Contents

# 3D Spheres Test

Test Purpose
The test uses simulation to evaluate the randomness of the triplets of sequential random numbers of uniform distribution. The stable response is the volume of the sphere. The radius of the sphere is equal to the minimal distance between the generated 3D points.
First Level Test
The test generates the vector
ui
of 12,000 random numbers (
i
= 0, 1, ..., 11999), which are uniformly distributed in the (0, 1000) interval. The test forms 4,000 triplets of random numbers
xk
= (
u
3
k
,
u
3
k
+1
,
u
3
k
+2
)(
k
= 0, 1, ..., 3999) situated in the cube
R
= (0, 1000)
х
(0, 1000)
х
(0, 1000). Further, the test calculates
dmin = d(xk, x
l
)(l ≠
k
), where
d(x, y)
is the Euclidean distance between
x
and
y
. In this case, the volume of the sphere with the
d
min
radius should have the distribution close to the exponential one with
a
= 0,
b
= 40
p
parameters. Thus, the distribution of the
p
= 1 - exp(-(
d
min
)3/30) value should be close to the uniform distribution. The result of the first level test is the p-value.
Second Level Test
The second level test performs the first level test ten times. The p-value
pj
,
j
= 1, 2, ..., 10 is the result of each first level test. The test applies the Kolmogorov-Smirnov goodness-of-fit test with Anderson-Darling statistic to the obtained set of
pj
(
j
= 1, 2, ..., 10). If the resulting p-value is
p
< 0.05 or
p
> 0.95, the test fails.
Final Result Interpretation
The final result is the FAIL percentage for the failed first level tests. The test performs the second level test ten times. The acceptable result is the value of FAIL < 50%.
Tested Generators
Function Name
Application
vsRngUniform
applicable
vdRngUniform
applicable
viRngUniform
not applicable
viRngUniformBits
applicable
Note
The test transforms the integer output into the real output within the interval (0, 1) for the function
viRngUniformBits
. For detailed information about the normalization of the integer output see the description of the given BRNG.

#### Product and Performance Information

1

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Notice revision #20110804