This class computes the same discrepancy in [84] (eq. More...
Public Member Functions | |
| DiscShiftBaker1Lattice (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions, and with all weights \(\gamma_r =1\). More... | |
| DiscShiftBaker1Lattice (double[][] points, int n, int s, double[] gamma) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions, with weights \(\gamma_r = \) gamma[r-1]. More... | |
| DiscShiftBaker1Lattice (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(r\)], \(r = 0, 1, …, (s-1)\). More... | |
| DiscShiftBaker1Lattice (Rank1Lattice set) | |
Constructor with the point set set. More... | |
| DiscShiftBaker1Lattice () | |
| Empty constructor. More... | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( shiftBaker1lat ) 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 ( shiftBaker1lat ) 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 ( shiftBaker1latdim1 ) with weight \(\gamma=1\) for the 1-dimensional lattice of \(n\) points \(T\). | |
| double | compute (double[] T, int n, double gamma) |
Computes the discrepancy ( shiftBaker1latdim1 ) for the 1-dimensional lattice of \(n\) points \(T\), with weight \(\gamma=\) gamma. | |
 Public Member Functions inherited from DiscShiftBaker1 | |
| DiscShiftBaker1 (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions, with all the weights \(\gamma_r = 1\). More... | |
| DiscShiftBaker1 (double[][] points, int n, int s, double[] gamma) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions, with weights \(\gamma_r = \) gamma[r-1]. More... | |
| DiscShiftBaker1 (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(r\)], \(r = 0, 1, …, (s-1)\). More... | |
| DiscShiftBaker1 (PointSet set) | |
Constructor with the point set set. More... | |
| DiscShiftBaker1 () | |
| Empty constructor. More... | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( baker1 ) for the \(s\)-dimensional points of set points, containing \(n\) points. More... | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
Computes the discrepancy ( baker1 ) for the first \(n\) points of points in dimension \(s\) and with weight \(\gamma_r = \) gamma[r-1]. | |
| double | compute (double[] T, int n) |
| Computes the discrepancy ( baker1dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma= 1\). | |
| double | compute (double[] T, int n, double gamma) |
Computes the discrepancy ( baker1dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma=\) gamma. | |
| 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. | |
Static Package Attributes | |
| static final double | TRENTEUN24 = 31.0/24.0 |
| static final double | SEPT24 = 7.0/24.0 |
| 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 |
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 DiscShiftBaker1 | |
| static double [] | setC (double gam) |
| static void | setC (double[] C1, double[] C2, double[] C3, 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 } |
Detailed Description
This class computes the same discrepancy in [84] (eq.
16) for the randomly shifted points of a set \(\mathcal{L}\) as given in eq. ( baker1 ) for class DiscShiftBaker1, but for the special case when the points are the nodes of an integration lattice. It is given by
\begin{align} [\mathcal{D}(\mathcal{L})]^2 & = -1 + \frac{1}{n} \sum_{i=1}^n \prod_{r=1}^s \left[1 - \frac{4\gamma_r^2}{3} \left[B_4(x_{ir}) - B_4(\{x_{ir}-1/2\})\right]\right. - \nonumber \\ & \frac{\gamma_r^4}{9} \left[7B_4(x_{ir}) - 2B_4(\{x_{ir}-1/2\})\right] \left. {} - \frac{16\gamma_r^4}{45} \left[B_6(x_{ir}) -B_6(\{x_{ir}-1/2\})\right] \right], \tag{shiftBaker1lat} \end{align}
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. The \(B_{\alpha}(x)\) are the Bernoulli polynomials [1] (chap. 23) of degree \(\alpha\). The discrepancy is much faster to calculate for a lattice than for a general point set. For a 1-dimensional lattice, the discrepancy becomes
\begin{align} [\mathcal{D}(\mathcal{L})]^2 & = - \frac{1}{n} \sum_{i=1}^n \left[\frac{4\gamma^2}{3} \left[B_4(x_i) - B_4(\{x_i-1/2\})\right]\right. + \nonumber \\ & \frac{\gamma^4}{9} \left[7B_4(x_i) - 2B_4(\{x_i-1/2\})\right] \left. {} + \frac{16\gamma^4}{45} \left[B_6(x_i) -B_6(\{x_i-1/2\})\right] \right], \tag{shiftBaker1latdim1} \end{align}
Constructor & Destructor Documentation
◆ DiscShiftBaker1Lattice() [1/5]
| DiscShiftBaker1Lattice | ( | double | points[][], |
| int | n, | ||
| int | s | ||
| ) |
Constructor with the \(n\) points points[i] in \(s\) dimensions, and with all weights \(\gamma_r =1\).
points[i][r] is the r-th coordinate of point i. Indices i and r start at 0.
◆ DiscShiftBaker1Lattice() [2/5]
| DiscShiftBaker1Lattice | ( | double | points[][], |
| int | n, | ||
| int | s, | ||
| double [] | gamma | ||
| ) |
Constructor with the \(n\) points points[i] in \(s\) dimensions, with weights \(\gamma_r = \) gamma[r-1].
points[i][r] is the r-th coordinate of point i. Indices i and r start at 0.
◆ DiscShiftBaker1Lattice() [3/5]
| DiscShiftBaker1Lattice | ( | int | n, |
| int | s, | ||
| double [] | gamma | ||
| ) |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(r\)], \(r = 0, 1, …, (s-1)\).
The \(n\) points will be chosen later.
◆ DiscShiftBaker1Lattice() [4/5]
Constructor with the point set set.
All the points are copied in an internal array.
◆ DiscShiftBaker1Lattice() [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 ( shiftBaker1lat ) 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:
- DiscShiftBaker1Lattice.java
