NAG C Library Function Document

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)

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 3.3.1.3 in How to Use the NAG Library and its Documentation 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: $quas [\dim]$ – doubleOutput

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: $pdquas$ – 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: $iref [\dim]$ – IntegerCommunication 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_quasi_rand_uniform (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 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

order = Nag_ColMajor

. The Faure generator is serial.

9

Further Comments

None.

10

Example

This example calls nag_quasi_init (g05ylc) and nag_quasi_rand_uniform (g05ymc) to estimate the value of the integral

\int_{0}^{1} \dots \int_{0}^{1} \prod_{i = 1}^{s} |4 x_{i} - 2| d x_{1}, d x_{2}, \dots, d x_{s} = 1 .

In this example the number of dimensions

S

is set to

8

10.1

Program Text

Program Text (g05ymce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (g05ymce.r)

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g05 Chapter Contents

g05 Chapter Introduction

NAG C 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