This class computes the same discrepancy in [81] (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 \(\gamma_r =1\). | |
| 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]. | |
| DiscShiftBaker1Lattice (int n, int s, double[] gamma) | |
| The number of points is \(n\), the dimension \(s\), and the. | |
| DiscShiftBaker1Lattice (Rank1Lattice set) | |
| Constructor with the point set set. | |
| DiscShiftBaker1Lattice () | |
| Empty constructor. | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( shiftBaker1lat ) for the \(s\)-dimensional points of lattice points, containing \(n\) points. | |
| 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 umontreal.ssj.discrepancy.DiscShiftBaker1 | |
| DiscShiftBaker1 (double[][] points, int n, int s) | |
| Constructor with the \(n\) points points[i] in \(s\) dimensions, with all the weights \(\gamma_r = 1\). | |
| 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]. | |
| DiscShiftBaker1 (int n, int s, double[] gamma) | |
| The number of points is \(n\), the dimension \(s\), and the. | |
| DiscShiftBaker1 (PointSet set) | |
| Constructor with the point set set. | |
| DiscShiftBaker1 () | |
| Empty constructor. | |
| Public Member Functions inherited from umontreal.ssj.discrepancy.Discrepancy | |
| Discrepancy (double[][] points, int n, int s) | |
| Constructor with the \(n\) points points[i] in \(s\) dimensions. | |
| 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[ \(j\)], \(j = 0, 1,
…, (s-1)\). | |
| Discrepancy (int n, int s, double[] gamma) | |
| The number of points is \(n\), the dimension \(s\), and the. | |
| Discrepancy (PointSet set) | |
| Constructor with the point set set. | |
| Discrepancy () | |
| Empty constructor. | |
| 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) |
| Computes the discrepancy of all the points of points in maximum dimension. | |
| double | compute (double[] T) |
| Computes the discrepancy of all the points of T in 1 dimension. | |
| 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. | |
| 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 \(s\). | |
| void | setPoints (double[][] points) |
| Sets the points to points. | |
| 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 umontreal.ssj.discrepancy.Discrepancy | |
| static double[][] | toArray (PointSet set) |
| Returns all the \(n\) points ( \(s\)-dimensional) of. | |
| static DoubleArrayList | sort (double[] T, int n) |
| Sorts the first \(n\) points of \(T\). | |
Detailed Description
This class computes the same discrepancy in [81] (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}
Definition at line 61 of file DiscShiftBaker1Lattice.java.
Constructor & Destructor Documentation
◆ DiscShiftBaker1Lattice() [1/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.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.
Definition at line 98 of file DiscShiftBaker1Lattice.java.
◆ DiscShiftBaker1Lattice() [2/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.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.
Definition at line 107 of file DiscShiftBaker1Lattice.java.
◆ DiscShiftBaker1Lattice() [3/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.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.
Definition at line 117 of file DiscShiftBaker1Lattice.java.
◆ DiscShiftBaker1Lattice() [4/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.DiscShiftBaker1Lattice | ( | Rank1Lattice | set | ) |
Constructor with the point set set.
All the points are copied in an internal array.
Definition at line 125 of file DiscShiftBaker1Lattice.java.
◆ DiscShiftBaker1Lattice() [5/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.DiscShiftBaker1Lattice | ( | ) |
Empty constructor.
The points and parameters must be defined before calling methods of this class.
Definition at line 133 of file DiscShiftBaker1Lattice.java.
Member Function Documentation
◆ compute() [1/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.compute | ( | double[] | T, |
| int | n ) |
Computes the discrepancy ( shiftBaker1latdim1 ) with weight \(\gamma=1\) for the 1-dimensional lattice of \(n\) points \(T\).
Reimplemented from umontreal.ssj.discrepancy.DiscShiftBaker1.
Definition at line 182 of file DiscShiftBaker1Lattice.java.
◆ compute() [2/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.compute | ( | double[] | T, |
| int | n, | ||
| double | gamma ) |
Computes the discrepancy ( shiftBaker1latdim1 ) for the 1-dimensional lattice of \(n\) points \(T\), with weight \(\gamma=\) gamma.
Reimplemented from umontreal.ssj.discrepancy.DiscShiftBaker1.
Definition at line 192 of file DiscShiftBaker1Lattice.java.
◆ compute() [3/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.compute | ( | double | points[][], |
| int | n, | ||
| int | s ) |
Computes the discrepancy ( shiftBaker1lat ) for the \(s\)-dimensional points of lattice points, containing \(n\) points.
All weights \(\gamma_r = 1\).
Reimplemented from umontreal.ssj.discrepancy.DiscShiftBaker1.
Definition at line 142 of file DiscShiftBaker1Lattice.java.
◆ compute() [4/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.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].
Reimplemented from umontreal.ssj.discrepancy.DiscShiftBaker1.
Definition at line 153 of file DiscShiftBaker1Lattice.java.
The documentation for this class was generated from the following file:
- src/main/java/umontreal/ssj/discrepancy/DiscShiftBaker1Lattice.java
