LatBuilder::Kernel::RAlpha Class Reference

One-dimensional merit function for the \(\mathcal R_\alpha\) figure of merit for any \(\alpha \geq 0\). More...

#include <RAlpha.h>

Inherits LatBuilder::Kernel::Base< RAlpha >.

Public Member Functions
	RAlpha (Real alpha)
	Constructor. More...
Real	pointValue (const Real &x, uInteger n) const
	Returns the one-dimensional function evaluated at x. More...
template<LatticeType LR, EmbeddingType L, Compress C, PerLevelOrder P>
RealVector	valuesVector (const Storage< LR, L, C, P > &storage) const
	Creates a new vector of kernel values. More...
Real	alpha () const
	Returns the value of \(\alpha\).
bool	symmetric () const
std::string	name () const
Public Member Functions inherited from LatBuilder::Kernel::Base< RAlpha >
RealVector	valuesVector (const Storage< LR, L, C, P > &storage) const
	Creates a new vector of kernel values. More...
bool	symmetric () const
	Returns true if the kernel takes the same value at points \(x\) and \(1 - x\) for \(x \in [0,1)\).
std::string	name () const
	Returns the name of the kernel.
RAlpha &	derived ()
const RAlpha &	derived () const

Static Public Member Functions
static constexpr Compress	suggestedCompression ()

Static Public Attributes
static constexpr Real	CUPower = 2

Detailed Description

One-dimensional merit function for the \(\mathcal R_\alpha\) figure of merit for any \(\alpha \geq 0\).

This merit function is defined as

\[ \omega(x) = r_{\alpha,n}(x) = \sum_{h = -\lfloor (n-1)/2 \rfloor}^{\lfloor n/2 \rfloor} |\max(1, h)|^{-\alpha} e^{2 \pi i h x} - 1 \]

For even \(n\), the sum is over \(h=-n/2+1,\dots,n/2\); for odd \(n\), the sum is over \(h=-(n-1)/2,\dots,(n-1)/2\). By replacing \(h\) with \(-h\) in the part of the sum that is over the negative values of \(h\), we obtain:

\[ r_{\alpha,n}(x) = 2 \sum_{h=1}^{\lfloor (n-1)/2 \rfloor} h^{-\alpha} \cos(2\pi h x) + \mathbb 1[\text{$n$ is even}] \, (n/2)^{-\alpha} e^{i\pi n x} \]

This is the expression implemented by pointValue(). Note that the last term is nonzero only if \(n\) is even and, for \(x=j/n\), is equal to \((-1)^j\) for any integer \(j\).

Remarks

This functor only returns the real part of the last term.

Alternatively, by replacing \(h\) with \(h-n\) in the part of the sum that is over the negative values of \(h\), we obtain:

\[ r_{\alpha,n}(x) = \sum_{h=0}^{n-1} \hat r_{\alpha,n}(h) e^{2 \pi i h x}, \]

where

\[ \hat r_{\alpha,n}(h) = \begin{cases} 0 & \text{if $h=0$} \\ h^{-\alpha} & \text{if $0 < h \leq n/2$} \\ (n-h)^{-\alpha} & \text{if $n/2 < h < n $} \\ \end{cases} \]

The values of \(r_{\alpha,n}(j/n)\) for \(j=0,\dots,n-1\) are the \(n\)-point inverse discrete Fourier transform of \(\hat r_{\alpha,n}(h)\) at \(h=0,\dots,n-1\). This is how valuesVector() computes these values.

Template Parameters

ET	Type of lattice.

Constructor & Destructor Documentation

RAlpha()

LatBuilder::Kernel::RAlpha::RAlpha ( Real  alpha )

inline

Constructor.

Parameters

alpha

Value of \(\alpha\).

Member Function Documentation

pointValue()

Real LatBuilder::Kernel::RAlpha::pointValue ( const Real &  x, uInteger  n  ) const

inline

Returns the one-dimensional function evaluated at x.

Parameters

x	Point at which the function must be evaluated.
n	Number of terms in the sum (see class documentation).

Remarks

Returns only the real part of the kernel value.

References alpha().

valuesVector()

template<LatticeType LR, EmbeddingType L, Compress C, PerLevelOrder P>

RealVector LatBuilder::Kernel::RAlpha::valuesVector ( const Storage< LR, L, C, P > &  storage ) const

inline

Creates a new vector of kernel values.

The values of the kernel evaluated at sizeParam.numPoints() regular intervals in \([0,1)\) are stored in a linear vector. The intervals are of size 1/sizeParam.numPoints() and the first point is at 0.

the \(i^{\text{th}}\) element \(\omega_i\) is:

• \(\omega(i/n)\) in the case of an ordinary lattice with modulus \(n\).

• \(\omega((\nu_m(\frac{i(z)}{P(z)}))\) in the case of a polynomial lattice of modulus \(P(z)\) ( \( i(z) = \sum a_iz^i\) where \(i =\sum a_i2^i\)).

  Returns

  The newly created vector.

  Creates a new vector of kernel values using fast Fourier transforms.

References alpha(), and fftw< T >::ifft().

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

• include/latbuilder/Kernel/RAlpha.h