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 . 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...

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

template<>
double	evaluate (double r)

template<>
double	evaluate (double r)

template<>
double	evaluate (double r)

template<>
double	evaluate (double r)

template<>
double	evaluate (double r)

template<>
double	evaluate (double r)

template<>
double	evaluateGradient (double r)

template<>
double	evaluateGradient (double r)

template<>
double	evaluateGradient (double r)

template<>
double	evaluateGradient (double r)

template<>
double	evaluateGradient (double r)

template<>
double	evaluateGradient (double r)

template<>
double	evaluateIntegral (double a, double b, double t)

template<>
double	evaluateIntegral (double a, double b, double t)

template<>
double	evaluateIntegral (double a, double b, double t)

template<>
double	evaluateIntegral (double a, double b, double t)

template<>
double	evaluateIntegral (double a, double b, double t)

template<>
double	evaluateIntegral (double a, double b, double t)

Private Member Functions
void	finish_set_polynomial ()
	Normalizes the polynomial coefficients such that the integral over the unit sphere of the axisymmetric function is unity. More...

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

Mdouble	evaluate_1D (Mdouble r)
	Returns the value of the polynomial averaged over 2 dimensions. For StatType=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 . See evaluate_2D. More...

Mdouble	evaluateGradient_1D (Mdouble r)
	Returns the value of the gradient averaged over 2 dimensions. More...

Mdouble	evaluateGradient_2D (Mdouble r)
	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
	Access to the coefficients. More...

template<>
void	set_average ()

template<>
void	set_average ()

template<>
void	set_average ()

template<>
void	set_average ()

template<>
void	set_average ()

template<>
void	set_average ()

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< Mdouble >	coefficients
	Stores the coefficients . More...

std::vector< Mdouble >	averaged_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.

References NORMALIZED_POLYNOMIAL< T >::coefficients, NORMALIZED_POLYNOMIAL< T >::dim, and NORMALIZED_POLYNOMIAL< T >::setName().

     {
         setName("Polynomial");
         coefficients.resize(0);
         dim = 0;
     }

Member Function Documentation

template<StatType T>

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

Todo:: I originally forgot to put a return statement here but the compiler did not give me a warning. Can we give a compiler option to throw warnings if there are no return statements?

Definition at line 130 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients.

 {
     unsigned int N = coefficients.size();
     double value = coefficients[0];
     for (unsigned int i = 1; i < N; i++)
         value = value * r + coefficients[i];
     return value;
 }

template<>

double NORMALIZED_POLYNOMIAL< X >::evaluate ( double r )

Definition at line 365 of file NormalisedPolynomial.hcc.

 {
     return evaluate_1D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< Y >::evaluate ( double r )

Definition at line 369 of file NormalisedPolynomial.hcc.

 {
     return evaluate_1D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< Z >::evaluate ( double r )

Definition at line 373 of file NormalisedPolynomial.hcc.

 {
     return evaluate_1D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< XY >::evaluate ( double r )

Definition at line 377 of file NormalisedPolynomial.hcc.

 {
     return evaluate_2D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< XZ >::evaluate ( double r )

Definition at line 381 of file NormalisedPolynomial.hcc.

 {
     return evaluate_2D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< YZ >::evaluate ( double r )

Definition at line 385 of file NormalisedPolynomial.hcc.

 {
     return evaluate_2D(r);
 }

template<StatType T>

double 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.

Definition at line 152 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::averaged_coefficients.

 {
     unsigned int N = averaged_coefficients.size();
     double value = averaged_coefficients[0];
     for (unsigned int i = 1; i < N; i++)
         value = value * r + averaged_coefficients[i];
     return value;
 }

template<StatType T>

double NORMALIZED_POLYNOMIAL< T >::evaluate_2D ( Mdouble r )

private

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

Definition at line 183 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients.

