This class implements the sequence of Halton [73] , which is essentially a modification of Hammersley nets for producing an infinite sequence of points having low discrepancy. More...

Inheritance diagram for umontreal.ssj.hups.HaltonSequence:

Public Member Functions
	HaltonSequence (int dim)
	Constructs a new Halton sequence in dim dimensions.
void	setStart (double[] x0)
	Initializes the Halton sequence starting at point x0.
void	init (double[] x0)
	Initializes the Halton sequence starting at point x0.
void	addFaureLemieuxPermutations ()
	Permutes the digits using permutations from [57] for all coordinates.
void	addFaurePermutations ()
	Permutes the digits using Faure permutations for all coordinates.
void	ErasePermutations ()
	Erases the permutations: from now on, the digits will not be permuted.
int	getNumPoints ()
	Returns the number of points.
double	getCoordinate (int i, int j)
	Returns \(u_{i,j}\), the coordinate \(j\) of the point \(i\).
Public Member Functions inherited from umontreal.ssj.hups.PointSet
int	getDimension ()
	Returns the dimension (number of available coordinates) of the points.
PointSetIterator	iterator ()
	Constructs and returns a point set iterator.
void	randomize (PointSetRandomization rand)
	Randomizes this point set using the given rand.
void	addRandomShift (int d1, int d2, RandomStream stream)
	By default, this method generates a random shift in the protected double[] array shift, to be used eventually for a random shift modulo 1.
void	addRandomShift (RandomStream stream)
	Same as `addRandomShift(0, dim, stream)`, where dim is the dimension of the point set.
void	addRandomShift (int d1, int d2)
	Refreshes the random shift (generates new uniform values for the random shift coordinates) for coordinates d1 to d2-1, using the saved shiftStream.
void	addRandomShift ()
	Same as `addRandomShift(0, dim)`, where dim is the dimension of the point set.
void	clearRandomShift ()
	Erases the current random shift, if any.
String	toString ()
	Formats a string that contains information about the point set.
String	formatPoints ()
	Same as invoking `formatPoints(n, d)` with \(n\) and \(d\) equal to the number of points and the dimension of this object, respectively.
String	formatPoints (int n, int d)
	Formats a string that displays the same information as returned by toString, together with the first \(d\) coordinates of the first.
String	formatPoints (PointSetIterator iter)
	Same as invoking `formatPoints(iter, n, d)` with \(n\) and \(d\) equal to the number of points and the dimension, respectively.
String	formatPoints (PointSetIterator iter, int n, int d)
	Same as invoking `formatPoints(n, d)`, but prints the points by calling iter repeatedly.
String	formatPointsBase (int b)
	Similar to `formatPoints()`, but the points coordinates are printed in base \(b\).
String	formatPointsBase (int n, int d, int b)
	Similar to `formatPoints(n, d)`, but the points coordinates are printed in base \(b\).
String	formatPointsBase (PointSetIterator iter, int b)
	Similar to `formatPoints(iter)`, but the points coordinates are printed in base.
String	formatPointsBase (PointSetIterator iter, int n, int d, int b)
	Similar to `formatPoints(iter, n, d)`, but the points coordinates are printed in base \(b\).
String	formatPointsNumbered ()
	Same as invoking `formatPointsNumbered(n, d)` with \(n\) and \(d\) equal to the number of points and the dimension, respectively.
String	formatPointsNumbered (int n, int d)
	Same as invoking `formatPoints(n,d)`, except that the points are numbered.

Additional Inherited Members
Static Public Member Functions inherited from umontreal.ssj.hups.PointSet
static int	getMaxBits ()
	Returns the maximum number of usable bits.
Protected Attributes inherited from umontreal.ssj.hups.PointSet
double	EpsilonHalf = 1.0 / Num.TWOEXP[55]
	To avoid 0 for nextCoordinate when random shifting, we add this to each coordinate.
int	dim = 0
	Dimension of the points.
int	numPoints = 0
	Number of points.
int	dimShift = 0
	Current dimension of the shift.
int	capacityShift = 0
	Number of array elements in the shift vector, always >= dimShift.
double[]	shift
	This is the shift vector as a double[] array, which contains the current random shift in case we apply a random shift modulo 1.
RandomStream	shiftStream
	Stream used to generate the random shifts.
Static Protected Attributes inherited from umontreal.ssj.hups.PointSet
static final int	MAXBITS = 31
	Since Java has no unsigned type, the 32nd bit cannot be used efficiently, so we have only 31 bits.

Detailed Description

This class implements the sequence of Halton [73] , which is essentially a modification of Hammersley nets for producing an infinite sequence of points having low discrepancy.

