g05ym 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 methods g05yl or g05yn.

Syntax

C#
public static void g05ym( int n, int rcord, double[,] quas, int[] iref, out int ifail )

Visual Basic
Public Shared Sub g05ym ( _ n As Integer, _ rcord As Integer, _ quas As Double(,), _ iref As Integer(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g05ym( int n, int rcord, array<double,2>^ quas, array<int>^ iref, [OutAttribute] int% ifail )

F#
static member g05ym : n : int * rcord : int * quas : float[,] * iref : int[] * ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: the number of quasi-random numbers required.

Constraint: $n \geq 0$ and $n + previous number of generated values \leq 2^{31} - 1$ .

rcord: Type: System..::..Int32
On entry: the order in which the generated values are returned.

Constraint: $rcord = 1$ or $2$ .

quas

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1,

tdquas

]

Note: dim1 must satisfy the constraint:

if $rcord = 1$ , $dim1 \geq idim$ ;
if $rcord = 2$ , $dim1 \geq n$ .

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

rcord = 1

quas [i - 1, j - 1]

holds the

j

th value for the

i

th dimension.

rcord = 2

quas [i - 1, j - 1]

holds the

i

th value for the

j

th dimension.

iref: Type: array<System..::..Int32>[]()[][]
An array of size [ $liref$ ]
On entry: contains information on the current state of the sequence.
On exit: contains updated information on the state of the sequence.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

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.

g05ym 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 g05ym depends on the initialization method called and can include those proposed by Sobol, Faure or Niederreiter. If the initialization method g05yn was used then the sequence will be scrambled (see [Description] in g05yn for details).

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

Error Indicators and Warnings

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDQUAS) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$: On entry, either $n < 0$ or there have been too many calls to the generator.

$ifail = 2$: On entry, $rcord \neq 1$ or $2$ .

$ifail = 4$: On entry, $rcord = 1$ and $ldquas < idim$ .
On entry, $rcord = 2$ and $ldquas < n$ .

$ifail = 5$: Incorrect initialization. g05yl must be called prior to g05ym and iref must remain unaltered after this call.

$ifail = -9000$: An error occured, see message report.
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

Not applicable.

Parallelism and Performance

None.

Further Comments

None.

Example

In this example the number of dimensions

S

is set to

8

Example program (C#): g05yme.cs

Example program data: g05yme.d

Example program results: g05yme.r