A function class that defines a Chebyshev polynomial. More...

#include <Chebyshev.h>

Inheritance diagram for casacore::Chebyshev< T >:

Public Member Functions
	Chebyshev ()
	create a zero-th order Chebyshev polynomial with the first coefficient equal to zero. More...

	Chebyshev (const uInt n)
	create an n-th order Chebyshev polynomial with the coefficients equal to zero. More...

	Chebyshev (const T &min, const T &max, const typename ChebyshevEnums::OutOfIntervalMode mode=ChebyshevEnums::CONSTANT, const T &defval=T(0))
	create a zero-th order Chebyshev polynomical with the first coefficient equal to one. More...

	Chebyshev (const Vector< T > &coeffs, const T &min, const T &max, const typename ChebyshevEnums::OutOfIntervalMode mode=ChebyshevEnums::CONSTANT, const T &defval=T(0))
	create a fully specified Chebyshev polynomial. More...

	Chebyshev (uInt order, const RecordInterface &mode)
	create a fully specified Chebyshev polynomial. More...

	Chebyshev (const Vector< T > &coeffs, const RecordInterface &mode)

	Chebyshev (const Chebyshev &other)
	create a deep copy of another Chebyshev polynomial More...

Chebyshev< T > &	operator= (const Chebyshev< T > &other)
	make this instance a (deep) copy of another Chebyshev polynomial More...

virtual	~Chebyshev ()
	Destructor. More...

virtual T	eval (const typename FunctionTraits< T >::ArgType *x) const
	Evaluate the Chebyshev at `x`. More...

Chebyshev< T >	derivative () const
	Return the Chebyshev polynomial which is the derivative of this one (with respect to the argument `x`). More...

virtual Function< T > *	clone () const
	Create a new copy of this object. More...

Public Member Functions inherited from casacore::ChebyshevParamModeImpl< T >
	ChebyshevParamModeImpl ()

	ChebyshevParamModeImpl (const uInt n)

	ChebyshevParamModeImpl (const T &min, const T &max, typename ChebyshevEnums::OutOfIntervalMode mode=ChebyshevEnums::CONSTANT, const T &defval=T(0))

	ChebyshevParamModeImpl (const Vector< T > &coeffs, const T &min, const T &max, typename ChebyshevEnums::OutOfIntervalMode mode=ChebyshevEnums::CONSTANT, const T &defval=T(0))

	ChebyshevParamModeImpl (uInt order, const RecordInterface &mode)

	ChebyshevParamModeImpl (const Vector< T > &coeffs, const RecordInterface &mode)

	ChebyshevParamModeImpl (const ChebyshevParamModeImpl &other)

virtual void	setMode (const RecordInterface &mode)
	get/set the function mode. More...

virtual void	getMode (RecordInterface &mode) const

virtual Bool	hasMode () const
	return True if the implementing function supports a mode. More...

Public Member Functions inherited from casacore::ChebyshevParam< T >
	ChebyshevParam ()
	create a zero-th order Chebyshev polynomial with the first coefficient equal to zero. More...

	ChebyshevParam (const uInt n)
	create an n-th order Chebyshev polynomial with the coefficients equal to zero. More...

	ChebyshevParam (const T &min, const T &max, ChebyshevEnums::OutOfIntervalMode mode=ChebyshevEnums::CONSTANT, const T &defval=T(0))
	create a zero-th order Chebyshev polynomical with the first coefficient equal to one. More...

	ChebyshevParam (const Vector< T > &coeffs, const T &min, const T &max, ChebyshevEnums::OutOfIntervalMode mode=ChebyshevEnums::CONSTANT, const T &defval=T(0))
	create a fully specified Chebyshev polynomial. More...

	ChebyshevParam (uInt order, const RecordInterface &mode)
	create a fully specified Chebyshev polynomial. More...

