This class computes the same discrepancy for randomly shifted points of a set \(\mathcal{L}\) as given in eq. More...
Public Member Functions | |
| DiscShift1Lattice (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions with all weights \(\gamma_r =1\). More... | |
| DiscShift1Lattice (double[][] points, int n, int s, double[] gamma) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions with the weights \(\gamma_r = \) gamma[r-1], \(r = 1, …, s\). More... | |
| DiscShift1Lattice (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| DiscShift1Lattice (Rank1Lattice set) | |
Constructor with the lattice set. More... | |
| DiscShift1Lattice () | |
| Empty constructor. More... | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( shift1lat ) for the \(s\)-dimensional points of lattice points, containing \(n\) points. More... | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
Computes the discrepancy ( shift1lat ) for the \(s\)-dimensional points of lattice points, containing \(n\) points, with weights \(\gamma_r = \) gamma[r-1]. | |
| double | compute (double[] T, int n) |
| Computes the discrepancy ( shift1dim1lat ) for the 1-dimensional lattice of \(n\) points \(T\). | |
 Public Member Functions inherited from DiscShift1 | |
| DiscShift1 (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions and with all weights \(\gamma_r =1\). More... | |
| DiscShift1 (double[][] points, int n, int s, double[] gamma) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions, and with the weights \(\gamma_r = \) gamma[r-1], \(r = 1, …, s\). More... | |
| DiscShift1 (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| DiscShift1 (PointSet set) | |
Constructor with the point set set. More... | |
| DiscShift1 () | |
| Empty constructor. More... | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( shift1 ) for the first \(n\) points of set points in dimension \(s\). More... | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
Computes the discrepancy ( shift1 ) in dimension \(s\) with \(\gamma_r = \) gamma[r-1]. | |
| double | compute (double[] T, int n) |
| Computes the discrepancy ( shift1dim1 ) for the 1-dimensional set of \(n\) points \(T\). | |
| Public Member Functions inherited from Discrepancy | |
| Discrepancy (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions. More... | |
| Discrepancy (double[][] points, int n, int s, double[] gamma) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions and the \(s\) weight factors gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| Discrepancy (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| Discrepancy (PointSet set) | |
Constructor with the point set set. More... | |
| Discrepancy () | |
| Empty constructor. More... | |
| double | compute () |
| Computes the discrepancy of all the points in maximal dimension (dimension of the points). | |
| double | compute (int s) |
| Computes the discrepancy of all the points in dimension \(s\). | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
Computes the discrepancy of the first n points of points in dimension s with weights gamma. | |
| abstract double | compute (double[][] points, int n, int s) |
Computes the discrepancy of the first n points of points in dimension s with weights \(=1\). | |
| double | compute (double[][] points) |
Computes the discrepancy of all the points of points in maximum dimension. More... | |
| double | compute (double[] T, int n) |
Computes the discrepancy of the first n points of T in 1 dimension. More... | |
| double | compute (double[] T) |
Computes the discrepancy of all the points of T in 1 dimension. More... | |
| double | compute (double[] T, int n, double gamma) |
Computes the discrepancy of the first n points of T in 1 dimension with weight gamma. | |
| double | compute (PointSet set, double[] gamma) |
Computes the discrepancy of all the points in set in the same dimension as the point set and with weights gamma. | |
| double | compute (PointSet set) |
Computes the discrepancy of all the points in set in the same dimension as the point set. More... | |
| int | getNumPoints () |
| Returns the number of points \(n\). | |
| int | getDimension () |
| Returns the dimension of the points \(s\). | |
| void | setPoints (double[][] points, int n, int s) |
