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

#include <nag.h>

void	g05ymc (Nag_OrderType order, Integer n, double quas[], Integer pdquas, Integer iref[], NagError *fail)

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

x^{1}, x^{2}, \dots, x^{N}

with high uniformity in the

S

-dimensional unit cube

I^{S} = {[0, 1]}^{S}

Let

G

be a subset of

I^{S}

and define the counting function

S_{N} (G)

as the number of points

x^{i} \in G

. For each

x = (x_{1}, x_{2}, \dots, x_{S}) \in I^{S}

, let

G_{x}

be the rectangular

S

-dimensional region

G_{x} = [0, x_{1}) \times [0, x_{2}) \times \dots \times [0, x_{S})

with volume

x_{1}, x_{2}, \dots, x_{S}

. Then one measure of the uniformity of the points

x^{1}, x^{2}, \dots, x^{N}

is the discrepancy:

D_{N}^{*} (x^{1}, x^{2}, \dots, x^{N}) = \sup_{x \in I^{S}} | S_{N} (G_{x}) - N x_{1}, x_{2}, \dots, x_{S} | .

which has the form

D_{N}^{*} (x^{1}, x^{2}, \dots, x^{N}) \leq C_{S} {(\log N)}^{S} + O ({(\log N)}^{S - 1}) for all N \geq 2 .

The principal aim in the construction of low-discrepancy sequences is to find sequences of points in

I^{S}

with a bound of this form where the constant

C_{S}

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:

$idim = idim$ , the number of dimensions required, see g05ylc or g05ync
$liref = liref$ , the length of iref as supplied to the initialization function g05ylc or g05ync

1: $order$ – 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

order = Nag_RowMajor

. 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:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – Integer Input

On entry: the number of quasi-random numbers required.

Constraint:

n \geq 0

and

n + previous number of generated values \leq 2^{31} - 1

3: $quas [\dim]$ – double Output

Note: where

QUAS (i, j)

appears in this document, it refers to the array element

$quas [(j - 1) \times pdquas + i - 1]$ when $order = Nag_ColMajor$ ;
$quas [(i - 1) \times pdquas + j - 1]$ when $order = Nag_RowMajor$ .

On exit:

QUAS (i, j)

holds the

i

th value for the

j

th dimension.

4: $pdquas$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array quas.

Constraints:

if $order = Nag_RowMajor$ , $pdquas \geq idim$ ;
if $order = Nag_ColMajor$ , $pdquas \geq n$ .

5: $iref [\dim]$ – Integer Communication Array

Note: the dimension, dim, of the array iref must be at least

liref

On entry: contains information on the current state of the sequence.

On exit: contains updated information on the state of the sequence.

6: $fail$ – 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 $⟨ value ⟩$ 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, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdquas = ⟨ value ⟩$ , $idim = ⟨ value ⟩$ .
Constraint: if $order = Nag_RowMajor$ , $pdquas \geq idim$ .

On entry, $pdquas = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $order = Nag_ColMajor$ , $pdquas \geq n$ .
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: $n = ⟨ value ⟩$ , generator has previously been called $⟨ value ⟩$ 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