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NORMALIZED_POLYNOMIAL< T > Class Template Reference

This class is used to define polynomial axisymmetric coarse-graining functions. More...

#include <NormalisedPolynomial.h>

## Public Member Functions

NORMALIZED_POLYNOMIAL ()
Basic constructor; note that this does not determine the particular polynomial; one needs to call set_polynomial to define the coefficients. More...

void set_polynomial (std::vector< Mdouble > new_coefficients, unsigned int new_dim)
Use this function to set the polynomial coefficients $$c_i$$. This function calls finish_set_polynomial to normalize the coefficients. More...

void set_polynomial (Mdouble *new_coefficients, unsigned int num_coeff, unsigned int new_dim)
Some as set_polynomial, but avoids the use of a vector. More...

void setName (const char *new_name)
Use this function to change the name of the polynomial. More...

std::string getName ()
Returns name of the polynomial. More...

Mdouble evaluate (Mdouble r)
Returns the value of the polynomial, $$p(r)=\sum_{i=0}^N c_i r^{N-i}$$. More...

Returns the gradient of the polynomial, $$\partial_\alpha p(x,y,z)=\sum_{i=0}^N c_{i,\alpha} r^{N-i},\ \alpha=x,y,z$$. More...

Mdouble evaluateIntegral (Mdouble a, Mdouble b, Mdouble t)
Returns the value of the line integral along the normal P1P2 "from a to b" over the axisymmetric function. More...

int getOrder (void)
Returns the order of the polynomial. More...

## Private Member Functions

void finish_set_polynomial ()
Normalizes the polynomial coefficients $$c_i$$ such that the integral over the unit sphere of the axisymmetric function $$p(r)$$ is unity. More...

Mdouble get_volume ()
Returns the integral over the unit sphere of the axisymmetric function $$p(r)$$. More...

Mdouble evaluate_1D (Mdouble r)
Returns the value of the polynomial averaged over 2 dimensions. For StatType=X, $$r=|x|$$. See also set_average_1D. More...

Mdouble evaluate_2D (Mdouble r)
Returns the value of the polynomial averaged over 1 dimension. For StatType=XY, $$r=\sqrt{x^2+y^2}$$. More...

void set_average ()
Sets averaged_coefficients. More...

void set_average_1D ()
Sets averaged_coefficients $$\bar{c}_i$$ for StatType=X,Y,Z such that $$\sum_{i=0}^N \bar{c}_i x^{N-i} = \int\int_{|\vec{x}|\leq 1} p(|\vec{x}|) dy dz$$. See evaluate_1D. More...

void set_average_2D ()
For StatType=XY,XZ,XZ, averaged_coefficients is not used since $$\bar{p}(r)$$ can be evaluated as a function of $$c_i$$. See evaluate_2D. More...

Returns the value of the gradient averaged over 2 dimensions. More...

Returns the value of the gradient averaged over 1 dimensions. More...

Mdouble evaluateIntegral_1D (Mdouble a, Mdouble b, Mdouble t)
Returns the value of the line integral along the normal P1P2 "from a to b" over the axisymmetric function averaged over 2 dimensions. More...

Mdouble evaluateIntegral_2D (Mdouble a, Mdouble b, Mdouble t)
Returns the value of the line integral along the normal P1P2 "from a to b" over the axisymmetric function averaged over 1 dimensions. More...

Mdouble operator[] (int i) const

## Private Attributes

std::string name
Contains the name of the polynomial which will be displayed as CGtype by the statistical code. More...

unsigned int dim
The system dimension. More...

std::vector< Mdoublecoefficients
Stores the coefficients $$c_i$$. More...

std::vector< Mdoubleaveraged_coefficients
Stores some coefficients used in evaluate and evaluateIntegral for StatTypes different from XYZ. More...

## Friends

std::ostream & operator<< (std::ostream &os, const NORMALIZED_POLYNOMIAL &P)
Returns a text description of the polynomial. More...

## Detailed Description

### template<StatType T> class NORMALIZED_POLYNOMIAL< T >

This class is used to define polynomial axisymmetric coarse-graining functions.

This class stores a polynomial, $$p(r)=\sum_{i=0}^N c_i r^{N-i}$$, which is normalized such that the integral over the unit sphere of the axisymmetric function $$p(|\vec{x}|)$$ is unity.
Use set_polynomial to define the polynomial. Use evaluate to evaluate the polynomial. Use evaluateGradient to evaluate the polynomial's gradient.
Calculations can be found in src/docs/Polynomials.nb
This is used to define polynomial axisymmetric coarse-graining functions (see StatisticsVector).
Note: not everything is implemented yet: only dim=3 is working, no gradients are computed.