 {
     //~ std::cout << "eval_2D" << std::endl;
     unsigned int N = coefficients.size();
     double value1 = 2. * coefficients[N - 1];
     double value2 = 0;
     double r2 = r * r;
     switch (N)
     {
         case 7:
             value1 += coefficients[N - 7] * 2. / 35. * (5. + r2 * (6. + r2 * (8 * r2 + 16)));
             //[[clang::fallthrough]];
         case 6:
             value1 += coefficients[N - 6] * (8. + (10. + 15. * r2) * r2) / 24.;
             value2 += coefficients[N - 6] * (15. * r2 * r2 * r2) / 24.;
             //[[clang::fallthrough]];
         case 5:
             value1 += coefficients[N - 5] * .4 * (1. + r2 * (4. / 3. + r2 * 8. / 3.));
             //[[clang::fallthrough]];
         case 4:
             value1 += coefficients[N - 4] * (2. + r2 * 3.) / 4.;
             value2 += coefficients[N - 4] * .75 * r2 * r2;
             //[[clang::fallthrough]];
         case 3:
             value1 += coefficients[N - 3] * 2. / 3. * (1. + r2 * 2.);
             //[[clang::fallthrough]];
         case 2:
             value1 += coefficients[N - 2] * 1.;
             value2 += coefficients[N - 2] * r2;
             //[[clang::fallthrough]];
         case 1:
             break;
         default:
             std::cerr << "Error: no rules set for high-order polynomials" << std::endl;
             exit(-1);
     }
     double root = sqrt(1 - r2);
     double invz = root / r;
     double arccsch = log(sqrt(1. + invz * invz) + invz);
     return value1 * root + value2 * arccsch;
 }

template<StatType T>

double NORMALIZED_POLYNOMIAL< T >::evaluateGradient ( Mdouble r )

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

Definition at line 142 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients.

 {
     unsigned int N = coefficients.size();
     double value = (N - 1) * coefficients[0];
     for (unsigned int i = 1; i < N - 1; i++)
         value = value * r + (N - 1 - i) * coefficients[i];
     return value;
 }

template<>

double NORMALIZED_POLYNOMIAL< X >::evaluateGradient ( double r )

Definition at line 390 of file NormalisedPolynomial.hcc.

 {
     return evaluateGradient_1D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< Y >::evaluateGradient ( double r )

Definition at line 394 of file NormalisedPolynomial.hcc.

 {
     return evaluateGradient_1D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< Z >::evaluateGradient ( double r )

Definition at line 398 of file NormalisedPolynomial.hcc.

 {
     return evaluateGradient_1D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< XY >::evaluateGradient ( double r )

Definition at line 402 of file NormalisedPolynomial.hcc.

 {
     return evaluateGradient_2D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< XZ >::evaluateGradient ( double r )

Definition at line 406 of file NormalisedPolynomial.hcc.

 {
     return evaluateGradient_2D(r);
 }

template<>

double NORMALIZED_POLYNOMIAL< YZ >::evaluateGradient ( double r )

Definition at line 410 of file NormalisedPolynomial.hcc.

 {
     return evaluateGradient_2D(r);
 }

template<StatType T>

double NORMALIZED_POLYNOMIAL< T >::evaluateGradient_1D ( Mdouble r )

private

Returns the value of the gradient averaged over 2 dimensions.

Definition at line 162 of file NormalisedPolynomial.hcc.

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

 {
     unsigned int N = coefficients.size();
     double value = coefficients[0];
     double value2 = coefficients[0];
     for (unsigned int i = 1; i < N - 1; i++)
     {
         value = value * r + coefficients[i];
         value2 += coefficients[i];
     }
     return 2. * constants::pi * r * (value2 - value * r);
 }

template<StatType T>

double NORMALIZED_POLYNOMIAL< T >::evaluateGradient_2D ( Mdouble r )

private

Returns the value of the gradient averaged over 1 dimensions.

Todo:

Definition at line 176 of file NormalisedPolynomial.hcc.

 {
     return 0.0;
 }

template<StatType T>

double 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.