The \(i\)th point in \(s\) dimensions is

\[ \mathbf{u}_i = (\psi_{b_1}(i),\psi_{b_2}(i),…, \psi_{b_s}(i)), \tag{Halton-point2} \]

for \(i=0,1,2,…\), where \(\psi_b\) is the radical inverse function in base \(b\), defined in class RadicalInverse, and where \(2 = b_1 < \cdots< b_s\) are the \(s\) smallest prime numbers in increasing order.

A fast method is implemented to generate randomized Halton sequences [218], [225] , starting from an arbitrary point \(x_0\).

       The points can be "scrambled" by applying a permutation to the
       digits of

\(i\) before computing each coordinate via ( Halton-point ), in the same way as for the class HammersleyPointSet, for all coordinates \(j\ge0\).

Definition at line 53 of file HaltonSequence.java.

Constructor & Destructor Documentation

◆ HaltonSequence()

umontreal.ssj.hups.HaltonSequence.HaltonSequence ( int dim )

Constructs a new Halton sequence in dim dimensions.

Parameters

dim dimension

Definition at line 66 of file HaltonSequence.java.

Member Function Documentation

◆ addFaureLemieuxPermutations()

void umontreal.ssj.hups.HaltonSequence.addFaureLemieuxPermutations ( )

Permutes the digits using permutations from [57] for all coordinates.

After the method is called, the coordinates

\(u_{i,j}\) are generated via

\[ u_{i,j} = \sum_{r=0}^{k-1} \pi_j[a_r] b_j^{-r-1}, \]

for \(j=0,…,s-1\), where \(\pi_j\) is the Faure-Lemieux (2008) permutation of \(\{0,…,b_j-1\}\).

Definition at line 109 of file HaltonSequence.java.

◆ addFaurePermutations()

void umontreal.ssj.hups.HaltonSequence.addFaurePermutations ( )

Permutes the digits using Faure permutations for all coordinates.

After the method is called, the coordinates \(u_{i,j}\) are generated via

\[ u_{i,j} = \sum_{r=0}^{k-1} \pi_j[a_r] b_j^{-r-1}, \]

for \(j=0,…,s-1\), where \(\pi_j\) is the Faure permutation of \(\{0,…,b_j-1\}\).

Definition at line 124 of file HaltonSequence.java.

◆ ErasePermutations()

void umontreal.ssj.hups.HaltonSequence.ErasePermutations ( )

Erases the permutations: from now on, the digits will not be permuted.

Definition at line 136 of file HaltonSequence.java.

◆ getCoordinate()

double umontreal.ssj.hups.HaltonSequence.getCoordinate	(	int	i,
		int	j )

Returns \(u_{i,j}\), the coordinate \(j\) of the point \(i\).

When the points are randomized (e.g., a random shift is added), the values returned by this method should incorporate the randomizations. Richard: La méthode getCoordinate de certaines classes ne tient pas compte du random shift, contrairement à l’itérateur de la même classe. Faut-il que toutes les getCoordinate implémentent le random shift quand il existe? Oui.

Parameters

i	index of the point to look for
j	index of the coordinate to look for

Returns: the value of \(u_{i,j}\)

Reimplemented from umontreal.ssj.hups.PointSet.

Definition at line 145 of file HaltonSequence.java.

◆ getNumPoints()

int umontreal.ssj.hups.HaltonSequence.getNumPoints ( )

Returns the number of points.

If this number is actually infinite, Integer.MAX_VALUE is returned.

Returns: the number of points in the point set or Integer.MAX_VALUE if the point set has infinity cardinality.

Reimplemented from umontreal.ssj.hups.PointSet.

Definition at line 141 of file HaltonSequence.java.

◆ init()

void umontreal.ssj.hups.HaltonSequence.init ( double[] x0 )

Initializes the Halton sequence starting at point x0.

The dimension of x0 must be at least as large as the dimension of this object.

Parameters

x0	starting point of the Halton sequence

Definition at line 95 of file HaltonSequence.java.

◆ setStart()

void umontreal.ssj.hups.HaltonSequence.setStart ( double[] x0 )

Initializes the Halton sequence starting at point x0.

For each coordinate \(j\), the sequence starts at index \(i_j\) such that x0[ \(j\)] is the radical inverse of \(i_j\). The dimension of x0 must be at least as large as the dimension of this object.

Parameters

x0	starting point of the Halton sequence

Definition at line 84 of file HaltonSequence.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/hups/HaltonSequence.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ HaltonSequence()

Member Function Documentation

◆ addFaureLemieuxPermutations()

◆ addFaurePermutations()

◆ ErasePermutations()

◆ getCoordinate()

◆ getNumPoints()

◆ init()

◆ setStart()