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

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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 x_j 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 a_i (n) are integers:

The j-th coordinate of x_n 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

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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

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

Return Value

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A structure that contains information about the sequence.

Arguments

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Ndim

The dimension of the hyper-rectangle.

Keywords

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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

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6.4	Introduced

See Also

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IMSL_FAURE_NEXT_PT