Contents

# WH

This is a set of 273 Wichmann-Hill’s combined multiplicative congruential generators (
j
= 1, 2, ..., 273): WH is a set of 273 different basic generators. This generator is the second basic generator in the NAG libraries. The constants
ai,j
range from 112 to 127, the constants
mi,j
are prime numbers ranging from 16,718,909 to 16,776,971, close to 2
24
. These constants should show good results in the spectral test (see Knuth [Knuth81] and MacLaren [MacLaren89]). The period of each Wichmann-Hill generator may be equal to 2
92
if not for common factors between (
m
1
,j
-1), (
m
2
,j
-1), (
m
3
,j
-1) and (
m
4
,j
-1). However, each generator should still have a period of at least 2
80
. The generated pseudo-random sequences are essentially independent of one another according to the spectral test (for detailed information about properties of these generators see [MacLaren89]).
Real Implementation (Single and Double Precision)
The output vector is the sequence of the floating-point values
u
0
,
u
1
, ...
Integer Implementation
The output vector of 32-bit integers
x
0
,
y
0
,
z
0
,
w
0
,
x
1
,
y
1
,
z
1
,
w
1
Thus, the output vector stores practically every quadruple (
x, y, z, w
) of members of the integer output as four 32-bit integers. For example, to get a vector from n quadruples (
x, y, z, w
), the size of the output array should be large enough to store 4n 32-bit numbers.
Stream Initialization by Function
vslNewStream
WH generates the stream and initializes it specifying the 32-bit input integer parameter seed :
1. Assume
x
0
=
seed mod m1
. If
x
0
= 0, assume
x
0
= 1.
2. Assume
y
0
= 1,
z
0
= 1,
w
0
= 1.
The test selects a WH generator adding an offset to the named constant
VSL_BRNG_WH: VSL_BRNG_WH+0, VSL_BRNG_WH+1, ... , VSL_BRNG_WH+272
. The following example illustrates the initialization of the seventh (of 273) WH generator:
```
vslNewStream (&stream, VSL_BRNG_WH+6, seed);
```
Stream Initialization of Function
vslNewStreamEx
WH generates the stream and initializes it specifying the the array
params[]
of
n
32-bit integers:
1. If
n
= 0, assume
x
0
= 1,
y
0
= 1,
z
0
= 1,
w
0
= 1.
2. If
n
= 1, assume
x
0
=
params mod m1
,
y
0
= 1,
z
0
= 1,
w
0
= 1. If
x
0
= 0, assume
x
0
=1.
3. If
n
= 2, assume
x
0
=
params mod m1
,
y
0
=
params mod m2
,
z
0
= 1,
w
0
= 1. If
x
0
= 0, assume
x
0
= 1. If
y
0
= 0, assume
y
0
= 1.
4. If
n
= 3, assume
x
0
=
params mod m1
,
y
0
=
params mod m2
,
z
0
=
params mod m3
,
w
0
= 1. If
x
0
= 0, assume
x
0
= 1. If
y
0
= 0, assume
y
0
= 1. If
z
0
= 0, assume
z
0
= 1.
5. If n ≥ 4, assume
x
0
=
params mod m1
,
y
0
=
params mod m2
,
z
0
=
params mod m3
,
w
0
=
params mod m4
. If
x
0 = 0, assume
x
0
= 1. If
y
0
= 0, assume
y
0
= 1. If
z
0
= 0, assume
z
0
= 1. If
w
0
= 0, assume
w
0
= 1.
Subsequences Selection Methods
Generator Period Empirical Testing Results Summary
Test Name
vsRngUniform
vdRngUniform
viRngUniform
viRngUniformBits
3D Spheres Test
OK (0% errors)
OK (0% errors)
Not applicable
OK (0% errors)
Birthday Spacing Test
Not applicable
Not applicable
Not applicable
FAIL (60% errors)
Bitstream Test
Not applicable
Not applicable
Not applicable
OK (10% errors)
Rank of 31x31 Binary Matrices Test
Not applicable
Not applicable
Not applicable
N/A
Rank of 32x32 Binary Matrices Test
Not applicable
Not applicable
Not applicable
N/A
Rank of 6x8 Binary Matrices Test
Not applicable
Not applicable
Not applicable
OK (0% errors)
Counts-the-1’s Test (stream of bits)
Not applicable
Not applicable
Not applicable
OK (10% errors)
Counts-the-1’s Test (stream of specific bytes)
Not applicable
Not applicable
Not applicable
OK (0% errors)
Craps Test
OK (20% errors)
OK (20% errors)
OK (20% errors)
OK (10% errors)
Parking Lot Test
OK (10% errors)
OK (10% errors)
Not applicable
OK (10% errors)
2D Self-Avoiding Random Walk Test
OK (10% errors)
OK (0% errors)
Not applicable
OK (20% errors)
Note
1. The tabulated data is obtained using the one-level (threshold) testing technique. The OK result indicates FAIL < 50%. The run fails when p-value falls outside the interval [0.05, 0.95].
2. The stream tested is generated by calling the function
vslNewStream
with seed=7,777,777.

#### Product and Performance Information

1

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Notice revision #20110804