	ChebyshevParam (const Vector< T > &coeffs, const RecordInterface &mode)

	ChebyshevParam (const ChebyshevParam &other)
	create a deep copy of another Chebyshev polynomial More...

template<class W >
	ChebyshevParam (const ChebyshevParam< W > &other)

ChebyshevParam< T > &	operator= (const ChebyshevParam< T > &other)
	make a (deep) copy of another Chebyshev polynomial More...

virtual	~ChebyshevParam ()
	Destructor. More...

void	setCoefficients (const Vector< T > &coeffs)
	set the Chebyshev coefficients. More...

void	setCoefficient (const uInt which, const T &value)
	set a particular Chebyshev coefficient. More...

const Vector< T > &	getCoefficients () const
	return the current set of coefficients into a given Vector. More...

T	getCoefficient (const uInt which) const
	return a particular coefficient. More...

uInt	nCoefficients () const
	return the number of coeefficients currently loaded. More...

void	setInterval (T xmin, T xmax)
	set the Chebyshev interval for this function. More...

T	getIntervalMin () const
	return the minimum value for the currently Chebyshev interval. More...

T	getIntervalMax () const
	return the maximum value for the currently Chebyshev interval. More...

void	setOutOfIntervalMode (ChebyshevEnums::OutOfIntervalMode mode)
	set the behavior of this function when it is evaluated outside its Chebyshev interval More...

ChebyshevEnums::OutOfIntervalMode	getOutOfIntervalMode () const
	return the behavior of this function when it is evaluated outside of its Chebyshev interval. More...

void	setDefault (const T &val)
	set the default value of this function. More...

const T &	getDefault () const
	return the currently set default value. More...

uInt	order () const
	return the order of this polynomial. More...

virtual const String &	name () const
	Give name of function. More...

Public Member Functions inherited from casacore::Function1D< T >
	Function1D ()
	Constructors. More...

	Function1D (const uInt n)

	Function1D (const Vector< T > &in)

	Function1D (const FunctionParam< T > &other)

	Function1D (const Function1D< W, X > &other)

virtual	~Function1D ()
	Destructor. More...

virtual uInt	ndim () const
	Returns the number of dimensions of function. More...

Public Member Functions inherited from casacore::Function< T, T >
	Function ()
	Constructors. More...

	Function (const uInt n)

	Function (const Vector< T > &in)

	Function (const FunctionParam< T > &other)

	Function (const Function< W, X > &other)

virtual	~Function ()
	Destructor. More...

uInt	nparameters () const
	Returns the number of parameters. More...

virtual T	eval (FunctionArg x) const =0
	Evaluate the function object. More...

T &	operator[] (const uInt n)
	Manipulate the nth parameter (0-based) with no index check. More...

const T &	operator[] (const uInt n) const

virtual T	operator() () const
	Evaluate this function object at `x`or at `x, y`. More...

virtual T	operator() (const ArgType &x) const

virtual T	operator() (const Vector< ArgType > &x) const

virtual T	operator() (FunctionArg x) const

virtual T	operator() (const ArgType &x, const ArgType &y) const

virtual T	operator() (const ArgType &x, const ArgType &y, const ArgType &z) const

Bool &	mask (const uInt n)
	Manipulate the mask associated with the nth parameter (e.g. More...

const Bool &	mask (const uInt n) const

const FunctionParam< T > &	parameters () const
	Return the parameter interface. More...

FunctionParam< T > &	parameters ()

const Vector< ArgType > &	argp () const
	Get `arg_p` and `parset_p`. More...

Bool	parsetp () const

void	lockParam ()
	Compiler cannot always find the correct 'const' version of parameter access. More...

void	unlockParam ()

ostream &	print (ostream &os) const
	Print the function (i.e. More...

virtual Function< typename FunctionTraits< T >::DiffType > *	cloneAD () const

virtual Function< typename FunctionTraits< T >::BaseType > *	cloneNonAD () const

Public Member Functions inherited from casacore::Functional< FunctionTraits< T >::ArgType, T >
virtual	~Functional ()
	Destructor. More...

virtual T	operator() (const FunctionTraits< T >::ArgType &x) const =0
	Map a Domain `x` into a Range `y` value. More...

Public Member Functions inherited from casacore::Functional< Vector< FunctionTraits< T >::ArgType >, T >
virtual	~Functional ()
	Destructor. More...

virtual T	operator() (const Vector< FunctionTraits< T >::ArgType > &x) const =0
	Map a Domain `x` into a Range `y` value. More...

