- The linear congruential method produces a sequence of
integers
between zero and
*m-1*according to the following recursive relationship:

- The initial value is called the seed;
*a*is called the constant multiplier;*c*is the increment*m*is the modulus

*a, c, m*and drastically affects the statistical properties such as mean and variance, and the cycle length. - When , the form is called the
*mixed congruential method*; When*c = 0*, the form is known as the*multiplicative congruential method*. - Example 8.1 on page 292
- Issues to consider:
- The numbers generated from the example can only assume
values from the set
*I = {0, 1/m, 2/m, ..., (m-1)/m}*. If*m*is very large, it is of less problem. Values of and are in common use. - To achieve maximum density for a given range, proper
choice of
*a, c, m*and is very important. Maximal period can be achieved by some proven selection of these values.- For
*m*a power of 2, i.e. , and , the longest possible period is , when*c*is relatively prime to*m*and*a = 1 + 4 k*where*k*is an integer. - For
*m*a power of 2, i.e. , and , the longest possible period is , when is odd and the multiplier,*a*is given by or where*k*is an integer. - For
*m*a prime number and*c = 0*, the longest possible period is*P = m - 1*when*a*satisfies the property that the smallest*k*such that is divisible by*m*is*k = m - 1*.For example, we choose

*m = 7*and*a = 3*, the above conditions satisfy. Here*k*has to be 6.- when k = 6, which is divisible by
*m* - when k = 5, which is not divisible by
*m* - when k = 4, which is not divisible by
*m* - when k = 3, which is not divisible by
*m*

- when k = 6, which is divisible by

- For

- The numbers generated from the example can only assume
values from the set
- Examples 8.2, 8.3 and 8.4 on page 294 and page 295.