Public Types
typedef npstat::PreciseType< double >::type	precise_type

typedef std::pair< double, double >	MeasurePoint

typedef GaussLegendreQuadrature	PreciseQuadrature

typedef ContOrthoPoly1DDeg	degree_functor

Public Member Functions
template<typename Numeric1 , typename Numeric2 >
	ContOrthoPoly1D (unsigned maxDegree, const std::vector< std::pair< Numeric1, Numeric2 > > &measure, OrthoPolyMethod m=OPOLY_STIELTJES, bool shiftMeasureCoords=true)

template<typename Numeric >
	ContOrthoPoly1D (unsigned maxDegree, const std::vector< Numeric > &coords, OrthoPolyMethod m=OPOLY_STIELTJES, bool shiftMeasureCoords=true)

template<typename Numeric1 , typename Numeric2 , class OrthoPoly >
	ContOrthoPoly1D (unsigned maxDegree, const std::vector< std::pair< Numeric1, Numeric2 > > &measure, const OrthoPoly &ops, bool shiftMeasureCoords=false)

template<typename Numeric , class OrthoPoly >
	ContOrthoPoly1D (unsigned maxDegree, const std::vector< Numeric > &coords, const OrthoPoly &ops, bool shiftMeasureCoords=false)

ContOrthoPoly1D *	clone () const

unsigned	maxDegree () const

unsigned long	measureLength () const

double	weight (const unsigned long index) const

precise_type	sumOfWeights () const

precise_type	sumOfWeightSquares () const

bool	areAllWeightsEqual () const

double	minCoordinate () const

double	maxCoordinate () const

precise_type	measureShift () const

precise_type	meanCoordinate () const

precise_type	location () const

precise_type	scale () const

MeasurePoint	measure (const unsigned long index) const

double	coordinate (const unsigned long index) const

double	effectiveSampleSize () const

double	poly (unsigned deg, double x) const

std::pair< double, double >	polyPair (unsigned deg1, unsigned deg2, double x) const

template<typename Numeric >
void	allPolys (const double x, const unsigned maxdeg, Numeric *polyValues) const

void	allpoly (const long double x, long double *values, const unsigned maxdeg) const

template<typename Numeric >
void	sampleAverages (const Numeric coords, unsigned long lenCoords, double averages, unsigned maxdeg) const

template<class Pair >
void	weightedPointsAverages (const Pair points, unsigned long numPoints, double averages, unsigned maxdeg) const

template<typename Numeric >
Matrix< double >	sampleProductAverages (const Numeric *coords, unsigned long lenCoords, unsigned maxdeg) const

template<typename Numeric >
double	series (const Numeric *coeffs, unsigned deg, double x) const

template<class Functor , typename Numeric >
void	calculateCoeffs (const Functor &fcn, Numeric *coeffs, unsigned maxdeg) const

template<class Functor >
double	constrainedCoeffs (const Functor &fcn, double *coeffs, unsigned maxdeg, double xmin, double xmax, double integralValue=1.0) const

void	weightCoeffs (double *coeffs, unsigned maxdeg) const

template<class Functor >
double	integratedSquaredError (const Functor &fcn, const double *coeffs, unsigned maxdeg, double xmin, double xmax, unsigned integrationRulePoints=0U) const

template<class Functor >
double	weightedSquaredError (const Functor &fcn, const double *coeffs, unsigned maxdeg) const

precise_type	empiricalKroneckerDelta (unsigned deg1, unsigned deg2) const

Matrix< precise_type >	empiricalKroneckerMatrix (unsigned maxdeg) const

double	empiricalKroneckerCovariance (unsigned deg1, unsigned deg2, unsigned deg3, unsigned deg4) const

Matrix< precise_type >	jacobiMatrix (unsigned n) const

precise_type	integratePoly (unsigned degree, unsigned power, double xmin, double xmax) const

precise_type	integrateTripleProduct (unsigned deg1, unsigned deg2, unsigned deg3, double xmin, double xmax) const