Definition at line 51 of file NormalisedPolynomial.h.

## Constructor & Destructor Documentation

template<StatType T>
 NORMALIZED_POLYNOMIAL< T >::NORMALIZED_POLYNOMIAL ( )
inline

Basic constructor; note that this does not determine the particular polynomial; one needs to call set_polynomial to define the coefficients.

Definition at line 85 of file NormalisedPolynomial.h.

86  {
87  setName("Polynomial");
88  coefficients.resize(0);
89  dim = 0;
90  }
void setName(const char *new_name)
Use this function to change the name of the polynomial.
unsigned int dim
The system dimension.
std::vector< Mdouble > coefficients
Stores the coefficients .

## Member Function Documentation

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluate ( Mdouble r )

Returns the value of the polynomial, $$p(r)=\sum_{i=0}^N c_i r^{N-i}$$.

For averaged StatType this function is templated. If averaging statistics are used, then an averaged function is stored as well; for averaging a over certain dimensions is stored as well.

For averaging over two dimensions, $$(y_{max}-y_{min})\cdot (z_{max}-z_{min})\cdot \bar{p}(x)=\int_{\vec{x}\leq1} p(|\vec{x}|) dy\,dz = \sum_{i=0}^N \bar{c}_i r^{N-i}$$.

For averaging over one dimensions, $$(z_{max}-z_{min})\cdot \bar{p}(x,y)=\int_{\vec{x}\leq1} p(|\vec{x}|) dz = \sum_{i=0}^N \bar{c}_i r^{N-i}$$.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluate_1D ( Mdouble r )
private

Returns the value of the polynomial averaged over 2 dimensions. For StatType=X, $$r=|x|$$. See also set_average_1D.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluate_2D ( Mdouble r )
private

Returns the value of the polynomial averaged over 1 dimension. For StatType=XY, $$r=\sqrt{x^2+y^2}$$.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluateGradient ( Mdouble r )

Returns the gradient of the polynomial, $$\partial_\alpha p(x,y,z)=\sum_{i=0}^N c_{i,\alpha} r^{N-i},\ \alpha=x,y,z$$.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluateGradient_1D ( Mdouble r )
private

Returns the value of the gradient averaged over 2 dimensions.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluateGradient_2D ( Mdouble r )
private

Returns the value of the gradient averaged over 1 dimensions.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluateIntegral ( Mdouble a, Mdouble b, Mdouble t )

Returns the value of the line integral along the normal P1P2 "from a to b" over the axisymmetric function.

circle denotes the cutoff radius of the cg function around P

For averaged StatType this function is templated.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluateIntegral_1D ( Mdouble a, Mdouble b, Mdouble t )
private

Returns the value of the line integral along the normal P1P2 "from a to b" over the axisymmetric function averaged over 2 dimensions.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::evaluateIntegral_2D ( Mdouble a, Mdouble b, Mdouble t )
private

Returns the value of the line integral along the normal P1P2 "from a to b" over the axisymmetric function averaged over 1 dimensions.

template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::finish_set_polynomial ( )
private

Normalizes the polynomial coefficients $$c_i$$ such that the integral over the unit sphere of the axisymmetric function $$p(r)$$ is unity.

$$\int_0^1 f(r) p(r) dr = 1$$, with $$f(r)=4\pi r^2$$ for 3D, $$f(r)=2\pi r$$ for 2D, $$f(r)=2$$ for 1D systems.

Also sets averaged_coefficients

Assumes that dim and coefficients are already set.

template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::get_volume ( )
private

Returns the integral over the unit sphere of the axisymmetric function $$p(r)$$.

$$\int_{|\vec{x}|\leq1} p(|\vec{x}|) d\vec{x} = \int_0^1 f(r) p(r) dr = 1$$, with $$f(r)=4\pi r^2$$ for 3D, $$f(r)=2\pi r$$ for 2D, $$f(r)=2$$ for 1D systems.

For $$p(r)=\sum_{i=0}^{N-1} c_i r^{N-1-i}$$, we obtain $$V = \sum_{i=0}^{N-1} 4\pi c_i/(2+N-i)$$ for 3D, $$V = \sum_{i=0}^{N-1} 2\pi c_i/(1+N-i)$$ for 2D, $$V = \sum_{i=0}^{N-1} 2 c_i/(N-i)$$ for 1D systems.

template<StatType T>
 std::string NORMALIZED_POLYNOMIAL< T >::getName ( )
inline

Returns name of the polynomial.

Definition at line 113 of file NormalisedPolynomial.h.

References NORMALIZED_POLYNOMIAL< T >::name.

