorithm is set up as follows:

1. The seed for the first LCG, , should be selected in the range of [1, 2147483562].

The seed for the second LCG, , should be selected in the range of [1, 2147483398].
Set:

2. The two LCGs are evaluated as follows:

3. The CLCG equation is solved as shown below:

4. Calculate the random number:

5. Increment the counter (i=i+1) then return to step 2 and repeat.

The maximum period of the two LCGs used is calculated using the formula:.[1]

This equates to 2.1x109 for the two LCGs used.

This CLCG shown in this example has a maximum period of:

This represents a tremendous improvement over the period of the individual LCGs. It can be seen that the combined method increases the period by 9 orders of magnitude.

Surprisingly the period of this CLCG may not be sufficient for all applications:.[1] Other algorithms using the CLCG method have been used to create pseudo-random number generators with periods as long as 3x1057.[4][5][6]

See also[edit]

References[edit]

  1. ^ a b c d e f Banks 2010, Sec. 7.3.2
  2. ^ a b c d L'Ecuyer 1988
  3. ^ a b Pandey 2008, Sec. 2.2
  4. ^ L'Ecuyer 1996
  5. ^ L'Ecuyer 1999
  6. ^ L'Ecuyer 2002

External links[edit]