HaltonSequence (Java Libraries for Stochastic Simulation)

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

$\displaystyle \mbox{\boldmath$u$}$ _i = (ψ_b₁(i), ψ_b₂(i),..., ψ_{b_s}(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.

Constructors
Constructor and Description
`HaltonSequence(int dim)` Constructs a new Halton sequence 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 u_{i, 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}

Class HaltonSequence

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

HaltonSequence

Method Detail

init

addFaurePermutations

ErasePermutations

getNumPoints

getCoordinate