The IMSL_FAURE_NEXT_PT function computes 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 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_NEXT_PT(Npts, State [, /DOUBLE] [, SKIP=value])
Return Value
An array of size npts by state.dim containing the npts next points in the shuffled Faure sequence.
Arguments
Npts
The number of points to generate in the hyper-rectangle.
State
State structure created by a call to IMSL_FAURE_INIT.
Keywords
DOUBLE (optional)
If present and nonzero, double precision is used.
SKIP (optional)
The current point in the sequence. The sequence can be restarted by initializing a new sequence using this value for Skip, and using the same dimension for Ndim.
Version History
See Also
IMSL_FAURE_INIT