|
Lattice Builder Manual
Software Package for Constructing Rank-1 Lattices
|
Bound on the average weighted \(\mathcal P_\alpha\) discrepancy. More...
#include <PAlphaDPW08.h>
Inherits LatBuilder::Norm::PAlphaBase< PAlphaDPW08 >.
Public Member Functions | |
| PAlphaDPW08 (unsigned int alpha, const LatCommon::Weights &weights, Real normType) | |
| Constructor. More... | |
| template<LatType L> | |
| Real | value (Real lambda, const SizeParam< L > &sizeParam, Dimension dimension, Real norm=1.0) const |
| std::string | name () const |
Public Member Functions inherited from LatBuilder::Norm::PAlphaBase< PAlphaDPW08 > | |
| PAlphaBase (unsigned int alpha, Real normType) | |
| Constructor. More... | |
| unsigned | alpha () const |
| Real | normType () const |
| Real | minExp () const |
| Real | maxExp () const |
| Real | value (Real lambda, const SizeParam< L > &sizeParam, Dimension dimension, Real norm=1.0) const |
| Returns the value of the bound. More... | |
| Real | operator() (const SizeParam< L > &sizeParam, Dimension dimension, Real norm=1.0) const |
Returns the smallest value of the bound for dimension dimension. More... | |
| Real | minimum (const SizeParam< L > &sizeParam, Dimension dimension, Real norm) const |
| Returns the minimum value of the bound function. More... | |
Additional Inherited Members | |
Static Public Attributes inherited from LatBuilder::Norm::PAlphaBase< PAlphaDPW08 > | |
| static const unsigned | MINIMIZER_MAX_ITER |
| Maximum number of iterations to be used with the minimizer. | |
| static const int | MINIMIZER_PREC_BITS |
| Relative precision on the minimum value to be used with the minimizer. | |
Bound on the average weighted \(\mathcal P_\alpha\) discrepancy.
This is the bound
\[ \min_{1/\alpha < \lambda \leq 1} \overline{E}^2_{n,s,\alpha,\boldsymbol\gamma}(\lambda) \]
from Theorem 10 (with the minimization step proposed in Algorithm 2) in [5] .
| LatBuilder::Norm::PAlphaDPW08::PAlphaDPW08 | ( | unsigned int | alpha, |
| const LatCommon::Weights & | weights, | ||
| Real | normType | ||
| ) |
Constructor.
| alpha | Smoothness level \(\alpha\) of the class of functions. |
| weights | Projection-dependent weights \( \gamma_{\mathfrak u} \). |
| normType | Type of cross-projection norm used by the figure of merit. |