Matrix< precise_type >	tripleProductMatrix (const std::vector< std::pair< unsigned, unsigned > > &pairs, unsigned maxdeg, double xmin, double xmax) const

template<class Functor >
double	integrateEWPolyProduct (const Functor &weight, unsigned deg1, unsigned deg2, double xmin, double xmax, unsigned integrationRulePoints) const

template<class Functor >
void	extWeightAverages (const Functor &weight, const AbsIntervalQuadrature1D &quad, unsigned maxdeg, double xmin, double xmax, double *averages) const

template<class Functor >
Matrix< double >	extWeightProductAverages (const Functor &weight, const AbsIntervalQuadrature1D &quad, unsigned maxdeg, double xmin, double xmax) const

double	powerAverage (unsigned deg, unsigned power) const

double	jointPowerAverage (unsigned deg1, unsigned power1, unsigned deg2, unsigned power2) const

double	jointAverage (const unsigned *degrees, unsigned nDegrees, bool degreesSortedIncreasingOrder=false) const

double	cachedJointAverage (unsigned deg1, unsigned deg2, unsigned deg3, unsigned deg4) const

double	cachedJointAverage (unsigned deg1, unsigned deg2, unsigned deg3, unsigned deg4, unsigned deg5, unsigned deg6) const

double	cachedJointAverage (unsigned deg1, unsigned deg2, unsigned deg3, unsigned deg4, unsigned deg5, unsigned deg6, unsigned deg7, unsigned deg8) const

double	cov4 (unsigned a, unsigned b, unsigned c, unsigned d) const

double	cov6 (unsigned a, unsigned b, unsigned c, unsigned d, unsigned e, unsigned f) const

double	cov8 (unsigned a, unsigned b, unsigned c, unsigned d, unsigned e, unsigned f, unsigned g, unsigned h) const

double	covCov4 (unsigned a, unsigned b, unsigned c, unsigned d, unsigned e, unsigned f, unsigned g, unsigned h) const

double	slowCov8 (unsigned a, unsigned b, unsigned c, unsigned d, unsigned e, unsigned f, unsigned g, unsigned h) const

double	epsExpectation (unsigned m, unsigned n, bool highOrder) const

double	epsCovariance (unsigned i, unsigned j, unsigned m, unsigned n, bool highOrder) const

Matrix< double >	epsCovarianceMatrix (const std::vector< std::pair< unsigned, unsigned > > &pairs, bool highOrder) const

CPP11_auto_ptr< StorablePolySeries1D >	makeStorablePolySeries (double xmin, double xmax, const double *coeffs=0, unsigned maxdeg=0) const

std::pair< precise_type, precise_type >	recurrenceCoeffs (const unsigned k) const

std::pair< precise_type, precise_type >	monicRecurrenceCoeffs (const unsigned k) const

Friends
class	PolyFcn

class	TripleProdFcn

template<class Funct >
class	EWPolyProductFcn

Constructor & Destructor Documentation

◆ ContOrthoPoly1D() [1/4]

template<typename Numeric1 , typename Numeric2 >

npstat::ContOrthoPoly1D::ContOrthoPoly1D	(	unsigned	maxDegree,
		const std::vector< std::pair< Numeric1, Numeric2 > > &	measure,
		OrthoPolyMethod	m = `OPOLY_STIELTJES`,
		bool	shiftMeasureCoords = `true`
	)

Main constructor, with obvious arguments. The first element of the measure pair is the coordinate and the second element is the weight (all weights must be non-negative). Internally, the measure will be sorted in the order of increasing weight and the measure coordinates will normally be shifted so that their weighted mean is at 0.

◆ ContOrthoPoly1D() [2/4]

template<typename Numeric >

npstat::ContOrthoPoly1D::ContOrthoPoly1D	(	unsigned	maxDegree,
		const std::vector< Numeric > &	coords,
		OrthoPolyMethod	m = `OPOLY_STIELTJES`,
		bool	shiftMeasureCoords = `true`
	)

Constructor which assumes that all weights are 1.0

◆ ContOrthoPoly1D() [3/4]

template<typename Numeric1 , typename Numeric2 , class OrthoPoly >

npstat::ContOrthoPoly1D::ContOrthoPoly1D	(	unsigned	maxDegree,
		const std::vector< std::pair< Numeric1, Numeric2 > > &	measure,
		const OrthoPoly &	ops,
		bool	shiftMeasureCoords = `false`
	)

