g05ymf 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 routines g05ylf or g05ynf.

2 Specification

Fortran Interface

Subroutine g05ymf (

n, rcord, quas, ldquas, iref, ifail)

Integer, Intent (In)	::	n, rcord, ldquas
Integer, Intent (Inout)	::	iref(*), ifail
Real (Kind=nag_wp), Intent (Inout)	::	quas(ldquas,*)

C Header Interface

#include <nag.h>

void	g05ymf_ (const Integer n, const Integer rcord, double quas[], const Integer ldquas, Integer iref[], Integer ifail)

The routine may be called by the names g05ymf or nagf_rand_quasi_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.

g05ymf 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 g05ymf depends on the initialization routine called and can include those proposed by Sobol, Faure or Niederreiter. If the initialization routine g05ynf was used then the sequence will be scrambled (see Section 3 in g05ynf 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 g05ylf or g05ynf
$liref = liref$ , the length of iref as supplied to the initialization routine g05ylf or g05ynf

1: $n$ – 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

2: $rcord$ – Integer Input

On entry: the order in which the generated values are returned.

Constraint:

rcord = 1

2

3: $quas (ldquas, *)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array quas must be at least

n

rcord = 1

and at least

idim

rcord = 2

On exit: contains the n quasi-random numbers of dimension idim.

rcord = 1

quas (i, j)

holds the

j

th value for the

i

th dimension.

rcord = 2

quas (i, j)

holds the

i

th value for the

j

th dimension.

4: $ldquas$ – Integer Input

On entry: the first dimension of the array quas as declared in the (sub)program from which g05ymf is called.

Constraints:

if $rcord = 1$ , $ldquas \geq idim$ ;
if $rcord = 2$ , $ldquas \geq n$ .

5: $iref (*)$ – Integer array Communication Array

Note: the dimension 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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, value of n would result in too many calls to the generator: $n = ⟨ value ⟩$ , generator has previously been called $⟨ value ⟩$ times.

$ifail = 2$: On entry, $rcord = ⟨ value ⟩$ .
Constraint: $rcord = 1$ or $2$ .

$ifail = 4$: On entry, $ldquas = ⟨ value ⟩$ , $idim = ⟨ value ⟩$ .
Constraint: if $rcord = 1$ , $ldquas \geq idim$ .

On entry, $ldquas = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $rcord = 2$ , $ldquas \geq n$ .

$ifail = 5$: On entry, iref has either not been initialized or has been corrupted.

$ifail = 84$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05ymf 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 routine. 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 g05ymf have been parallelized, but require quite large problem sizes to see any significant performance gain. Parallelism is only enabled when

rcord = 2

. The Faure generator is serial.

9 Further Comments

None.

10 Example

This example calls g05ylf and g05ymf 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

g05ym: FL CL CPP AD PY MB

NAG FL Interfaceg05ymf (quasi_​uniform)

▸▿ 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

NAG FL Interface
g05ymf (quasi_uniform)