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

#include <Chebyshev.h>

Inheritance diagram for casa::Chebyshev< T >:

Public Member Functions
	Chebyshev ()
	create a zero-th order Chebyshev polynomial with the first coefficient equal to zero.
	Chebyshev (const uInt n)
	create an n-th order Chebyshev polynomial with the coefficients equal to zero.
	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.
	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.
	Chebyshev (uInt order, const RecordInterface &mode)
	create a fully specified Chebyshev polynomial.
	Chebyshev (const Vector< T > &coeffs, const RecordInterface &mode)
	Chebyshev (const Chebyshev &other)
	create a deep copy of another Chebyshev polynomial
Chebyshev< T > &	operator= (const Chebyshev< T > &other)
	make this instance a (deep) copy of another Chebyshev polynomial
virtual	~Chebyshev ()
	Destructor.
virtual T	eval (const typename FunctionTraits< T >::ArgType *x) const
	Evaluate the Chebyshev at `x`.
Chebyshev< T >	derivative () const
	Return the Chebyshev polynomial which is the derivative of this one (with respect to the argument `x`).
virtual Function< T > *	clone () const
	Create a new copy of this object.

Detailed Description

template<class T>
class casa::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>

casa::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>

casa::Chebyshev< T >::Chebyshev ( const uInt n ) [inline, explicit]

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>

casa::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>

casa::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>

casa::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>

casa::Chebyshev< T >::Chebyshev	(	const Vector< T > &	coeffs,
		const RecordInterface &	mode
	)		`[inline]`

Definition at line 288 of file Chebyshev.h.

template<class T>

casa::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 casa::Chebyshev< T >::~Chebyshev ( ) [inline, virtual]

Destructor.

Definition at line 302 of file Chebyshev.h.

Member Function Documentation

template<class T>

virtual Function<T>* casa::Chebyshev< T >::clone ( ) const [inline, virtual]

Create a new copy of this object.

The caller is responsible for deleting the pointer.

Implements casa::Function< T, T >.

Reimplemented in casa::MarshallableChebyshev< T >.

Definition at line 316 of file Chebyshev.h.

template<class T>

Chebyshev<T> casa::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 casa::Chebyshev< T >::eval ( const typename FunctionTraits< T >::ArgType * x ) const [virtual]

Evaluate the Chebyshev at x.

template<class T>

Chebyshev<T>& casa::Chebyshev< T >::operator= ( const Chebyshev< T > & other ) [inline]

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

Reimplemented in casa::MarshallableChebyshev< T >.

Definition at line 298 of file Chebyshev.h.

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

casacore/scimath/Functionals/Chebyshev.h

Public Member Functions

Detailed Description

template<class T> class casa::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 casa::Chebyshev< T >