fr.cnes.sirius.patrius.math.analysis.polynomials

Class PolynomialChebyshevFunction

java.lang.Object

fr.cnes.sirius.patrius.math.analysis.polynomials.PolynomialChebyshevFunction

All Implemented Interfaces:

ParametricUnivariateFunction, PolynomialFunctionInterface, UnivariateFunction, Serializable

public class PolynomialChebyshevFunction
extends Object
implements UnivariateFunction, ParametricUnivariateFunction, PolynomialFunctionInterface

Immutable representation of a Chebyshev polynomial with real Chebyshev coefficients.

The following methods and algorithms are based on the article [NUMERICAL RECIPES], §5.8 (Chebyshev Approximation) & §5.9 (Derivatives or Integrals of a Chebyshev).

Important notice: two conventions exist for defining Chebyshev polynomials:

• Defining the first coefficient similarly to other coefficients. Chebyshev Polynomial is then Σc_iT_i (for i in [0, n])
• Adding a 0.5 multiplier to first coefficient. Chebyshev Polynomial is then c₀/2 + Σc_iT_i (for i in [1, n])

PATRIUS uses the first convention (convention used in SPICE library).

Author:

bonitt

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor and Description
PolynomialChebyshevFunction(double start, double end, double[] c) Construct a Chebyshev polynomial with the given coefficients.

Method Summary

Modifier and Type	Method and Description
PolynomialChebyshevFunction	add(PolynomialChebyshevFunction p) Add a Chebyshev polynomial to the instance.
PolynomialChebyshevFunction	derivative() Return a new PolynomialChebyshevFunction object that approximates the derivative of the existing function over the same range [start; end].
boolean	equals(Object obj)
protected static double	evaluate(double start, double end, double[] coefficients, double argument) The Chebyshev polynomial is evaluated at a point $y\; =\; [x\; -\; (b\; +\; a)\; /\; 2]\; /\; [(b\; -\; a)\; /\; 2]$ and the result is returned as the function value.
double[]	getChebyshevAbscissas(int n) Compute the N Chebyshev abscissas on the range [start ; end] in a chronological (increasing) order..
double[]	getCoefficients() Returns a copy of the coefficients array.
int	getDegree() Returns the degree of the Chebyshev polynomial.
double	getEnd() Getter for the end range.
PolynomialType	getPolynomialType() Return the type of this polynomial function.
double	getStart() Getter for the start range.
double[]	gradient(double x, double... parameters) Compute the gradient of the function with respect to its parameters.
int	hashCode()
PolynomialChebyshevFunction	negate() Negate the instance coefficients.
PolynomialChebyshevFunction	primitive(double x0, double y0) Returns the primitive of the Chebyshev polynomial.
PolynomialChebyshevFunction	subtract(PolynomialChebyshevFunction p) Subtract a Chebyshev polynomial from the instance.
String	toString() Returns a string representation of the Chebyshev polynomial.
UnivariateFunction	univariateDerivative() Return a new Chebyshev polynomial object as a UnivariateFunction that approximates the derivative of the existing function over the same range [start; end].
double	value(double x) Compute the value of the function for the given argument.
double	value(double x, double... parameters) Compute the value of the function.

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

PolynomialChebyshevFunction

public PolynomialChebyshevFunction(double start,
                                   double end,
                                   double[] c)

Construct a Chebyshev polynomial with the given coefficients. The first element of the coefficients array is the constant term. Higher degree coefficients follow in sequence. The degree of the resulting polynomial is the index of the last non-null element of the array, or 0 if all elements are null.

The constructor makes a copy of the input array and assigns the copy to the coefficients property.

The Chebyshev polynomial is an approximation defined on the range [start ; end].

Parameters:

start - Start range

end - End range

c - Chebyshev polynomial coefficients

Throws:

NullArgumentException - if c is null

NoDataException - if c is empty

MathIllegalArgumentException - if start >= end

Method Detail

getStart

public double getStart()

Getter for the start range.

Returns:

the start range

getEnd

public double getEnd()

Getter for the end range.

Returns:

the end range

getDegree

public int getDegree()

Returns the degree of the Chebyshev polynomial.

Specified by:

getDegree in interface PolynomialFunctionInterface

Returns:

the degree of the Chebyshev polynomial

getCoefficients

public double[] getCoefficients()

Returns a copy of the coefficients array.

Changes made to the returned copy will not affect the coefficients of the Chebyshev polynomial.

Specified by:

getCoefficients in interface PolynomialFunctionInterface

Returns:

a fresh copy of the coefficients array

getChebyshevAbscissas

public double[] getChebyshevAbscissas(int n)

Compute the N Chebyshev abscissas on the range [start ; end] in a chronological (increasing) order..