114  {
115  return name;
116  }
std::string name
Contains the name of the polynomial which will be displayed as CGtype by the statistical code...
template<StatType T>
 int NORMALIZED_POLYNOMIAL< T >::getOrder ( void )
inline

Returns the order of the polynomial.

Definition at line 148 of file NormalisedPolynomial.h.

References NORMALIZED_POLYNOMIAL< T >::coefficients.

149  {
150  return coefficients.size() - 1;
151  }
std::vector< Mdouble > coefficients
Stores the coefficients .
template<StatType T>
 Mdouble NORMALIZED_POLYNOMIAL< T >::operator[] ( int i ) const
inlineprivate

Definition at line 247 of file NormalisedPolynomial.h.

References NORMALIZED_POLYNOMIAL< T >::coefficients, and constants::i.

248  {
249  return coefficients[i];
250  }
const std::complex< Mdouble > i
Definition: ExtendedMath.h:50
std::vector< Mdouble > coefficients
Stores the coefficients .
template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::set_average ( )
private

This function is templated, with the default used only for StatType=XYZ, so it does nothing.

template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::set_average_1D ( )
private

Sets averaged_coefficients $$\bar{c}_i$$ for StatType=X,Y,Z such that $$\sum_{i=0}^N \bar{c}_i x^{N-i} = \int\int_{|\vec{x}|\leq 1} p(|\vec{x}|) dy dz$$. See evaluate_1D.

template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::set_average_2D ( )
private

For StatType=XY,XZ,XZ, averaged_coefficients is not used since $$\bar{p}(r)$$ can be evaluated as a function of $$c_i$$. See evaluate_2D.

template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::set_polynomial ( std::vector< Mdouble > new_coefficients, unsigned int new_dim )

Use this function to set the polynomial coefficients $$c_i$$. This function calls finish_set_polynomial to normalize the coefficients.

template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::set_polynomial ( Mdouble * new_coefficients, unsigned int num_coeff, unsigned int new_dim )

Some as set_polynomial, but avoids the use of a vector.

template<StatType T>
 void NORMALIZED_POLYNOMIAL< T >::setName ( const char * new_name )
inline

Use this function to change the name of the polynomial.

Definition at line 105 of file NormalisedPolynomial.h.

References NORMALIZED_POLYNOMIAL< T >::name.

Referenced by NORMALIZED_POLYNOMIAL< T >::NORMALIZED_POLYNOMIAL().

106  {
107  name = new_name;
108  }
std::string name
Contains the name of the polynomial which will be displayed as CGtype by the statistical code...

## Friends And Related Function Documentation

template<StatType T>
 std::ostream& operator<< ( std::ostream & os, const NORMALIZED_POLYNOMIAL< T > & P )
friend

Returns a text description of the polynomial.

Definition at line 156 of file NormalisedPolynomial.h.

157  {
158  unsigned int N = P.coefficients.size();
159  for (unsigned int i = 0; i < N; i++)
160  {
161  if (P[i] == 0.0)
162  continue;
163  if (P[i] >= 0)
164  os << "+";
165  os << std::setprecision(2) << P[i];
166  if (N - 1 - i > 1)
167  os << "r^" << N - 1 - i;
168  else if (N - 1 - i == 1)
169  os << "r";
170  }
171  return os;
172  }
const std::complex< Mdouble > i
Definition: ExtendedMath.h:50
std::vector< Mdouble > coefficients
Stores the coefficients .

## Member Data Documentation

template<StatType T>
 std::vector NORMALIZED_POLYNOMIAL< T >::averaged_coefficients
private

Stores some coefficients used in evaluate and evaluateIntegral for StatTypes different from XYZ.

Definition at line 76 of file NormalisedPolynomial.h.

template<StatType T>
 std::vector NORMALIZED_POLYNOMIAL< T >::coefficients
private

Stores the coefficients $$c_i$$.

Definition at line 71 of file NormalisedPolynomial.h.

template<StatType T>
 unsigned int NORMALIZED_POLYNOMIAL< T >::dim
private

The system dimension.

Definition at line 66 of file NormalisedPolynomial.h.

Referenced by NORMALIZED_POLYNOMIAL< T >::NORMALIZED_POLYNOMIAL().

template<StatType T>
 std::string NORMALIZED_POLYNOMIAL< T >::name
private

Contains the name of the polynomial which will be displayed as CGtype by the statistical code.

Definition at line 61 of file NormalisedPolynomial.h.

Referenced by NORMALIZED_POLYNOMIAL< T >::getName(), and NORMALIZED_POLYNOMIAL< T >::setName().

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