Constructor that uses the modified Chebyshev algorithm

◆ ContOrthoPoly1D() [4/4]

template<typename Numeric , class OrthoPoly >

npstat::ContOrthoPoly1D::ContOrthoPoly1D	(	unsigned	maxDegree,
		const std::vector< Numeric > &	coords,
		const OrthoPoly &	ops,
		bool	shiftMeasureCoords = `false`
	)

Constructor that uses the modified Chebyshev algorithm and assumes that all weights are 1.0

Member Function Documentation

◆ allpoly()

void npstat::ContOrthoPoly1D::allpoly	(	const long double	x,
		long double *	values,
		const unsigned	maxdeg
	)		const

inline

Function added for interface compatibility with AbsClassicalOrthoPoly1D

◆ allPolys()

template<typename Numeric >

void npstat::ContOrthoPoly1D::allPolys	(	const double	x,
		const unsigned	maxdeg,
		Numeric *	polyValues
	)		const

inline

Return the values of all orthonormal polynomials up to some maximum degree. Size of the "polyValues" array should be at least maxdeg + 1.

◆ cachedJointAverage()

double npstat::ContOrthoPoly1D::cachedJointAverage	(	unsigned	deg1,
		unsigned	deg2,
		unsigned	deg3,
		unsigned	deg4
	)		const

A measure-weighted average that will be remembered internally so that another call to this function with the same arguments will return the answer quickly

◆ calculateCoeffs()

template<class Functor , typename Numeric >

void npstat::ContOrthoPoly1D::calculateCoeffs	(	const Functor &	fcn,
		Numeric *	coeffs,
		unsigned	maxdeg
	)		const

Build the coefficients of the orthogonal polynomial series for the given function. The length of the array "coeffs" should be at least "maxdeg" + 1. Note that the coefficients are returned in the order of increasing degree (same order is used by the "series" function).

◆ clone()

ContOrthoPoly1D* npstat::ContOrthoPoly1D::clone ( ) const

inline

Function added for interface compatibility with AbsClassicalOrthoPoly1D

◆ constrainedCoeffs()

template<class Functor >

double npstat::ContOrthoPoly1D::constrainedCoeffs	(	const Functor &	fcn,
		double *	coeffs,
		unsigned	maxdeg,
		double	xmin,
		double	xmax,
		double	integralValue = `1.0`
	)		const

Build the coefficients of the orthogonal polynomial series for the given function in such a way that the integral of the truncated series on the [xmin, xmax] interval is constrained to "integralValue". The length of the array "coeffs" should be at least "maxdeg" + 1. Note that the coefficients are returned in the order of increasing degree (same order is used by the "series" function). The construction minimizes the ISE weighted by the empirical density.

The function returns the chi-square from the corresponding constrained least squares problem. This is the sum of (coeffs[i] - c[i])^2, 0 <= i <= maxdeg, where c[i] are determined by the "calculateCoeffs" method.

◆ cov4()

double npstat::ContOrthoPoly1D::cov4	(	unsigned	a,
		unsigned	b,
		unsigned	c,
		unsigned	d
	)		const

Covariance between v_{ab} and v_{cd}

◆ cov6()

double npstat::ContOrthoPoly1D::cov6	(	unsigned	a,
		unsigned	b,
		unsigned	c,
		unsigned	d,
		unsigned	e,
		unsigned	f
	)		const

Covariance between v_{ab}*v_{cd} and v_{ef}

◆ cov8()

double npstat::ContOrthoPoly1D::cov8	(	unsigned	a,
		unsigned	b,
		unsigned	c,
		unsigned	d,
		unsigned	e,
		unsigned	f,
		unsigned	g,
		unsigned	h
	)		const

Covariance between v_{ab}*v_{cd} and v_{ef}*v_{gh}

◆ covCov4()

double npstat::ContOrthoPoly1D::covCov4	(	unsigned	a,
		unsigned	b,
		unsigned	c,
		unsigned	d,
		unsigned	e,
		unsigned	f,
		unsigned	g,
		unsigned	h
	)		const