Additional Inherited Members
Public Types inherited from casacore::Function1D< T >
typedef const T *	FunctionArg

Public Types inherited from casacore::Function< T, T >
typedef FunctionTraits< T > ::ArgType	ArgType

typedef const ArgType *	FunctionArg

Static Public Member Functions inherited from casacore::ChebyshevParam< T >
static void	derivativeCoeffs (Vector< T > &coeffs, const T &xmin=T(-1), const T &xmax=T(1))
	transform a set of Chebyshev polynomial coefficients into a set representing the series' derivative. More...

static void	chebyshevToPower (Vector< T > &coeffs)
	convert a set of Chebyshev polynomial coefficients to power series coefficients. More...

static void	powerToChebyshev (Vector< T > &coeffs)
	convert a set of power series coefficients to Chebyshev polynomial coefficients. More...

Protected Attributes inherited from casacore::ChebyshevParam< T >
T	def_p
	Default value if outside interval. More...

T	minx_p
	Lowest interval bound. More...

T	maxx_p
	Highest inetrval bound. More...

ChebyshevEnums::OutOfIntervalMode	mode_p
	Out-of-interval handling type. More...

Protected Attributes inherited from casacore::Function< T, T >
FunctionParam< T >	param_p
	The parameters and masks. More...

Vector< ArgType >	arg_p
	Aid for non-contiguous argument storage. More...

Bool	parset_p
	Indicate parameter written. More...

Bool	locked_p
	Indicate that parameters are expected to be locked from changing. More...

Static Protected Attributes inherited from casacore::ChebyshevParam< T >
static Vector< String >	modes_s

Detailed Description

template<class T>
class casacore::Chebyshev< T >

A function class that defines a Chebyshev polynomial.

Intended use:

Public interface

Review Status

Reviewed By:: wbrouw
Date Reviewed:: 2001/11/12
Test programs:: tChebyshev

Prerequisite

Function

Etymology

This class is named after Chebyshev Type I polynomials

Synopsis

This class allows one to form and evaluate a function as a Chebyshev series, a linear combination of so-called Chebyshev polynomials.

This class's implementation is split into two parts: the parent class ChebyshevParam<T>, which manages the function's parameters, and this class, which handles how the function is evaluated. Thus, be sure to also consult ChebyshevParam<T> for the full interface of this function.

About Chebyshev Polynomials

Chebyshev polynomials are a special type of ultraspheric polynomials that are useful in such contexts as numerical analysis and circuit design. They form an orthogobnal set. A (type I) Chebyshev polynomial, T_n, is generated via the equation:

T_n(x) = cos n(arccos x)

Through clever use of trigometric identities, one can express T_n as a real polynomial expression of the form

n
T_n(x) = SUM C_i t^i
i=0

The low order polynomials look like this:

T_0 = 1
T_1 = x
T_2 = 2x^2 - 1
T_3 = 4x^3 - 3x
T_4 = 8x^4 - 8x^2 + 1
T_5 = 16x^5 - 20x^3 + 5x 

Higher order polynomials satisfy the recurrance relation,

T_(n+1) = 2xT_(n) - T_(n-1).

A common use of Chebyshev polynomials is in approximating functions. In particular, any function that is approximated by a power series,

f(x) ~ SUM P_i x^i,

over the interval [-1, 1] can be approximated by a linear combination of Chebyshev polynomials:

f(x) ~ SUM C_i T_i(x),

where C_i is the set of so-called Chebyshev coefficients.

Approximating a function with Chebyshev polynomials has some important advantages. For one, if the function is well approximated by a converging power series, one can obtain an equally accurate estimate using fewer terms of the corresponding Chebyshev series. More important, though, is the property over the interval [-1, 1], each polynomial has a domain of [-1, 1]; thus, the series is nicely bounded. And because of this bounded property, approximations calculated from a Chebyshev series are less susceptible to machine rounding errors than the equivalent power series.

Using the Chebyshev Function class

With a simple change of variable, it is possible to approximate a continuous function over any restricted interval using a Chebyshev series. This documention refers to this interval as the Chebyshev interval (set with the setInterval() function). The other important input parameters, of course, include the coefficients of the polynomials.

