umontreal.iro.lecuyer.hups

Class HammersleyPointSet

java.lang.Object

umontreal.iro.lecuyer.hups.PointSet

umontreal.iro.lecuyer.hups.HammersleyPointSet

public class HammersleyPointSet
extends PointSet

This class implements Hammersley point sets, which are defined as follows. Let 2 = b₁ < b₂ < ^... denote the sequence of all prime numbers by increasing order. The Hammersley point set with n points in s dimensions contains the points

u_i = (i/n, ψ_b1(i), ψ_b2(i),..., ψ_bs-1(i)),

for i = 0,..., n - 1, where ψ_b is the radical inverse function in base b, defined in RadicalInverse. This class is not a subclass of DigitalNet, because the basis is not the same for all coordinates. We do obtain a net in a generalized sense if n = b₁^k₁ = b₂^k₂ = ^... = b_s-1^k_s-1 for some integers k₁,..., k_s-1.

The points of a Hammersley point set can be ``scrambled'' by applying a permutation to the digits of i before computing each coordinate. If

i = a₀ + a₁b_j + ... + a_kj-1b_j^k_j-1,

and π_j is a permutation of the digits {0,..., b_j -1}, then

ψ_bj(i) = ∑_r=0^k_j-1a_rb_j^-r-1

is replaced by

u_{i, j} = ∑_r=0^k_j-1π_j[a_r]b_j^-r-1.

The permutations π_j can be deterministic or random. One (deterministic) possibility implemented here is to use the Faure permutation of {0,..., b_j} for π_j, for each coordinate j > 0.

Constructor Summary

Constructors

Constructor and Description
HammersleyPointSet(int n, int dim) Constructs a new Hammersley point set with n points in dim dimensions.

Method Summary

Methods

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.

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, getNumPoints, 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

HammersleyPointSet

public HammersleyPointSet(int n,
                  int dim)

Constructs a new Hammersley point set with n points in dim dimensions.

Parameters:

n - number of points

dim - dimension of the point set

Method Detail

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 = 1,..., s - 1 and u_{i, 0} = i/n, 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.

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}