6.1.6 L'Ecuyer's Combined Recursive Generator

The base generator referred to as L'Ecuyer's combined recursive generator is referred to as MRG32k3a in [12] and combines two multiple recursive generators: x(i) = [a1 * x(i - 1) + a2 * x(i - 2) + a3 * x(i - 3)] mod m1, y(i) = [b1 * y(i - 1) + b2 * y(i - 2) + b3 * y(i - 3)] mod m2, z(i) = [x(i) - y(i)] mod m1, u(i) = z_i / m1, where the u(i),i=1,2,... form the required sequence and a1 = 0, a2 = 1403580, a3 = 810728, m1 = 2^(32)-209, b1 = 527612, b2 = 0, b3 = 1370589 and m2 = 2^(32) - 22853.

Combining the two multiple recursive generators (MRG) results in sequences with better statistical properties in high dimensions and longer periods compared with those generated from a single MRG. The combined generator described above has a period length of approximately 2^(191)