NAG Library Function Document

nag_quasi_rand_uniform (g05ymc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_quasi_rand_uniform (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 nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync).

2 Specification

#include <nag.h>

#include <nagg05.h>

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

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.

nag_quasi_rand_uniform (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 nag_quasi_rand_uniform (g05ymc) depends on the initialization function called and can include those proposed by Sobol, Faure or Niederreiter. If the initialization function nag_quasi_init_scrambled (g05ync) was used then the sequence will be scrambled (see Section 3 in nag_quasi_init_scrambled (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 nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync)
$liref = liref$ , the length of iref as supplied to the initialization function nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync)
$tdquas = n$ if $order = Nag_RowMajor$ ; otherwise $tdquas = idim$

1: order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: n – IntegerInput

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[ $pdquas \times tdquas$ ] – doubleOutput

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

pdquas \times tdquas

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 – IntegerInput

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[ $liref$ ] – IntegerCommunication Array

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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.
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

nag_quasi_rand_uniform (g05ymc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

The Sobol, Sobol (A659) and Niederreiter quasi-random number generators in nag_quasi_rand_uniform (g05ymc) have been parallelized, but require quite large problem sizes to see any significant performance gain. Parallelism is only enabled when