Like all Functions, the Chebyshev series is evaluated via the function operator, operator(). If the input value is within the range set by setInterval(), it is transformed to the range [-1, 1] via,

y = x - (min + max)/2) / ((max - min)/2)

The series is then evaluated with the coefficients set either at construction or via setCoefficients(). The value that is returned when the input value is outside the Chebyshev interval depends on the out-of-interval mode (set via setOutOfIntervalMode()). The default mode is to return a default value which can be set via setDefault(). The supported modes are identified by the enumeration OutOfIntervalMode; see the documentation for ChebyshevParam for a detailed description of these modes. In practice, though, it is expected that this class will be configured for the interval of interest.

The derivative of a Chebyshev series with respect to the independent variable (i.e. the argument x) is easily calculated analytically and can be expressed as another Chebyshev series; this is what the derivative() function returns. However, the more general way to obtain derivatives is via the AutoDiff templated type.

Example

In this example, a 2nd order Chebyshev polynomial series is created.

// set coeffs to desired values
Vector<Double> coeffs(3, 1);   
// configure the function   
Chebyshev<Double> cheb;
cheb.setInterval(-0.8, 7.2);
cheb.setDefault(1.0);
cheb.setCoefficients(coeffs);
// evaluate the function as necessary
Double z = cheb(-0.5);    // -0.5 is within range, z = 0.78625
z = cheb(4.2);            // 4.2 is within range, z = 0.375
z = cheb(-3);             // -3 is out of the interval, z = 1

The next example illustrates how to use the AutoDiff class to simultaneously calculate derivatives. Here, we replace the Double type with AutoDiff<Double>.

Chebyshev<AutoDiffA<Double> > cheb;
cheb.setDefault(AutoDiffA<Double>(1));
cheb.setInterval(AutoDiffA<Double>(-0.8), AutoDiffA<Double>(7.2));
// we'll track derivatives with respect to x and each of our
// coefficients; for a second-order series, this makes 4
// derivatives total.  x will be the first variable; the
// coefficients will the 2nd-4th variables
cheb.setCoefficient(0, AutoDiffA<Double>(3.1, 4, 1));   // c0 = 3.1
cheb.setCoefficient(1, AutoDiffA<Double>(2.4, 4, 2));   // c1 = 2.4
cheb.setCoefficient(2, AutoDiffA<Double>(0.5, 4, 3));   // c2 = 0.5
// now evaluate the function
AutoDiffA<Double> x(1.2, 4, 0);    // x = 1.2
AutoDiffA<Double> y = cheb(x);     // y = 1.65
Double dydx = y.derivative(0);     // dy/dx = 0.35
Double dydc1 = y.derivative(2);    // dy/dc1 = -0.5

Motivation

This class was created to support systematic errors in the simulator tool. It can be used by Jones matrix classes to vary gains in a predictable way, mimicing natural processes of the atmosphere or instrumental effects.

Template Type Argument Requirements (T)

T should have standard numerical operators. Current implementation only tested for real types (and their AutoDiffs).

Thrown Exceptions

Assertion in debug mode if attempt is made to address incorrect coefficients

To Do

It would be helpful to be able to convert to and from the Polynomial<T> type; this would be supported via a function, Polynomial<T> polynomial(), and constructor, Chebyshev(Polynomial<T>)

Definition at line 234 of file Chebyshev.h.

Constructor & Destructor Documentation

template<class T>

casacore::Chebyshev< T >::Chebyshev ( )

inline

create a zero-th order Chebyshev polynomial with the first coefficient equal to zero.

The bounded domain is [T(-1), T(1)]. The OutOfDomainMode is CONSTANT, and the default value is T(0).

Definition at line 241 of file Chebyshev.h.

template<class T>

casacore::Chebyshev< T >::Chebyshev ( const uInt n )

inlineexplicit

create an n-th order Chebyshev polynomial with the coefficients equal to zero.

The bounded domain is [T(-1), T(1)]. The OutOfDomainMode is CONSTANT, and the default value is T(0).