Definition at line 226 of file NormalisedPolynomial.hcc.

References A, and NORMALIZED_POLYNOMIAL< T >::coefficients.

 {
     unsigned int N = coefficients.size();
     double value1 = coefficients[N - 1] * (b - a);
     double value2 = 0;
     double t2 = t * t;
     double a2 = a * a;
     double b2 = b * b;
     double roota = sqrt(a * a + t2);
     double rootb = sqrt(b * b + t2);
     switch (N)
     {
         case 7:
             value1 += coefficients[N - 7] * (b * (b2 * b2 * b2 / 7.0 + 0.6 * b2 * b2 * t2 + b2 * t2 * t2 + t2 * t2 * t2) - a * (a2 * a2 * a2 / 7.0 + 0.6 * a2 * a2 * t2 + a2 * t2 * t2 + t2 * t2 * t2));
             //[[clang::fallthrough]];
         case 6:
             value1 += coefficients[N - 6] * (b * rootb * (8. * b2 * b2 + 26. * b2 * t2 + 33. * t2 * t2) - a * roota * (8. * a2 * a2 + 26. * a2 * t2 + 33. * t2 * t2)) / 48.;
             value2 += coefficients[N - 6] * 15. / 48. * t2 * t2 * t2;
             //[[clang::fallthrough]];
         case 5:
             value1 += coefficients[N - 5] * (b * (b2 * b2 / 5. + 2. / 3. * b2 * t2 + t2 * t2) - a * (a2 * a2 / 5. + 2. / 3. * a2 * t2 + t2 * t2));
             //[[clang::fallthrough]];
         case 4:
             value1 += coefficients[N - 4] * (b * rootb * (2. * b2 + 5. * t2) - a * roota * (2. * a2 + 5. * t2)) / 8.;
             value2 += coefficients[N - 4] * 3. / 8. * t2 * t2;
             //[[clang::fallthrough]];
         case 3:
             value1 += coefficients[N - 3] * (b * (b2 / 3. + t2) - a * (a2 / 3. + t2));
             //[[clang::fallthrough]];
         case 2:
             value1 += coefficients[N - 2] * (b * rootb - a * roota) / 2.;
             value2 += coefficients[N - 2] * t2 / 2.;
             //[[clang::fallthrough]];
         case 1:
             break;
         default:
             std::cerr << "Error: no rules set for high-order polynomials" << std::endl;
             exit(-1);
     }
     double A = a / t;
     double B = b / t;
     double arcsinh = t < 1e-12 ? 0.0 : log(B + sqrt(1. + B * B)) - log(A + sqrt(1. + A * A));
     //~ std::cout 
     //~ << "value1 " << value1 << std::endl
     //~ << "value2 " << value2 << std::endl
     //~ << "arcsinh " << arcsinh << std::endl
     //~ << "a " << a << std::endl
     //~ << "b " << b << std::endl
     //~ << "t " << t << std::endl;
     return value1 + value2 * arcsinh;
 }

template<>

double NORMALIZED_POLYNOMIAL< X >::evaluateIntegral	(	double	a,
		double	b,
		double	t
	)

Definition at line 415 of file NormalisedPolynomial.hcc.

 {
     return evaluateIntegral_1D(a, b, t);
 }

template<>

double NORMALIZED_POLYNOMIAL< Y >::evaluateIntegral	(	double	a,
		double	b,
		double	t
	)

Definition at line 419 of file NormalisedPolynomial.hcc.

 {
     return evaluateIntegral_1D(a, b, t);
 }

template<>

double NORMALIZED_POLYNOMIAL< Z >::evaluateIntegral	(	double	a,
		double	b,
		double	t
	)

Definition at line 423 of file NormalisedPolynomial.hcc.

 {
     return evaluateIntegral_1D(a, b, t);
 }

template<>

double NORMALIZED_POLYNOMIAL< XY >::evaluateIntegral	(	double	a,
		double	b,
		double	t
	)

Definition at line 427 of file NormalisedPolynomial.hcc.

