Golden ratio versus pi as random sequence sources for Monte Carlo integration

The algebraic irrational number golden ratio φ = ( 1 + 5 ) / 2  = one of the two roots of the algebraic equation x 2 − x − 1 = 0 and the transcendental number π = 2 sin − 1 ( 1 )  = the ratio of the circumference and the diameter of any circle both have infinite number of digits with no apparent pattern. We discuss here the relative merits of these numbers as possible random sequence sources. The quality of these sequences is not judged directly based on the outcome of all known tests for the randomness of a sequence. Instead, it is determined implicitly by the accuracy of the Monte Carlo integration in a statistical sense. Since our main motive of using a random sequence is to solve real-world problems, it is more desirable if we compare the quality of the sequences based on their performances for these problems in terms of quality/accuracy of the output. We also compare these sources against those generated by a popular pseudo-random generator, viz., the Matlab rand and the quasi-random generator halton both in terms of error and time complexity. Our study demonstrates that consecutive blocks of digits of each of these numbers produce a good random sequence source. It is observed that randomly chosen blocks of digits do not have any remarkable advantage over consecutive blocks for the accuracy of the Monte Carlo integration. Also, it reveals that π is a better source of a random sequence than φ when the accuracy of the integration is concerned.