Covariance between two cov4 estimates (i.e., error on the error)

◆ effectiveSampleSize()

double npstat::ContOrthoPoly1D::effectiveSampleSize ( ) const

Kish's effective sample size for the measure

◆ empiricalKroneckerCovariance()

double npstat::ContOrthoPoly1D::empiricalKroneckerCovariance	(	unsigned	deg1,
		unsigned	deg2,
		unsigned	deg3,
		unsigned	deg4
	)		const

If the Kronecker deltas were calculated from a sample, they would be random and would have a covariance matrix. This is an estimate of the terms in that covariance matrix. The covariance is between delta_{deg1,deg2} and delta_{deg3,deg4}.

◆ empiricalKroneckerDelta()

precise_type npstat::ContOrthoPoly1D::empiricalKroneckerDelta	(	unsigned	deg1,
		unsigned	deg2
	)		const

This method is useful for testing the numerical precision of the orthonormalization procedure. It returns the scalar products between various polynomials.

◆ empiricalKroneckerMatrix()

Matrix<precise_type> npstat::ContOrthoPoly1D::empiricalKroneckerMatrix ( unsigned maxdeg ) const

A faster way to generate Kronecker deltas if the whole matrix of them is needed. The argument must not exceed the return value of the "maxDegree" function. The resulting matrix will be dimensioned (maxdeg + 1) x (maxdeg + 1).

◆ epsCovariance()

double npstat::ContOrthoPoly1D::epsCovariance	(	unsigned	i,
		unsigned	j,
		unsigned	m,
		unsigned	n,
		bool	highOrder
	)		const

Estimate of covariance between eps_{ij} and eps_{mn}.

The result should be identical with "empiricalKroneckerCovariance" in case "highOrder" argument is "false". However, this method caches results of various calculations of averages and should perform better inside cycles over indices.

◆ epsCovarianceMatrix()

Matrix<double> npstat::ContOrthoPoly1D::epsCovarianceMatrix	(	const std::vector< std::pair< unsigned, unsigned > > &	pairs,
		bool	highOrder
	)		const

Covariance matrix created by multiple calls to "epsCovariance"

◆ epsExpectation()

double npstat::ContOrthoPoly1D::epsExpectation	(	unsigned	m,
		unsigned	n,
		bool	highOrder
	)		const

Estimate of eps_{mn} expectation

◆ extWeightAverages()

template<class Functor >

void npstat::ContOrthoPoly1D::extWeightAverages	(	const Functor &	weight,
		const AbsIntervalQuadrature1D &	quad,
		unsigned	maxdeg,
		double	xmin,
		double	xmax,
		double *	averages
	)		const

Integral of the polynomials with some external weight function (e.g., reconstructed or oracle density). The length of array "averages" should be at least maxdeg + 1.

◆ extWeightProductAverages()

template<class Functor >

Matrix<double> npstat::ContOrthoPoly1D::extWeightProductAverages	(	const Functor &	weight,
		const AbsIntervalQuadrature1D &	quad,
		unsigned	maxdeg,
		double	xmin,
		double	xmax
	)		const

Integral of all products of two orthonormal polynomials with some external weight function (e.g., oracle density).

◆ integratedSquaredError()

template<class Functor >

double npstat::ContOrthoPoly1D::integratedSquaredError	(	const Functor &	fcn,
		const double *	coeffs,
		unsigned	maxdeg,
		double	xmin,
		double	xmax,
		unsigned	integrationRulePoints = `0U`
	)		const

Integrated squared error between the given function and the polynomial series. Parameter "integrationRulePoints" specifies the parameter "npoints" for the GaussLegendreQuadrature class. If "integrationRulePoints" is 0, the rule will be picked automatically in such a way that it integrates a polynomial of degree maxdeg*2 exactly.

◆ integrateEWPolyProduct()

template<class Functor >

double npstat::ContOrthoPoly1D::integrateEWPolyProduct	(	const Functor &	weight,
		unsigned	deg1,
		unsigned	deg2,
		double	xmin,
		double	xmax,
		unsigned	integrationRulePoints
	)		const

Integral of the product of two orthonormal polynomials with some external weight function (e.g., oracle density). "EW" in the method name stands for "externally weighted".

