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casacore::Chebyshev< T > Class Template Reference

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

#include <Chebyshev.h>

Inheritance diagram for casacore::Chebyshev< T >:
casacore::ChebyshevParamModeImpl< T > casacore::ChebyshevParam< T > casacore::Function1D< T > casacore::Function< T, U > casacore::Functional< FunctionTraits< T >::ArgType, U > casacore::Functional< Vector< FunctionTraits< T >::ArgType >, U > casacore::MarshallableChebyshev< 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...
 
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...
 
getIntervalMin () const
 return the minimum value for the currently Chebyshev interval. More...
 
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 Stringname () 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< T, 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, U >
 Function ()
 Constructors. More...
 
 Function (const uInt n)
 
 Function (const Vector< T > &in)
 
 Function (const FunctionParam< T > &other)
 
template<class W , class X >
 Function (const Function< W, X > &other)
 
 Function (const Function< T, U > &other)
 
virtual ~Function ()
 Destructor. More...
 
uInt nparameters () const
 Returns the number of parameters. More...
 
virtual U 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 U operator() () const
 Evaluate this function object at xor at x, y. More...
 
virtual U operator() (const ArgType &x) const
 
virtual U operator() (const Vector< ArgType > &x) const
 
virtual U operator() (FunctionArg x) const
 
virtual U operator() (const ArgType &x, const ArgType &y) const
 
virtual U operator() (const ArgType &x, const ArgType &y, const ArgType &z) const
 
Boolmask (const uInt n)
 Manipulate the mask associated with the nth parameter (e.g. More...
 
const Boolmask (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, U >
virtual ~Functional ()
 Destructor. More...
 
virtual U 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 >, U >
virtual ~Functional ()
 Destructor. More...
 
virtual U 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, U >
typedef FunctionTraits< T >
::ArgType 
ArgType
 
typedef const ArgTypeFunctionArg
 
- 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 >
def_p
 Default value if outside interval. More...
 
minx_p
 Lowest interval bound. More...
 
maxx_p
 Highest inetrval bound. More...
 
ChebyshevEnums::OutOfIntervalMode mode_p
 Out-of-interval handling type. More...
 
- Protected Attributes inherited from casacore::Function< T, U >
FunctionParam< T > param_p
 The parameters and masks. More...
 
Vector< ArgTypearg_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< Stringmodes_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

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)

Thrown Exceptions

To Do

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, U >.

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: