nag_quasi_random_uniform (g05yac) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_quasi_random_uniform (g05yac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

To generate multidimensional quasi-random sequences with a uniform probability distribution.

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_quasi_random_uniform (Nag_QuasiRandom_State state, Nag_QuasiRandom_Sequence seq, Integer iskip, Integer idim, double quasi[], Nag_QuasiRandom *gf, 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_random_uniform (g05yac) generates a set of points x1,x2,,xN with high uniformity in the S-dimensional unit cube IS=0,1S. One measure of the uniformity is the discrepancy which is defined as follows:
nag_quasi_random_uniform (g05yac) generates the low-discrepancy sequences proposed by Sobol, Faure and Niederreiter.

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

1:     stateNag_QuasiRandom_StateInput
On entry: the type of operation to perform.
state=Nag_QuasiRandom_Init
The first call for initialization, and there is no output via array quasi.
state=Nag_QuasiRandom_Cont
The sequence has been initialized by a prior call to nag_quasi_random_uniform (g05yac) with state=Nag_QuasiRandom_Init. Random numbers are output via array quasi.
state=Nag_QuasiRandom_Finish
The final call to release memory, and no further random numbers are required for output via array quasi.
Constraint: state=Nag_QuasiRandom_Init, Nag_QuasiRandom_Cont or Nag_QuasiRandom_Finish.
2:     seqNag_QuasiRandom_SequenceInput
On entry: the type of sequence to generate.
seq=Nag_QuasiRandom_Sobol
A Sobol sequence.
seq=Nag_QuasiRandom_Nied
A Niederreiter sequence.
seq=Nag_QuasiRandom_Faure
A Faure sequence.
Constraint: seq=Nag_QuasiRandom_Sobol, Nag_QuasiRandom_Nied or Nag_QuasiRandom_Faure.
3:     iskipIntegerInput
On entry: the number of terms in the sequence to skip on initialization.
iskip=0
All the terms of the sequence are generated.
iskip=k
The first k terms of the sequence are ignored and the first term of the sequence now corresponds to the kth term of the sequence when iskip=0.
If seq=Nag_QuasiRandom_Faure, iskip is not referenced.
Constraint: if seq=Nag_QuasiRandom_Nied or Nag_QuasiRandom_Sobol and state=Nag_QuasiRandom_Init, iskip0.
4:     idimIntegerInput
On entry: the number of dimensions required.
Constraint: 1idim40.
5:     quasi[idim]doubleOutput
On exit: the random numbers.
If state=Nag_QuasiRandom_Cont, quasi[k-1] contains the random number for the kth dimension.
6:     gfNag_QuasiRandom *Communication Structure
Workspace used to communicate information between calls to nag_quasi_random_uniform (g05yac). The contents of this structure should not be changed between calls.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, seq is not valid: seq=value.
NE_INITIALIZATION
Incorrect initialization.
NE_INT
On entry, idim=value.
Constraint: idim40.
On entry, idim=value.
Constraint: idim1.
On entry, iskip=value.
Constraint: iskip0.
On entry, value of skip too large: iskip=value.
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
Too many calls to generator.

7  Accuracy

Not applicable.

8  Further Comments

The maximum length of the generated sequences is 229-1, this should be adequate for practical purposes. In the case of the Niederreiter generator nag_quasi_random_uniform (g05yac) jumps to the appropriate starting point, while for the Sobol generator it simply steps through the sequence. In consequence the Sobol generator with large values of iskip will take a significant amount of time.

9  Example

This example approximates the integral
01 01 i=1 s 4xi-2 dx1, dx2, , dxs = 1 ,
where s is the number of dimensions.

9.1  Program Text

Program Text (g05yace.c)

9.2  Program Data

None.

9.3  Program Results

Program Results (g05yace.r)


nag_quasi_random_uniform (g05yac) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012