The IMSL_FAURE_INIT function initializes the structure used for computing a shuffled Faure sequence.

This routine requires an IDL Advanced Math and Stats license. For more information, contact your sales or technical support representative.

Discrepancy measures the deviation from uniformity of a point set.

The discrepancy of the point set:

is:

where the supremum is over all subsets of [0, 1]d of the form:

λ is the Lebesque measure, and:

is the number of the xj contained in E.

The sequence x1, x2, ... of points [0,1]d is a low-discrepancy sequence if there exists a constant c(d), depending only on d, such that:

for all n>1.

Generalized Faure sequences can be defined for any prime base b≥d. The lowest bound for the discrepancy is obtained for the smallest prime b≥d, so the keyword Base defaults to the smallest prime greater than or equal to the dimension. The generalized Faure sequence x1, x2, ..., is computed as follows:

Write the positive integer n in its b-ary expansion:

where ai (n) are integers:

The j-th coordinate of xn is:

The generator matrix for the series:

is defined to be:

and:

is an element of the Pascal matrix:

It is faster to compute a shuffled Faure sequence than to compute the Faure sequence itself. It can be shown that this shuffling preserves the low-discrepancy property.

The shuffling used is the b-ary Gray code. The function G(n) maps the positive integer n into the integer given by its b-ary expansion.

The sequence computed by this function is x(G(n)), where x is the generalized Faure sequence.

## Example

In this example, five points in the Faure sequence are computed. The points are in the three-dimensional unit cube.

Note that IMSL_FAURE_INIT is used to create a structure that holds the state of the sequence. Each call to IMSL_FAURE_NEXT_PT returns the next point in the sequence and updates the state structure.

`state = IMSL_FAURE_INIT(3)`
`p = IMSL_FAURE_NEXT_PT(5, state)`
`PM, p`

IDL prints:

`0.333689 0.492659 0.0640654`
`0.667022 0.825992 0.397399`
`0.778133 0.270436 0.175177`
`0.111467 0.603770 0.508510`
`0.444800 0.937103 0.841843`

## Syntax

Result = IMSL_FAURE_INIT(Ndim [, BASE=value] [, SKIP=value])

## Return Value

A structure that contains information about the sequence.

## Arguments

### Ndim

The dimension of the hyper-rectangle.

## Keywords

### BASE (optional)

The base of the Faure sequence. Default: The smallest prime greater than or equal to Ndim.

### SKIP (optional)

The number of points to be skipped at the beginning of the Faure sequence. Default:

where:

and B is the largest representable integer.

## Version History

 6.4 Introduced