umontreal.iro.lecuyer.hups

Class HaltonSequence

java.lang.Object

umontreal.iro.lecuyer.hups.PointSet

umontreal.iro.lecuyer.hups.HaltonSequence

public class HaltonSequence
extends PointSet

This class implements the sequence of Halton, which is essentially a modification of Hammersley nets for producing an infinite sequence of points having low discrepancy. The ith point in s dimensions is

_i = (ψ_b1(i), ψ_b2(i),..., ψ_bs(i)),

for i = 0, 1, 2,..., where ψ_b is the radical inverse function in base b, defined in class RadicalInverse, and where 2 = b₁ < ^... < b_s are the s smallest prime numbers in increasing order.

A fast method is implemented to generate randomized Halton sequences, starting from an arbitrary points x₀.

The points can be ``scrambled'' by applying a permutation to the digits of i before computing each coordinate, in the same way as for the class HammersleyPointSet, for all coordinates j >= 0.

Constructor Summary

Constructors

Constructor and Description
HaltonSequence(int dim) Constructs a new Halton sequence in dim dimensions.

Method Summary

Modifier and Type	Method and Description
void	addFaurePermutations() Permutes the digits using Faure permutations for all coordinates.
void	ErasePermutations() Erases the Faure permutations: from now on, the digits will not be Faure permuted.
double	getCoordinate(int i, int j) Returns ui, j, the coordinate j of the point i.
int	getNumPoints() Returns the number of points.
void	init(double[] x0) Initializes the Halton sequence starting at point x0.

Methods inherited from class umontreal.iro.lecuyer.hups.PointSet

addRandomShift, addRandomShift, addRandomShift, addRandomShift, clearRandomShift, formatPoints, formatPoints, formatPoints, formatPoints, formatPointsBase, formatPointsBase, formatPointsBase, formatPointsBase, formatPointsNumbered, formatPointsNumbered, getDimension, getStream, iterator, randomize, randomize, randomize, randomize, randomize, setStream, toString, unrandomize

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

HaltonSequence

public HaltonSequence(int dim)

Constructs a new Halton sequence in dim dimensions.

Parameters:

dim - dimension

Method Detail

init

public void 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

addFaurePermutations

public void 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} = ∑_r=0^k-1π_j[a_r]b_j^-r-1,

for j = 0,..., s - 1, where π_j is the Faure permutation of {0,..., b_j -1}.

ErasePermutations

public void ErasePermutations()

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

getNumPoints

public int getNumPoints()

Description copied from class: PointSet

Returns the number of points. If this number is actually infinite, Integer.MAX_VALUE is returned.

Overrides:

getNumPoints in class PointSet

Returns:

the number of points in the point set or Integer.MAX_VALUE if the point set has an infinity of points.

getCoordinate

public double getCoordinate(int i,
                            int j)

Description copied from class: PointSet

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

Specified by:

getCoordinate in class PointSet

Parameters:

i - index of the point to look for

j - index of the coordinate to look for

Returns:

the value of u_{i, j}