NAG CL Interface
g05ymc (quasi_uniform)
1
Purpose
g05ymc generates a uniformly distributed low-discrepancy sequence as proposed by Sobol, Faure or Niederreiter. It must be preceded by a call to one of the initialization functions
g05ylc or
g05ync.
2
Specification
The function may be called by the names: g05ymc, nag_rand_quasi_uniform or nag_quasi_rand_uniform.
3
Description
Low discrepancy (quasi-random) sequences are used in numerical integration, simulation and optimization. Like pseudorandom numbers they are uniformly distributed but they are not statistically independent, rather they are designed to give more even distribution in multidimensional space (uniformity). Therefore, they are often more efficient than pseudorandom numbers in multidimensional Monte Carlo methods.
g05ymc generates a set of points with high uniformity in the -dimensional unit cube .
Let
be a subset of
and define the counting function
as the number of points
. For each
, let
be the rectangular
-dimensional region
with volume
. Then one measure of the uniformity of the points
is the discrepancy:
which has the form
The principal aim in the construction of low-discrepancy sequences is to find sequences of points in with a bound of this form where the constant is as small as possible.
The type of low-discrepancy sequence generated by
g05ymc depends on the initialization function called and can include those proposed by Sobol, Faure or Niederreiter. If the initialization function
g05ync was used then the sequence will be scrambled (see
Section 3 in
g05ync for details).
4
References
Bratley P and Fox B L (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator ACM Trans. Math. Software 14(1) 88–100
Fox B L (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators ACM Trans. Math. Software 12(4) 362–376
5
Arguments
Note: the following variables are used in the parameter descriptions:
- , the number of dimensions required, see g05ylc or g05ync
- , the length of iref as supplied to the initialization function g05ylc or g05ync
-
1:
– Nag_OrderType
Input
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Integer
Input
-
On entry: the number of quasi-random numbers required.
Constraint:
and .
-
3:
– double
Output
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On exit: holds the th value for the th dimension.
-
4:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
quas.
Constraints:
- if , ;
- if , .
-
5:
– Integer
Communication Array
Note: the dimension,
dim, of the array
iref
must be at least
.
On entry: contains information on the current state of the sequence.
On exit: contains updated information on the state of the sequence.
-
6:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INITIALIZATION
-
On entry,
iref has either not been initialized or has been corrupted.
On entry, the specified dimensions are out of sync.
A different number of values have been generated from at least one of the specified dimensions.
- NE_INT
-
On entry, .
Constraint: .
- NE_INT_2
-
On entry, , .
Constraint: if , .
On entry, and .
Constraint: if , .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_TOO_MANY_CALLS
-
On entry, value of
n would result in too many calls to the generator:
, generator has previously been called
times.
7
Accuracy
Not applicable.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
g05ymc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The Sobol, Sobol (A659) and Niederreiter quasi-random number generators in g05ymc have been parallelized, but require quite large problem sizes to see any significant performance gain. Parallelism is only enabled when . The Faure generator is serial.
None.
10
Example
This example calls
g05ylc and
g05ymc to estimate the value of the integral
In this example the number of dimensions is set to .
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results