Definition at line 246 of file Chebyshev.h.

template<class T>

casacore::Chebyshev< T >::Chebyshev	(	const T &	min,
		const T &	max,
		const typename ChebyshevEnums::OutOfIntervalMode	mode = `ChebyshevEnums::CONSTANT`,
		const T &	defval = `T(0)`
	)

inline

create a zero-th order Chebyshev polynomical with the first coefficient equal to one.

min is the minimum value of its Chebyshev interval, and max is the maximum value. mode sets the behavior of the function outside the Chebyshev interval (see setOutOfIntervalMode() and OutOfIntervalMode enumeration definition for details). defval is the value returned when the function is evaluated outside the Chebyshev interval and mode=CONSTANT.

Definition at line 257 of file Chebyshev.h.

template<class T>

casacore::Chebyshev< T >::Chebyshev	(	const Vector< T > &	coeffs,
		const T &	min,
		const T &	max,
		const typename ChebyshevEnums::OutOfIntervalMode	mode = `ChebyshevEnums::CONSTANT`,
		const T &	defval = `T(0)`
	)

inline

create a fully specified Chebyshev polynomial.

coeffs holds the coefficients of the Chebyshev polynomial (see setCoefficients() for details). min is the minimum value of its canonical range, and max is the maximum value. mode sets the behavior of the function outside the Chebyshev interval (see setOutOfIntervalMode() and OutOfIntervalMode enumeration definition for details). defval is the value returned when the function is evaluated outside the canonical range and mode=CONSTANT.

Definition at line 273 of file Chebyshev.h.

template<class T>

casacore::Chebyshev< T >::Chebyshev	(	uInt	order,
		const RecordInterface &	mode
	)

inline

create a fully specified Chebyshev polynomial.

config is a record that contains the non-coefficient data that configures this class. The fields recognized by this class are those documented for the ChebyshevPara::setMode() function.

Definition at line 286 of file Chebyshev.h.

template<class T>

casacore::Chebyshev< T >::Chebyshev	(	const Vector< T > &	coeffs,
		const RecordInterface &	mode
	)

inline

Definition at line 288 of file Chebyshev.h.

template<class T>

casacore::Chebyshev< T >::Chebyshev ( const Chebyshev< T > & other )

inline

create a deep copy of another Chebyshev polynomial

Definition at line 294 of file Chebyshev.h.

template<class T>

virtual casacore::Chebyshev< T >::~Chebyshev ( )

inlinevirtual

Destructor.

Definition at line 302 of file Chebyshev.h.

Member Function Documentation

template<class T>

virtual Function<T>* casacore::Chebyshev< T >::clone ( ) const

inlinevirtual

Create a new copy of this object.

The caller is responsible for deleting the pointer.

Implements casacore::Function< T, T >.

Reimplemented in casacore::MarshallableChebyshev< T >.

Definition at line 316 of file Chebyshev.h.

template<class T>

Chebyshev<T> casacore::Chebyshev< T >::derivative ( ) const

Return the Chebyshev polynomial which is the derivative of this one (with respect to the argument x).

template<class T>

virtual T casacore::Chebyshev< T >::eval ( const typename FunctionTraits< T >::ArgType * x ) const

virtual

Evaluate the Chebyshev at x.

template<class T>

Chebyshev<T>& casacore::Chebyshev< T >::operator= ( const Chebyshev< T > & other )

inline

make this instance a (deep) copy of another Chebyshev polynomial

Definition at line 298 of file Chebyshev.h.

References casacore::ChebyshevParam< T >::operator=().

Referenced by casacore::MarshallableChebyshev< T >::operator=().

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

scimath/Functionals/Chebyshev.h

Public Member Functions

Additional Inherited Members

Detailed Description

template<class T> class casacore::Chebyshev< T >

Intended use:

Review Status

Prerequisite

Etymology

Synopsis

About Chebyshev Polynomials

Using the Chebyshev Function class

Example

Motivation

Template Type Argument Requirements (T)

Thrown Exceptions

To Do

Constructor & Destructor Documentation

Member Function Documentation

template<class T>
class casacore::Chebyshev< T >