"integrationRulePoints" argument will be passed to the GaussLegendreQuadrature constructor and must be supported by that class.

◆ integratePoly()

precise_type npstat::ContOrthoPoly1D::integratePoly	(	unsigned	degree,
		unsigned	power,
		double	xmin,
		double	xmax
	)		const

Integral of an orthonormal polynomials to some power without the weight (so this is not an inner product with the poly of degree 0).

◆ integrateTripleProduct()

precise_type npstat::ContOrthoPoly1D::integrateTripleProduct	(	unsigned	deg1,
		unsigned	deg2,
		unsigned	deg3,
		double	xmin,
		double	xmax
	)		const

Integral of the product of three orthonormal polynomials without the weight (so this is not an inner product)

◆ jacobiMatrix()

Matrix<precise_type> npstat::ContOrthoPoly1D::jacobiMatrix ( unsigned n ) const

Generate principal minor of order n of the Jacobi matrix. n must not exceed "maxDegree". Note that, if you calculate roots using this Jacobi matrix, they will be shifted by meanCoordinate() (that is, add meanCoordinate() to all such roots).

◆ jointAverage()

double npstat::ContOrthoPoly1D::jointAverage	(	const unsigned *	degrees,
		unsigned	nDegrees,
		bool	degreesSortedIncreasingOrder = `false`
	)		const

A measure-weighted average of the product of several orthonormal poly values

◆ jointPowerAverage()

double npstat::ContOrthoPoly1D::jointPowerAverage	(	unsigned	deg1,
		unsigned	power1,
		unsigned	deg2,
		unsigned	power2
	)		const

A measure-weighted average of the product of two orthonormal poly values to some powers

◆ makeStorablePolySeries()

CPP11_auto_ptr<StorablePolySeries1D> npstat::ContOrthoPoly1D::makeStorablePolySeries	(	double	xmin,
		double	xmax,
		const double *	coeffs = `0`,
		unsigned	maxdeg = `0`
	)		const

Construct a StorablePolySeries1D object out of this one

◆ maxDegree()

unsigned npstat::ContOrthoPoly1D::maxDegree ( ) const

inline

A simple inspector of object properties

◆ measure()

MeasurePoint npstat::ContOrthoPoly1D::measure ( const unsigned long index ) const

inline

Note that this returns the internal, shifted coordinate. Add the mean coordinate to restore the original coordinate.

◆ monicRecurrenceCoeffs()

std::pair<precise_type,precise_type> npstat::ContOrthoPoly1D::monicRecurrenceCoeffs ( const unsigned k ) const

inline

Examine recurrence coefficients for the corresponding monic polynomials

◆ poly()

double npstat::ContOrthoPoly1D::poly	(	unsigned	deg,
		double	x
	)		const

Return the value of one of the normalized polynomials

◆ polyPair()

std::pair<double,double> npstat::ContOrthoPoly1D::polyPair	(	unsigned	deg1,
		unsigned	deg2,
		double	x
	)		const

Return the values of two of the normalized polynomials

◆ powerAverage()

double npstat::ContOrthoPoly1D::powerAverage	(	unsigned	deg,
		unsigned	power
	)		const

A measure-weighted average of orthonormal poly values to some power

◆ recurrenceCoeffs()

std::pair<precise_type,precise_type> npstat::ContOrthoPoly1D::recurrenceCoeffs ( const unsigned k ) const

inline

Examine recurrence coefficients

◆ sampleAverages()

template<typename Numeric >

void npstat::ContOrthoPoly1D::sampleAverages	(	const Numeric *	coords,
		unsigned long	lenCoords,
		double *	averages,
		unsigned	maxdeg
	)		const

Average values of the polynomials for a sample of points. Size of the "averages" array should be at least maxdeg + 1. Note that the resulting averages are not weighted by the measure.

