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Lattice Builder Manual
Software Package for Constructing Rank-1 Lattices
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Lattice Builder Manual
Software Package for Constructing Rank-1 Lattices
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Bound on the weighted \(\mathcal P_\alpha\) discrepancy. More...
#include <PAlphaSL10.h>
Inherits LatBuilder::Norm::PAlphaBase< PAlphaSL10 >.
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... | |
Additional Inherited Members | |
| 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. | |
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 \(y_0(\lambda) = 1\).
| LatBuilder::Norm::PAlphaSL10::PAlphaSL10 | ( | unsigned int | alpha, |
| const LatCommon::Weights & | weights, | ||
| Real | normType = 2 |
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| ) |
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. |
1.8.10 
