Editing RANDU (section)

{{Short description|Pseudorandom number generator}}
{{Use dmy dates|date=October 2020}}
[[Image:Randu.png|thumb|right|[[Three-dimensional space|Three-dimensional plot]] of 100,000 values generated with RANDU. Each point represents 3 consecutive pseudorandom values. It is clearly seen that the points fall in 15 [[2D geometric model|two-dimensional]] [[Plane (mathematics)|planes]].]]

'''RANDU'''<ref name=":0">Compaq Fortran Language Reference Manual (Order Number: AA-Q66SD-TK) September 1999 (formerly DIGITAL Fortran and DEC Fortran&nbsp;90).</ref> is a [[linear congruential generator|linear congruential]] [[pseudorandom number generator]] (LCG) of the [[Park–Miller random number generator|Park–Miller type]], which was used primarily in the 1960s and 1970s.<ref name="Entacher-2000">
{{cite web
 | last = Entacher
 | first =  Karl
 | title = A collection of classical pseudorandom number generators with linear structures – advanced version
 | date = June 2000
 | url=http://random.mat.sbg.ac.at/results/karl/server/server.html
 | url-status=dead
 | archive-url=https://web.archive.org/web/20181118052935if_/http://random.mat.sbg.ac.at/results/karl/server/server.html
 | archive-date=2018-11-18
}}</ref> It is defined by the [[Recursion|recurrence]]

: <math>V_{j+1} = 65539 \cdot V_j \bmod 2^{31}</math>

with the initial seed number <math>V_0</math> as an [[even and odd numbers|odd number]]. It generates pseudorandom [[integer number|integers]] <math>V_j</math> which are [[uniform distribution (discrete)|uniformly distributed]] in the interval {{nowrap|[1, 2<sup>31</sup> − 1]}}, but in practical applications are often mapped into pseudorandom [[rational number|rationals]] <math>X_j</math> in the interval {{nowrap|(0, 1)}}, by the formula
: <math>X_j = \frac{V_j}{2^{31}}.</math>

IBM's RANDU is widely considered to be one of the most ill-conceived random number generators ever designed,<ref>[[Donald Knuth|Knuth D. E.]] ''[[The Art of Computer Programming]]'', Volume&nbsp;2: ''Seminumerical Algorithms'', 2nd edition. Addison-Wesley, 1981. {{ISBN|0-201-03822-6}}. Section&nbsp;3.3.4, p.&nbsp;104: "its very name RANDU is enough to bring dismay into the eyes and stomachs of many computer scientists!" [Extensive coverage of statistical tests for non-randomness.]</ref> and was described as "truly horrible" by [[Donald Knuth]].<ref>Knuth (1998), p. 188.{{full citation needed|date=February 2024}}</ref> It fails the [[spectral test]] badly for dimensions greater than 2, as shown below.

The reason for choosing these particular values for the multiplier and modulus had been that with a 32-bit-integer word size, the arithmetic of mod 2<sup>31</sup> and <math>65539 = 2^{16} + 3</math> calculations could be done quickly, using [[bitwise operator]]s in hardware, but the values were chosen for computational convenience, not statistical quality.