 {
     return evaluateIntegral_2D(a, b, t);
 }

template<>

double NORMALIZED_POLYNOMIAL< XZ >::evaluateIntegral	(	double	a,
		double	b,
		double	t
	)

Definition at line 431 of file NormalisedPolynomial.hcc.

 {
     return evaluateIntegral_2D(a, b, t);
 }

template<>

double NORMALIZED_POLYNOMIAL< YZ >::evaluateIntegral	(	double	a,
		double	b,
		double	t
	)

Definition at line 435 of file NormalisedPolynomial.hcc.

 {
     return evaluateIntegral_2D(a, b, t);
 }

template<StatType T>

double 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.

Definition at line 287 of file NormalisedPolynomial.hcc.

References A, and NORMALIZED_POLYNOMIAL< T >::averaged_coefficients.

 {
     unsigned int N = averaged_coefficients.size();
     double value1 = averaged_coefficients[N - 1] * (b - a);
     double value2 = 0;
     double t2 = t * t;
     double a2 = a * a;
     double b2 = b * b;
     double roota = sqrt(a * a + t2);
     double rootb = sqrt(b * b + t2);
     switch (N)
     {
         case 7:
             value1 += averaged_coefficients[N - 7] * (b * (b2 * b2 * b2 / 7.0 + 0.6 * b2 * b2 * t2 + b2 * t2 * t2 + t2 * t2 * t2) - a * (a2 * a2 * a2 / 7.0 + 0.6 * a2 * a2 * t2 + a2 * t2 * t2 + t2 * t2 * t2));
             //[[clang::fallthrough]];
         case 6:
             value1 += averaged_coefficients[N - 6] * (b * rootb * (8. * b2 * b2 + 26. * b2 * t2 + 33. * t2 * t2) - a * roota * (8. * a2 * a2 + 26. * a2 * t2 + 33. * t2 * t2)) / 48.;
             value2 += averaged_coefficients[N - 6] * 15. / 48. * t2 * t2 * t2;
             //[[clang::fallthrough]];
         case 5:
             value1 += averaged_coefficients[N - 5] * (b * (b2 * b2 / 5. + 2. / 3. * b2 * t2 + t2 * t2) - a * (a2 * a2 / 5. + 2. / 3. * a2 * t2 + t2 * t2));
             //[[clang::fallthrough]];
         case 4:
             value1 += averaged_coefficients[N - 4] * (b * rootb * (2. * b2 + 5. * t2) - a * roota * (2. * a2 + 5. * t2)) / 8.;
             value2 += averaged_coefficients[N - 4] * 3. / 8. * t2 * t2;
             //[[clang::fallthrough]];
         case 3:
             value1 += averaged_coefficients[N - 3] * (b * (b2 / 3. + t2) - a * (a2 / 3. + t2));
             //[[clang::fallthrough]];
         case 2:
             value1 += averaged_coefficients[N - 2] * (b * rootb - a * roota) / 2.;
             value2 += averaged_coefficients[N - 2] * t2 / 2.;
             //[[clang::fallthrough]];
         case 1:
             break;
         default:
             std::cerr << "Error: no rules set for high-order polynomials" << std::endl;
             exit(-1);
     }
     double A = a / t;
     double B = b / t;
     double arcsinh = t < 1e-12 ? 0.0 : log(B + sqrt(1. + B * B)) - log(A + sqrt(1. + A * A));
     return value1 + value2 * arcsinh;
     
     // 3D Heaviside:  ( b-a );
     // 3D Polynomial: evaluateIntegral(max(a,-wn)/w,min(b,wn)/w,tangential.GetLength()/w)*w;
     // 1D Polynomial: evaluateIntegral(max(a,-wn)/w,min(b,wn)/w,tangential.GetLength()/w)*w;
     // 1D Heaviside:  ((b*pi*(wn2-mathsFunc::square(b)/3.0)))-((a*pi*(wn2-mathsFunc::square(a)/3.0)));
     //~ averaged_coefficients[N+1] += coefficients[i]*2*pi/(N-1-i+2);
     //~ averaged_coefficients[i] -= coefficients[i]*2*pi/(N-1-i+2);
     