Parameters:

n - Number of points to evaluate

Returns:

the N Chebyshev abscissas

Throws:

NotStrictlyPositiveException - if n <= 0

MathIllegalArgumentException - if start >= end

value

public double value(double x)

Compute the value of the function for the given argument.

The value returned is
coefficients[0] + coefficients[1] * T1 + ... + coefficients[n] * Tn

Note: the argument must be in the Chebyshev polynomial range [start; end].

Specified by:

value in interface UnivariateFunction

Parameters:

x - Argument for which the function value should be computed

Returns:

the value of the Chebyshev polynomial at the given point

Throws:

IllegalArgumentException - if the argument isn't in the Chebyshev polynomial range

See Also:

UnivariateFunction.value(double)

evaluate

protected static double evaluate(double start,
                                 double end,
                                 double[] coefficients,
                                 double argument)

The Chebyshev polynomial is evaluated at a point $y\; =\; [x\; -\; (b\; +\; a)\; /\; 2]\; /\; [(b\; -\; a)\; /\; 2]$ and the result is returned as the function value.

Parameters:

start - Start range

end - End range

coefficients - Coefficients of the polynomial to evaluate.

argument - Input value

Returns:

the value of the Chebyshev polynomial

Throws:

NoDataException - if coefficients is empty

NullArgumentException - if coefficients is null

MathIllegalArgumentException - if start >= end

IllegalArgumentException - if the argument isn't in the Chebyshev polynomial range

add

public PolynomialChebyshevFunction add(PolynomialChebyshevFunction p)

Add a Chebyshev polynomial to the instance.

Note: the Chebyshev polynomial to add must be defined on the same range as the instance.

Parameters:

p - Chebyshev polynomial to add

Returns:

a new Chebyshev polynomial which is the sum of the instance and p

Throws:

IllegalArgumentException - if the given Chebyshev polynomial hasn't the same range as the instance

subtract

public PolynomialChebyshevFunction subtract(PolynomialChebyshevFunction p)

Subtract a Chebyshev polynomial from the instance.

Note: the Chebyshev polynomial to subtract must be defined on the same range as the instance.

Parameters:

p - Chebyshev polynomial to subtract

Returns:

a new Chebyshev polynomial which is the difference of the instance minus p

Throws:

IllegalArgumentException - if the given Chebyshev polynomial hasn't the same range as the instance

negate

public PolynomialChebyshevFunction negate()

Negate the instance coefficients.

Returns:

a new Chebyshev polynomial

derivative

public PolynomialChebyshevFunction derivative()

Return a new PolynomialChebyshevFunction object that approximates the derivative of the existing function over the same range [start; end].

Specified by:

derivative in interface PolynomialFunctionInterface

Returns:

a new Chebyshev derivative polynomial

univariateDerivative

public UnivariateFunction univariateDerivative()

Return a new Chebyshev polynomial object as a UnivariateFunction that approximates the derivative of the existing function over the same range [start; end].

Returns:

a new Chebyshev derivative polynomial

primitive

public PolynomialChebyshevFunction primitive(double x0,
                                             double y0)

Returns the primitive of the Chebyshev polynomial. The constant term is computed so the primitive polynomial is matching the given (x0,y0) point.

Specified by:

primitive in interface PolynomialFunctionInterface

Parameters:

x0 - the absicssa at which the polynomial value is known

y0 - the value of the polynomial at x0

Returns:

the primitive Chebyshev polynomial

gradient

public double[] gradient(double x,
                         double... parameters)

Compute the gradient of the function with respect to its parameters.

Specified by:

gradient in interface ParametricUnivariateFunction

Parameters:

x - Point for which the function value should be computed.

parameters - Function parameters.

Returns:

the value.

value

public double value(double x,
                    double... parameters)

Compute the value of the function.

Specified by:

value in interface ParametricUnivariateFunction

Parameters:

x - Point for which the function value should be computed.

parameters - Function parameters.

Returns:

the value.

toString

public String toString()

Returns a string representation of the Chebyshev polynomial.

The representation is user oriented. Terms are displayed lowest degrees first. The multiplications signs, coefficients equals to one and null terms are not displayed (except if the polynomial is 0, in which case the 0 constant term is displayed). Addition of terms with negative coefficients are replaced by subtraction of terms with positive coefficients except for the first displayed term (i.e. we display -3 for a constant negative polynomial, but 1 - 3 T1 + T2 if the negative coefficient is not the first one displayed).

Overrides:

toString in class Object

Returns:

a string representation of the polynomial.

getPolynomialType

public PolynomialType getPolynomialType()

Return the type of this polynomial function.

Specified by:

getPolynomialType in interface PolynomialFunctionInterface

Returns:

the type of this polynomial function

hashCode

public int hashCode()

Overrides:

hashCode in class Object

equals

public boolean equals(Object obj)

Overrides:

equals in class Object