This file contains all the functions related to computing the Gauss-Legendre integration variables. More...

#include <iostream>
#include <cmath>
#include <vector>
#include <rvapi_interface.h>

Namespaces
	ProSHADE_legendre
	This namespace contains the functions related to the Gauss-Legendre integration variables.

Functions
void	ProSHADE_internal_legendre::getPolyAtZero (unsigned int order, double polyValue, double deriValue)
	This function obtains the Legendre polynomial values and its derivative at zero for any positive integer order polynomial. More...

double	ProSHADE_internal_legendre::advancePolyValue (double from, double to, double valAtFrom, unsigned int noSteps)
	This function finds the next value of the polynomial. More...

double	ProSHADE_internal_legendre::evaluateSeries (double *series, double target, unsigned int terms)
	This function evaluates the Taylor expansion. More...

void	ProSHADE_internal_legendre::getFirstEvenRoot (double polyAtZero, unsigned int order, double abscAtZero, double weighAtZero)
	This function finds the first root for Legendre polynomials of odd order. More...

void	ProSHADE_internal_legendre::completeLegendreSeries (unsigned int order, std::vector< double > abscissa, std::vector< double > weights)
	This function completes the Legendre polynomial series assuming you have obtained the first values. More...

void	ProSHADE_internal_legendre::getLegendreAbscAndWeights (unsigned int order, std::vector< double > abscissa, std::vector< double > weights, ProSHADE::ProSHADE_settings *settings)
	This function takes everytging together and computes the abscissas and weights for the Gauss-Legendre integration given only the order. More...

Detailed Description

This file contains functions required to compute the Gauss-Legendre integration variables such as the abscissas and the weights for any integer order of integration. This code was adapted from John Burkardt's web. It is build around finding the zeroes of the Legendre polynomial of given order, expending the series and then finding the roots. These can then be used to obtain the values of interest - abscissas and weights.

This file is part of the ProSHADE library for calculating shape descriptors and symmetry operators of protein structures. This is a prototype code, which is by no means complete or fully tested. Its use is at your own risk only. There is no quarantee that the results are correct.

Code adapted from: John Burkardt (http://people.sc.fsu.edu/~jburkardt/cpp_src/legendre_rule_fast/legendre_rule_fast.html)

Author: Michal Tykac; Garib N. Murshudov

Version: 0.6.5

Date: NOV 2018

Definition in file ProSHADE_legendre.h.

Function Documentation

§ advancePolyValue()

double ProSHADE_internal_legendre::advancePolyValue	(	double	from,
		double	to,
		double	valAtFrom,
		unsigned int	noSteps
	)

inline

Given the previous value of the polynomial, the distance to proceed and the number of steps to take, this function finds the next value of the polynomial using Taylor series.

Parameters

[in]	from	Current polynomial position.
[in]	to	Polynomial position to move to.
[in]	valAtFrom	The current value of the polynomial at the <from> position.
[in]	noSteps	Number of steps in which to reach the <to> position.
[out]	X	The polynomial value at the <to> position.

Warning: This is an internal function which should not be used by the user.

Definition at line 95 of file ProSHADE_legendre.h.

§ completeLegendreSeries()

void ProSHADE_internal_legendre::completeLegendreSeries	(	unsigned int	order,
		std::vector< double > *	abscissa,
		std::vector< double > *	weights
	)

inline

Given that the polynomial value at zero is known, this function will complete the Legendre polynomial and with it the absicassas and weights for the Gauss-Legendre integration using the other functions defined above.

Parameters

[in]	order	The positive integer value of the polynomial order.
[in]	abscissa	Pointer to a vector of abscissas containing the first value.
[in]	weights	Pointer to a vector of weights containing the first value.

Warning: This is an internal function which should not be used by the user.

Definition at line 239 of file ProSHADE_legendre.h.

§ evaluateSeries()

double ProSHADE_internal_legendre::evaluateSeries	(	double *	series,
		double	target,
		unsigned int	terms
	)

inline

This function takes the series array, the target value and the cap on Taylor expansion and proceeds to evaluate the series. The main use of this is to evaluate the series twice, one where the series evaluation 'overshoots' and once where it 'undershoots' and taking value in between those, thus adding accuracy.

Parameters

[in]	series	Pointer to array with the series values.
[in]	target	The target location on the series value.
[in]	terms	The Taylor expansion cap.
[out]	X	The value of the series at the target location.

Warning: This is an internal function which should not be used by the user.

Definition at line 144 of file ProSHADE_legendre.h.

§ getFirstEvenRoot()

void ProSHADE_internal_legendre::getFirstEvenRoot	(	double	polyAtZero,
		unsigned int	order,
		double *	abscAtZero,
		double *	weighAtZero
	)

inline

The Legendre polynomials with odd order have zero as the first root, but the even oder polenomials have different value and this function serves the purpose of finding this value (i.e. the first root of the polynomial if the order is even).

Parameters

[in]	polyAtZero	The value of the polynomial at zero.
[in]	order	The positive integer value of the polynomial order.
[in]	abscAtZero	Pointer to variable storing the abscissa value at zero.
[in]	weightAtZero	Pointer to variable storing the weight value at zero.

Warning: This is an internal function which should not be used by the user.

Definition at line 176 of file ProSHADE_legendre.h.

§ getLegendreAbscAndWeights()

void ProSHADE_internal_legendre::getLegendreAbscAndWeights	(	unsigned int	order,
		std::vector< double > *	abscissa,
		std::vector< double > *	weights,
		ProSHADE::ProSHADE_settings *	settings
	)

inline

For the comfort of usage, this function takes everything together and computes the abscissas and weights for the Gauss-Legendre integration by first finding the polynomial value at zero, then finding the root and completing the series.

Parameters

[in]	order	The positive integer value of the polynomial order required.
[in]	abscissa	Pointer to a vector where abscissas will be saved.
[in]	weights	Pointer to a vector where weights will be saved.
[in]	settings	ProSHADE_settings container with infotmation about HTML reporting.

Warning: This is an internal function which should not be used by the user.

Definition at line 337 of file ProSHADE_legendre.h.

§ getPolyAtZero()

void ProSHADE_internal_legendre::getPolyAtZero	(	unsigned int	order,
		double *	polyValue,
		double *	deriValue
	)

inline

This function takes the positive integer order of the Legendre polynomial and uses the recursive properties of the polynomials to work up to the order, computing the value at zero and its derivative for all lesser orders. It then returns the final values.

Parameters

[in]	order	Positive integer order of the Legendre polynomial which value at zero we want.
[in]	polyValue	Pointer to variable which will store the resulting polynomial value at zero.
[in]	deriValue	Pointer to variable which will store the derivative of the zero value.

Warning: This is an internal function which should not be used by the user.

Definition at line 55 of file ProSHADE_legendre.h.

Namespaces

Functions

Detailed Description

Function Documentation

§ advancePolyValue()

§ completeLegendreSeries()

§ evaluateSeries()

§ getFirstEvenRoot()

§ getLegendreAbscAndWeights()

§ getPolyAtZero()