 }

template<StatType T>

double NORMALIZED_POLYNOMIAL< T >::evaluateIntegral_2D	(	Mdouble a	UNUSED,
		Mdouble b	UNUSED,
		Mdouble t	UNUSED
	)

private

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

Todo:: {Thomas: has yet to be implemented}

Definition at line 280 of file NormalisedPolynomial.hcc.

 {   
     std::cerr << "evaluateIntegral_2D is not implemented yet" << std::endl;
     exit(-1);
 }

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.

Definition at line 45 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients, NORMALIZED_POLYNOMIAL< T >::get_volume(), NORMALIZED_POLYNOMIAL< T >::set_average(), and NORMALIZED_POLYNOMIAL< T >::setName().

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

 {
     //normalizes the Polynomial.
     double volume = get_volume();
     if (fabs(volume - 1.0) > 1e-12)
         std::cout << "Warning: Polynomial has to be normalized by " << volume << std::endl;
     for (unsigned int i = 0; i < coefficients.size(); i++)
         coefficients[i] /= volume;
     
     //sets #averaged_coefficients   
     set_average();
     
     //std::couts the Polynomial
     std::stringstream sstr;
     sstr << *this;
     setName(sstr.str().c_str());
     std::cout << "p=" << *this << std::endl;
 }

template<StatType T>

double 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.

Definition at line 65 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients, NORMALIZED_POLYNOMIAL< T >::dim, and constants::pi.

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

 {
     double volume = 0;
     unsigned int N = coefficients.size();
     if (dim == 3)
     {
         for (unsigned int i = 0; i < N; i++)
             volume += coefficients[i] / (2. + N - i);
         volume *= 4. * constants::pi;
     }
     else if (dim == 2)
     {
         std::cerr << "dim=2 is not working yet" << std::endl;
         exit(-1);
         for (unsigned int i = 0; i < coefficients.size(); i++)
             volume += coefficients[i] / (1. + N - i);
         volume *= 2. * constants::pi;
     }
     else if (dim == 1)
     {
         std::cerr << "dim=1 is not working yet" << std::endl;
         exit(-1);
         for (unsigned int i = 0; i < coefficients.size(); i++)
             volume += coefficients[i] / (0. + N - i);
         volume *= 2.;
     }
     else
     {
         std::cerr << "Error in get_volume: dim=" << dim << std::endl;
         exit(-1);
     }
     return volume;
 }

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.

     {
         return name;
     }

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.

     {
         return coefficients.size() - 1;
     }

template<StatType T>

Mdouble NORMALIZED_POLYNOMIAL< T >::operator[] ( int i ) const

inlineprivate

Access to the coefficients.

Definition at line 247 of file NormalisedPolynomial.h.

References NORMALIZED_POLYNOMIAL< T >::coefficients.

     {
         return coefficients[i];
     }

template<StatType T>

void NORMALIZED_POLYNOMIAL< T >::set_average ( )

private

Sets averaged_coefficients.

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

Definition at line 100 of file NormalisedPolynomial.hcc.

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

 {
     std::cout << "set_average" << std::endl;
 }

template<>

void NORMALIZED_POLYNOMIAL< X >::set_average ( )

private

Definition at line 340 of file NormalisedPolynomial.hcc.

 {
     set_average_1D();
 }

template<>

void NORMALIZED_POLYNOMIAL< Y >::set_average ( )

private

Definition at line 344 of file NormalisedPolynomial.hcc.

 {
     set_average_1D();
 }

template<>

void NORMALIZED_POLYNOMIAL< Z >::set_average ( )

private

Definition at line 348 of file NormalisedPolynomial.hcc.

