Lattice Builder Manual Software Package for Constructing Rank-1 Lattices
LatBuilder::Norm::PAlphaSL10 Class Reference

Bound on the weighted $$\mathcal P_\alpha$$ discrepancy. More...

#include <PAlphaSL10.h>

## Public Member Functions

PAlphaSL10 (unsigned int alpha, const LatCommon::Weights &weights, Real normType=2)
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< PAlphaSL10 >
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...

Static Public Attributes inherited from LatBuilder::Norm::PAlphaBase< 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 [18] 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

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

Constructor.

Parameters
 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.

The documentation for this class was generated from the following file:
• latbuilder/include/latbuilder/Norm/PAlphaSL10.h