# rand.f

          FUNCTION RAND (R)
    C***BEGIN PROLOGUE  RAND
    C***PURPOSE  Generate a uniformly distributed random number.
    C***LIBRARY   SLATEC (FNLIB)
    C***CATEGORY  L6A21
    C***TYPE      SINGLE PRECISION (RAND-S)
    C***KEYWORDS  FNLIB, RANDOM NUMBER, SPECIAL FUNCTIONS, UNIFORM
    C***AUTHOR  Fullerton, W., (LANL)
    C***DESCRIPTION
    C
    C      This pseudo-random number generator is portable among a wide
    C variety of computers.  RAND(R) undoubtedly is not as good as many
    C readily available installation dependent versions, and so this
    C routine is not recommended for widespread usage.  Its redeeming
    C feature is that the exact same random numbers (to within final round-
    C off error) can be generated from machine to machine.  Thus, programs
    C that make use of random numbers can be easily transported to and
    C checked in a new environment.
    C
    C      The random numbers are generated by the linear congruential
    C method described, e.g., by Knuth in Seminumerical Methods (p.9),
    C Addison-Wesley, 1969.  Given the I-th number of a pseudo-random
    C sequence, the I+1 -st number is generated from
    C             X(I+1) = (A*X(I) + C) MOD M,
    C where here M = 2**22 = 4194304, C = 1731 and several suitable values
    C of the multiplier A are discussed below.  Both the multiplier A and
    C random number X are represented in double precision as two 11-bit
    C words.  The constants are chosen so that the period is the maximum
    C possible, 4194304.
    C
    C      In order that the same numbers be generated from machine to
    C machine, it is necessary that 23-bit integers be reducible modulo
    C 2**11 exactly, that 23-bit integers be added exactly, and that 11-bit
    C integers be multiplied exactly.  Furthermore, if the restart option
    C is used (where R is between 0 and 1), then the product R*2**22 =
    C R*4194304 must be correct to the nearest integer.
    C
    C      The first four random numbers should be .0004127026,
    C .6750836372, .1614754200, and .9086198807.  The tenth random number
    C is .5527787209, and the hundredth is .3600893021 .  The thousandth
    C number should be .2176990509 .
    C
    C      In order to generate several effectively independent sequences
    C with the same generator, it is necessary to know the random number
    C for several widely spaced calls.  The I-th random number times 2**22,
    C where I=K*P/8 and P is the period of the sequence (P = 2**22), is
    C still of the form L*P/8.  In particular we find the I-th random
    C number multiplied by 2**22 is given by
    C I   =  0  1*P/8  2*P/8  3*P/8  4*P/8  5*P/8  6*P/8  7*P/8  8*P/8
    C RAND=  0  5*P/8  2*P/8  7*P/8  4*P/8  1*P/8  6*P/8  3*P/8  0
    C Thus the 4*P/8 = 2097152 random number is 2097152/2**22.
    C
    C      Several multipliers have been subjected to the spectral test
    C (see Knuth, p. 82).  Four suitable multipliers roughly in order of
    C goodness according to the spectral test are
    C    3146757 = 1536*2048 + 1029 = 2**21 + 2**20 + 2**10 + 5
    C    2098181 = 1024*2048 + 1029 = 2**21 + 2**10 + 5
    C    3146245 = 1536*2048 +  517 = 2**21 + 2**20 + 2**9 + 5
    C    2776669 = 1355*2048 + 1629 = 5**9 + 7**7 + 1
    C
    C      In the table below LOG10(NU(I)) gives roughly the number of
    C random decimal digits in the random numbers considered I at a time.
    C C is the primary measure of goodness.  In both cases bigger is better.
    C
    C                   LOG10 NU(I)              C(I)
    C       A       I=2  I=3  I=4  I=5    I=2  I=3  I=4  I=5
    C
    C    3146757    3.3  2.0  1.6  1.3    3.1  1.3  4.6  2.6
    C    2098181    3.3  2.0  1.6  1.2    3.2  1.3  4.6  1.7
    C    3146245    3.3  2.2  1.5  1.1    3.2  4.2  1.1  0.4
    C    2776669    3.3  2.1  1.6  1.3    2.5  2.0  1.9  2.6
    C   Best
    C    Possible   3.3  2.3  1.7  1.4    3.6  5.9  9.7  14.9
    C
    C             Input Argument --
    C R      If R=0., the next random number of the sequence is generated.
    C        If R .LT. 0., the last generated number will be returned for
    C          possible use in a restart procedure.
    C        If R .GT. 0., the sequence of random numbers will start with
    C          the seed R mod 1.  This seed is also returned as the value of
    C          RAND provided the arithmetic is done exactly.
    C
    C             Output Value --
    C RAND   a pseudo-random number between 0. and 1.
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  (NONE)
    C***REVISION HISTORY  (YYMMDD)
    C   770401  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    C   890531  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C***END PROLOGUE  RAND