 {
     set_average_1D();
 }

template<>

void NORMALIZED_POLYNOMIAL< XY >::set_average ( )

private

Definition at line 352 of file NormalisedPolynomial.hcc.

 {
     set_average_2D();
 }

template<>

void NORMALIZED_POLYNOMIAL< XZ >::set_average ( )

private

Definition at line 356 of file NormalisedPolynomial.hcc.

 {
     set_average_2D();
 }

template<>

void NORMALIZED_POLYNOMIAL< YZ >::set_average ( )

private

Definition at line 360 of file NormalisedPolynomial.hcc.

 {
     set_average_2D();
 }

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.

Definition at line 106 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::averaged_coefficients, NORMALIZED_POLYNOMIAL< T >::coefficients, and constants::pi.

 {
     std::cout << "set_average_1D" << std::endl;
     unsigned int N = coefficients.size();
     averaged_coefficients.resize(N + 2);
     for (unsigned int i = 0; i < N + 2; i++)
     {
         averaged_coefficients[i] = 0;
     }
     for (unsigned int i = 0; i < N; i++)
     {
         averaged_coefficients[N + 1] += coefficients[i] * 2 * constants::pi / (N - 1 - i + 2);
         averaged_coefficients[i] -= coefficients[i] * 2 * constants::pi / (N - 1 - i + 2);
     }
 }

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.

Definition at line 124 of file NormalisedPolynomial.hcc.

125 {

126 }

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.

Definition at line 27 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients, NORMALIZED_POLYNOMIAL< T >::dim, and NORMALIZED_POLYNOMIAL< T >::finish_set_polynomial().

 {
     coefficients = new_coefficients;
     dim = new_dim;
     finish_set_polynomial();
 }

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.

Definition at line 35 of file NormalisedPolynomial.hcc.

References NORMALIZED_POLYNOMIAL< T >::coefficients, NORMALIZED_POLYNOMIAL< T >::dim, and NORMALIZED_POLYNOMIAL< T >::finish_set_polynomial().

 {
     coefficients.resize(num_coeff);
     for (unsigned int i = 0; i < num_coeff; i++)
         coefficients[i] = new_coefficients[i];
     dim = new_dim;
     finish_set_polynomial();
 }

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 >::finish_set_polynomial(), and NORMALIZED_POLYNOMIAL< T >::NORMALIZED_POLYNOMIAL().

     {
         name = new_name;
     }

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.

     {
         unsigned int N = P.coefficients.size();
         for (unsigned int i = 0; i < N; i++)
         {
             if (P[i]==0.0)
                 continue;
             if (P[i] >= 0)
                 os << "+";
             os << std::setprecision(2) << P[i];
             if (N - 1 - i > 1)
                 os << "r^" << N - 1 - i;
             else if (N - 1 - i == 1)
                 os << "r";
         }
         return os;
     }

Member Data Documentation

template<StatType T>

std::vector<Mdouble> 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.

Referenced by NORMALIZED_POLYNOMIAL< T >::evaluate_1D(), NORMALIZED_POLYNOMIAL< T >::evaluateIntegral_1D(), and NORMALIZED_POLYNOMIAL< T >::set_average_1D().

template<StatType T>

std::vector<Mdouble> 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 >::get_volume(), NORMALIZED_POLYNOMIAL< T >::NORMALIZED_POLYNOMIAL(), and NORMALIZED_POLYNOMIAL< T >::set_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 files:

/usr/local/www/mercury/repo/Release/0.11/Kernel/Math/NormalisedPolynomial.h
/usr/local/www/mercury/repo/Release/0.11/Kernel/Math/NormalisedPolynomial.hcc

Public Member Functions

Private Member Functions

Private Attributes

Friends

Detailed Description

template<StatType T> class NORMALIZED_POLYNOMIAL< T >

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

Member Data Documentation

template<StatType T>
class NORMALIZED_POLYNOMIAL< T >