From: krishna.myneni@ccreweb.org
On 9/12/24 21:10, Krishna Myneni wrote:
...
>
> I ran a simple Monte-Carlo integration of the area in a unit circle
> using the 64-bit LCG PRNG (from random.4th in kForth dist) and
> Marsaglia's 64-bit KISS PRNG. Hard to say which is better for this problem.
>
> --
> Krishna
>
> === Comparison of 64-bit LCG and KISS PRNGs ===
>
> PRNG: RANDOM (random.4th)
> Ntrials Area log(error)
> -------------------------------
> 10^2 3.28 -0.86
> 10^3 3.18 -1.42
> 10^4 3.1724 -1.51
> 10^5 3.14048 -2.95
> 10^6 3.14350 -2.72
> 10^7 3.14225 -3.18
> 10^8 3.14152 -4.14
>
> PRNG: RAN-KISS (kiss.4th)
> Ntrials Area log(error)
> -------------------------------
> 10^2 3.24 -1.01
> 10^3 3.10 -1.38
> 10^4 3.1252 -1.79
> 10^5 3.14084 -3.12
> 10^6 3.14073 -3.06
> 10^7 3.14184 -3.61
> 10^8 3.14167 -4.11
>
>
A bit more involved test of the same 64-bit PRNGs starts to show the
possibility of defects in Marsaglia's 64-bit kiss prng (RAN-KISS)
compared to a simple 64-bit LCG prng (RANDOM). Instead of drawing from a
uniform distribution, as assumed for these generators, the tests below
draw from a Maxwell-Boltzmann distribution, which describes the
probability distribution for speeds of ideal gas atoms at a given
temperature (non-interacting atoms except for hard sphere collisions in
three dimensions). The different "moments" of the speed may be computed
from the samples and compared with exact theoretical results: ,
, , etc., where v is a random sample of the speed for the M-B
distribution at given temperature and mass of atom. The parameter "a",
given by
a = m/(2*kB*T)
where m is the mass, T is the temperature, and kB is the Boltzmann
constant, determines the shape of the M-B distribution. Unsigned 64-bit
random numbers, generated by the PRNG under test, are converted to
floating point numbers in the interval [0, 1]. Ideally, the PRNG will
distribute these samples uniformly over the interval. The samples are
converted to random speed samples with an M-B distribution by an inverse
mapping of the cumulative probability distribution for M-B to the speed.
The moments for , , and are computed from these samples.
In this simulation, the temperature is set at 300 K, and the mass of the
He atom is used. Theory predictions for this m and T are
_th = sqrt( 4/(pi*a) ) = 1258.727 m/s
_th = 3/(2*a) = 1869538 (m/s)^2
_th = sqrt( 16/(pi*a^3) ) = 3140144 x 10^3 (m/s)^3
Below are the results for the two PRNGs. For N trials of 10^5 through
10^7, the simple 64-bit LCG prng consistently gives less relative error
than Marsaglia's 64-bit kiss prng with respect to the theory predictions.
Source code for these test results is given below.
--
Krishna
=== begin tests ===
Test of prng RANDOM 64-bit (random.4th):
' random test-prng
Moments of speed
N (m/s) (m/s)^2 (m/s)^3
10^2 1220.2444 1761665.1 2879481740.
10^3 1258.2315 1844889.5 3025976380.
10^4 1250.0696 1837249.9 3055346856.
10^5 1259.9899 1870367.4 3143506292.
10^6 1259.3923 1868383.4 3136761299.
10^7 1259.6343 1869366.9 3140010276.
Relative Errors
N |e1| |e2| |e3|
10^2 3.13e-02 5.77e-02 8.30e-02
10^3 1.19e-03 1.32e-02 3.64e-02
10^4 7.67e-03 1.73e-02 2.70e-02
10^5 2.08e-04 4.44e-04 1.07e-03
10^6 2.66e-04 6.18e-04 1.08e-03
10^7 7.39e-05 9.15e-05 4.27e-05
Test of prng RAN-KISS 64-bit (kiss.4th):
' ran-kiss test-prng
Moments of speed
N (m/s) (m/s)^2 (m/s)^3
10^2 1212.8900 1702106.0 2666594142.
10^3 1274.7097 1900519.4 3190481667.
10^4 1259.6892 1866276.3 3123781859.
10^5 1260.4578 1872500.5 3147907366.
10^6 1260.2589 1871249.6 3145073404.
10^7 1259.8296 1869882.9 3140954422.
Relative Errors
N |e1| |e2| |e3|
10^2 3.72e-02 8.96e-02 1.51e-01
10^3 1.19e-02 1.66e-02 1.60e-02
10^4 3.03e-05 1.74e-03 5.21e-03
10^5 5.80e-04 1.58e-03 2.47e-03
10^6 4.22e-04 9.16e-04 1.57e-03
10^7 8.11e-05 1.85e-04 2.58e-04
=== end tests ===
=== begin test-prng-mb.4th ===
\ test-prng-mb.4th
\
\ Test a pseudo-random number generator by computing the
\ moments of powers of speed using random samples from a
\ Maxwell-Boltzmann distribution of ideal gas speeds in 3D.
\
\ Copyright 2024 Krishna Myneni
\
\ Permission is granted to use this code for any purpose,
\ provided the original source is referenced.
\
\ Usage Example:
\
\ ' random TEST-PRNG
\
\ where RANDOM is a user-provided PRNG which returns
\ a cell-sized pseudo-random number.
\
\ Notes
\
\ 1. The xt for any PRNG may be passed to the word
\ TEST-PRNG provided it returns a cell-sized
\ bit pattern. All bits of the cell are assumed
\ to be part of the pseudo-random number.
\
\ 2. The default M-B distribution is computed for
\ helium atoms at 300K. If you change the default
\ temperature or atomic mass, the maximum velocity
\ used in the root finder call in CDF^-1 may need
\ to be changed.
\
include ans-words
include strings
include phyconsts
include fsl/fsl-util
include fsl/erf
include fsl/regfalsi
include random
include kiss
[UNDEFINED] FPICK [IF]
cr cr .( This program requires a separate FP stack! ) cr
quit
[THEN]
4.0e0 pi f* fconstant 4pi
-1 0 d>f fconstant F_MAX_U
300.0e0 fconstant DEF_TEMPERATURE \ K
1.0e4 fconstant DEF_VMAX \ m/s
1.0e-9 fconstant DEF_TOL \ root-finder tolerance
4.002602e0 fconstant m_He \ mass in amu for He atom
: fcube ( F: r -- r^3 ) fdup fsquare f* ;
: f3dup 2 fpick 2 fpick 2 fpick ;
[UNDEFINED] FTUCK [IF]
: ftuck ( F: a b -- b a b ) fswap fover ;
[THEN]
\ Relative error between a value and its reference value
: rel-error ( F: a aref -- e ) ftuck f- fswap f/ ;
\ The random number generator to be tested must return a
\ cell-sized bit pattern (unsigned single) -- all bits of
\ the cell are assumed to be randomized.
defer prngU
defer prngU-init
' random is prngU \ default PRNG
:noname ( -- ) 1000 0 do prngU drop loop ; is prngU-init
\ Generate a floating point random number between 0 and 1
\ using the selected unsigned single cell prng.
: randf ( F: -- r ) prngU 0 d>f F_MAX_U f/ ;
fvariable m_atom_kg
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