Lattice Builder
Software Package for Constructing Rank-1 Lattices
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LatBuilder::Norm::PAlphaDPW08 Class Reference

Bound on the weighted \(\mathcal P_\alpha\) discrepancy. More...

#include <PAlphaDPW08.h>

+ Inheritance diagram for LatBuilder::Norm::PAlphaDPW08:
+ Collaboration diagram for LatBuilder::Norm::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. More...
 
static const int MINIMIZER_PREC_BITS
 Relative precision on the minimum value to be used with the minimizer. More...
 

Detailed Description

Bound on the weighted \(\mathcal P_\alpha\) discrepancy.

A theorem in[1] states that, for \(\mathcal D^2(\boldsymbol a_s, n) = \mathcal P_\alpha(\boldsymbol a_s, n)\), there exists a generating vector \(\boldsymbol a_s \in \mathbb Z^s\) such that

\[ N_{n,s}(c, \lambda) = \dots \]

Todo:

Copy the definition of lambda.

Precise theorem number.

Constructor & Destructor Documentation

LatBuilder::Norm::PAlphaDPW08::PAlphaDPW08 ( unsigned int  alpha,
const LatCommon::Weights &  weights,
Real  normType 
)

Constructor.

Parameters
alphaSmoothness level \(\alpha\) of the class of functions.
weightsProjection-dependent weights \( \gamma_{\mathfrak u} \). normType Type of cross-projection norm used by the figure of merit.

The documentation for this class was generated from the following file: