FaureSequence Class |
Namespace: Imsl.Stat
The FaureSequence type exposes the following members.
Name | Description | |
---|---|---|
FaureSequence(Int32) |
Creates a Faure sequence with the default base.
| |
FaureSequence(Int32, Int32, Int32) |
Creates a Faure sequence.
|
Name | Description | |
---|---|---|
ComputeParameters |
Compute needed parameters.
| |
Equals | Determines whether the specified object is equal to the current object. (Inherited from Object.) | |
Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.) | |
GetHashCode | Serves as a hash function for a particular type. (Inherited from Object.) | |
GetType | Gets the Type of the current instance. (Inherited from Object.) | |
MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object.) | |
NextDouble |
Returns the first value of the next point in the sequence.
| |
NextPoint |
Returns the next point in the sequence.
| |
NextPrime |
Returns the smallest prime greater than or equal to n.
| |
ToString | Returns a string that represents the current object. (Inherited from Object.) |
Name | Description | |
---|---|---|
Base |
The base.
| |
Dimension |
Returns the dimension of the sequence.
| |
NumberOfProcessors |
Perform the parallel calculations with the maximum possible number of
processors set to NumberOfProcessors.
| |
Skip |
Returns the number of points skipped at the beginning of the
sequence.
|
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.