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The IMSL_FAURE_NEXT_PT function computes a shuffled Faure sequence.

Discrepancy measures the deviation from uniformity of a point set.

The discrepancy of the point set:


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:


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.

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