LatNet Builder Manual  2.0.1-11
Software Package for Constructing Highly Uniform Point Sets
LatBuilder::Norm::PAlphaSL10 Class Reference

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

#include <PAlphaSL10.h>

Inherits LatBuilder::Norm::NormAlphaBase< PAlphaSL10 >.

Public Member Functions

 PAlphaSL10 (unsigned int alpha, const LatticeTester::Weights &weights, Real normType=2)
 Constructor. More...
 
template<LatticeType LR, EmbeddingType L>
Real value (Real lambda, const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const
 
std::string name () const
 
- Public Member Functions inherited from LatBuilder::Norm::NormAlphaBase< PAlphaSL10 >
 NormAlphaBase (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< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const
 Returns the value of the bound. More...
 
Real operator() (const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const
 Returns the smallest value of the bound for dimension dimension. More...
 
Real minimum (const SizeParam< LR, 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::NormAlphaBase< PAlphaSL10 >
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.
 

Detailed Description

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

This is the general bound derived in Theorem 3 of [31] for projection-dependent weights. The theorem 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

\[ \mathcal D^2(\boldsymbol a_s, n) \leq N_{n,s}(c, \lambda) \]

for any \(\lambda \in (1/\alpha,1]\) and any \(c \in [0, 1]\), where

\[ N_{n,s}(c, \lambda) = \left[ \frac{1}{c \, \varphi(n)} \sum_{\emptyset \neq \mathfrak u \subseteq \{1,\dots,s\}} \gamma_{\mathfrak u}^\lambda \, \left( 2 \zeta(\alpha\lambda) \right)^{|\mathfrak u|} \right]^{1/\lambda}, \]

in which \(\zeta\) is the Riemann zeta function and \(\varphi\) is Euler's totient function. The normalization that is used is:

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

For order-dependent weights, the bound can be rewritten as:

\[ N_{n,s}(c, \lambda) = \left[ \frac{1}{c \, \varphi(n)} \sum_{\ell=1}^s \Gamma_\ell^\lambda \, \frac{s!}{\ell! (s-\ell)!} \, \left( 2 \zeta(\alpha\lambda) \right)^\ell \right]^{1/\lambda} = \left[ \frac{1}{c \, \varphi(n)} \sum_{\ell=1}^s \Gamma_\ell^\lambda \, y_\ell(\lambda) \right]^{1/\lambda}, \]

where

\[ y_\ell(\lambda) = \frac{s - \ell + 1}{\ell} \, 2 \zeta(\alpha\lambda) \times y_{\ell - 1}(\lambda) \]

for \(\ell \geq 1\) and \(y_0(\alpha) = 1\).

For product weights, it can be written as:

\[ N_{n,s}(c, \lambda) = \left\{ \frac{1}{c \, \varphi(n)} \left[ \prod_{j=1}^s \left( 1 + 2 \gamma_j^\lambda \, \zeta(\alpha\lambda) \right) - 1 \right] \right\}^{1/\lambda}, \]

For product and order-dependent (POD) weights, the bound can be written as:

\[ N_{n,s}(c, \lambda) = \left[ \frac{1}{c \, \varphi(n)} \sum_{\ell=1}^s \Gamma_\ell^\lambda \, y_\ell(\lambda) \right]^{1/\lambda}, \]

where

\[ y_\ell(\lambda) = \frac{s - \ell + 1}{\ell} \, 2 \gamma_\ell^\lambda \zeta(\alpha\lambda) \times y_{\ell - 1}(\lambda) \]

for \(\ell \geq 1\) and \(y_0(\lambda) = 1\).

Examples:
tutorial/FilteredCBC.cc, and tutorial/FilteredRCBC.cc.

Constructor & Destructor Documentation

◆ PAlphaSL10()

LatBuilder::Norm::PAlphaSL10::PAlphaSL10 ( unsigned int  alpha,
const LatticeTester::Weights weights,
Real  normType = 2 
)

Constructor.

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

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