◆ sampleProductAverages()

template<typename Numeric >

Matrix<double> npstat::ContOrthoPoly1D::sampleProductAverages	(	const Numeric *	coords,
		unsigned long	lenCoords,
		unsigned	maxdeg
	)		const

Average values of the pairwise polynomial products for a sample of points. The returned matrix will be symmetric and will have the dimensions (maxdeg + 1) x (maxdeg + 1). Note that the resulting averages are not weighted by the measure.

◆ series()

template<typename Numeric >

double npstat::ContOrthoPoly1D::series	(	const Numeric *	coeffs,
		unsigned	deg,
		double	x
	)		const

Return the value of the orthonormal polynomial series

◆ slowCov8()

double npstat::ContOrthoPoly1D::slowCov8	(	unsigned	a,
		unsigned	b,
		unsigned	c,
		unsigned	d,
		unsigned	e,
		unsigned	f,
		unsigned	g,
		unsigned	h
	)		const

Alternative implementation of "cov8". It is slower than "cov8" but easier to program and verify. Useful mainly for testing "cov8".

◆ tripleProductMatrix()

Matrix<precise_type> npstat::ContOrthoPoly1D::tripleProductMatrix	(	const std::vector< std::pair< unsigned, unsigned > > &	pairs,
		unsigned	maxdeg,
		double	xmin,
		double	xmax
	)		const

Unweighted triple product integrals arranged in a matrix. The returned matrix will be dimensioned pairs.size() x (maxdeg+1).

◆ weightCoeffs()

void npstat::ContOrthoPoly1D::weightCoeffs	(	double *	coeffs,
		unsigned	maxdeg
	)		const

Build the coefficients of the orthogonal polynomial series for the discrete weight values (that is, fcn(x_i) = w_i, using the terminology of the "calculateCoeffs" method). This can sometimes be useful for weighted measures. Of course, for unweighted measures this results in just delta_{deg,0}.

◆ weightedPointsAverages()

template<class Pair >

void npstat::ContOrthoPoly1D::weightedPointsAverages	(	const Pair *	points,
		unsigned long	numPoints,
		double *	averages,
		unsigned	maxdeg
	)		const

Averages of the polynomial values for the given sample of weighted points (not weighting by the measure). The first element of the pair will be treated as the coordinate and the second element will be treated as weight. Weights must not be negative. The length of array "averages" should be at least maxdeg + 1.

◆ weightedSquaredError()

template<class Functor >

double npstat::ContOrthoPoly1D::weightedSquaredError	(	const Functor &	fcn,
		const double *	coeffs,
		unsigned	maxdeg
	)		const

Squared error between the given function and the polynomial series, weighted according to the measure

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

nm/ContOrthoPoly1D.hh

Public Types

Public Member Functions

Friends

Constructor & Destructor Documentation

◆ ContOrthoPoly1D() [1/4]

◆ ContOrthoPoly1D() [2/4]

◆ ContOrthoPoly1D() [3/4]

◆ ContOrthoPoly1D() [4/4]

Member Function Documentation

◆ allpoly()

◆ allPolys()

◆ cachedJointAverage()

◆ calculateCoeffs()

◆ clone()

◆ constrainedCoeffs()

◆ cov4()

◆ cov6()

◆ cov8()

◆ covCov4()

◆ effectiveSampleSize()

◆ empiricalKroneckerCovariance()

◆ empiricalKroneckerDelta()

◆ empiricalKroneckerMatrix()

◆ epsCovariance()

◆ epsCovarianceMatrix()

◆ epsExpectation()

◆ extWeightAverages()

◆ extWeightProductAverages()

◆ integratedSquaredError()

◆ integrateEWPolyProduct()

◆ integratePoly()

◆ integrateTripleProduct()

◆ jacobiMatrix()

◆ jointAverage()

◆ jointPowerAverage()

◆ makeStorablePolySeries()

◆ maxDegree()

◆ measure()

◆ monicRecurrenceCoeffs()

◆ poly()

◆ polyPair()

◆ powerAverage()

◆ recurrenceCoeffs()

◆ sampleAverages()

◆ sampleProductAverages()

◆ series()

◆ slowCov8()

◆ tripleProductMatrix()

◆ weightCoeffs()

◆ weightedPointsAverages()

◆ weightedSquaredError()