Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0
Syntax
C# |
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[SerializableAttribute] public class FaureSequence : IRandomSequence |
Visual Basic (Declaration) |
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<SerializableAttribute> _ Public Class FaureSequence _ Implements IRandomSequence |
Visual C++ |
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[SerializableAttribute] public ref class FaureSequence : IRandomSequence |
Remarks
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 of the form is the Lebesque measure, and A(E;n) is the number of the contained in E.The sequence of points in is a low-discrepancy sequence if there exists a constant c(d), depending only on d, such that
for all .Generalized Faure sequences can be defined for any prime base . The lowest bound for the discrepancy is obtained for the smallest prime , so the base defaults to the smallest prime greater than or equal to the dimension.
The generalized Faure sequence , is computed as follows:
Write the positive integer n in its b-ary expansion,
where are integers, .The j-th coordinate of 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 , where is the generalized Faure sequence.