Computes a shuffled Faure sequence.

npts—The number of points to generate in the hyper-rectangle.

state—State structure created by a call to FAURE_INIT.

An array of size npts by state.dim containing the npts next points in the shuffled Faure sequence.

Double—If present and nonzero, double precision is used.

Skip—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.

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, λ]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 jth 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.

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

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

state = FAURE_INIT(3)

p = FAURE_NEXT_PT(5, state)

PM, p

; This results in the following output:

;  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