Sets the points to points and the dimension to \(s\). More... | |
| void | setPoints (double[][] points) |
Sets the points to points. More... | |
| void | setGamma (double[] gam, int s) |
Sets the weight factors to gam for each dimension up to \(s\). | |
| double [] | getGamma () |
Returns the weight factors gamma for each dimension up to \(s\). | |
| String | toString () |
| Returns the parameters of this class. | |
| String | formatPoints () |
| Returns all the points of this class. | |
| String | getName () |
| Returns the name of the Discrepancy. | |
Additional Inherited Members | |
| Static Public Member Functions inherited from Discrepancy | |
| static double [][] | toArray (PointSet set) |
Returns all the \(n\) points ( \(s\)-dimensional) of umontreal.ssj.hups.PointSet set as an array points[ \(n\)][ \(s\)]. | |
| static DoubleArrayList | sort (double[] T, int n) |
| Sorts the first \(n\) points of \(T\). More... | |
| Protected Member Functions inherited from Discrepancy | |
| void | appendGamma (StringBuffer sb, double[] gamma, int s) |
| Static Protected Member Functions inherited from DiscShift1 | |
| static void | setC (double[] C1, double[] gam, int s) |
| Static Protected Member Functions inherited from Discrepancy | |
| static void | setONES (int s) |
| Protected Attributes inherited from Discrepancy | |
| double [] | gamma |
| double [][] | Points |
| int | dim |
| int | numPoints |
| Static Protected Attributes inherited from Discrepancy | |
| static double [] | ONES = { 1 } |
| Static Package Attributes inherited from Discrepancy | |
| static final double | UNSIX = 1.0/6.0 |
| static final double | QUARAN = 1.0/42.0 |
| static final double | UNTRENTE = 1.0 / 30.0 |
| static final double | DTIERS = 2.0 / 3.0 |
| static final double | STIERS = 7.0 / 3.0 |
| static final double | QTIERS = 14.0 / 3.0 |
Detailed Description
This class computes the same discrepancy for randomly shifted points of a set \(\mathcal{L}\) as given in eq.
( shift1 ) for class DiscShift1, but for the special case when the points are the nodes of an integration lattice [84] (eq. 16). It is given by
\[ [\mathcal{D}(\mathcal{L})]^2 = -1 + \frac{1}{n} \sum_{i=1}^n \prod_{r=1}^s \left[1 + \gamma_r^2 B_2(x_{ir})\right], \tag{shift1lat} \]
where \(n\) is the number of points of \(\mathcal{L}\), \(s\) is the dimension of the points, \(x_{ir}\) is the \(r\)-th coordinate of point \(i\), and the \(\gamma_r\) are arbitrary positive weights. \(B_2(x)\) is the Bernoulli polynomials [1] (chap. 23) of degree \(2\). For a 1-dimensional lattice, the discrepancy becomes
\[ [\mathcal{D}(\mathcal{L})]^2 = \frac{1}{n} \sum_{i=1}^n B_2(x_i). \tag{shift1dim1lat} \]
Computing the discrepancy for a lattice is much faster than for a general point set.
Constructor & Destructor Documentation
◆ DiscShift1Lattice() [1/5]
| DiscShift1Lattice | ( | double | points[][], |
| int | n, | ||
| int | s | ||
| ) |
Constructor with the \(n\) points points[i] in \(s\) dimensions with all weights \(\gamma_r =1\).
Element points[i][j] is the j-th coordinate of point i. Indices i and j start at 0.
◆ DiscShift1Lattice() [2/5]
| DiscShift1Lattice | ( | double | points[][], |
| int | n, | ||
| int | s, | ||
| double [] | gamma | ||
| ) |
Constructor with the \(n\) points points[i] in \(s\) dimensions with the weights \(\gamma_r = \) gamma[r-1], \(r = 1, …, s\).
points[i][j] is the j-th coordinate of point i. Indices i and j start at 0.
◆ DiscShift1Lattice() [3/5]
| DiscShift1Lattice | ( | int | n, |
| int | s, | ||
| double [] | gamma | ||
| ) |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\).
The \(n\) points will be chosen later.
◆ DiscShift1Lattice() [4/5]
| DiscShift1Lattice | ( | Rank1Lattice | set | ) |
Constructor with the lattice set.
All the points are copied in an internal array.
◆ DiscShift1Lattice() [5/5]
Empty constructor.
The points and parameters must be defined before calling methods of this class.
Member Function Documentation
◆ compute()
| double compute | ( | double | points[][], |
| int | n, | ||
| int | s | ||
| ) |
Computes the discrepancy ( shift1lat ) for the \(s\)-dimensional points of lattice points, containing \(n\) points.
All weights \(\gamma_r = 1\).
The documentation for this class was generated from the following file:
- DiscShift1Lattice.java
