Sextic Class Reference

Sextic polynomials, also known as hextics. More...

#include <sextic.h>

Public Member Functions
	Sextic ()
	Empty.
	Sextic (const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a sextic.
	Sextic (const Quintic &)
	Creates a sextic given a quintic.
	Sextic (const Quadric &)
	Creates a sextic given a quadric.
	Sextic (const Cubic &)
	Creates a sextic given a cubic.
	Sextic (const Quartic &)
	Creates a sextic given a quartic.
	~Sextic ()
	Empty.
constexpr double &	operator[] (int)
	Access to the coefficients of the sextic.
constexpr double	operator[] (int) const
	Overloaded.
int	CheckDegree () const
	Check the degree of the sextic.
Sextic	operator- () const
	Unary.
Sextic &	operator+= (const Sextic &)
	Destructive sum of two sextics.
Sextic &	operator-= (const Sextic &)
	Destructive difference of two sextics.
Sextic &	operator*= (const double &)
	Scale a sextic by a double value.
Sextic &	operator/= (const double &)
	Scale a sextic by a double value.
Quintic	Prime () const
	Computes the first derivative of a sextic, which is a quintic.
Quartic	Second () const
	Computes the second derivative of a sextic, which is a quartic.
Cubic	Third () const
	Computes the third derivative of a sextic.
double	K (const double &, const double &) const
	Compute the Lipschitz constant of the sextic.
constexpr double	operator() (const double &) const
	Evaluates the sextic.
constexpr double	Derivative (const double &) const
	Computes the derivative of the sextic.
int	Solve (double *) const
	Solve the sextic equation over a given interval.
int	Solve (double *, const double &, const double &) const
	Search the roots of a sextic equation over a given interval.
void	Range (double &, double &, const double &=0.0, const double &=1.0) const
	Compute the range of values taken by a sextic over a given interval.

Static Public Member Functions
static Sextic	Compose (const Quadric &, const Cubic &)
	Compose the quadric by a cubic.
static Sextic	Compose (const Cubic &, const Quadric &)
	Compose the cubic by a quadric.

Static Public Attributes
static double	epsilon = 1.0e-10
	Epsilon value used to check discriminant terms in the root finding process.

Protected Attributes
double	c [7] = { 0.0,0.0,0.0,0.0,0.0,0.0,0.0 }
	Array of coefficients.

Friends
Sextic	operator+ (const Sextic &u, const Sextic &v)
	Overloaded.
Sextic	operator- (const Sextic &v, const Sextic &u)
	Overloaded.
Sextic	operator* (const Sextic &u, const double &e)
	Overloaded.
Sextic	operator* (const double &a, const Sextic &p)
	Overloaded.
Sextic	operator/ (const Sextic &p, const double &a)
	Overloaded.
Sextic	operator* (const Quartic &u, const Quadric &v)
	Multiply a quartic by a quadric.
Sextic	operator* (const Cubic &u, const Cubic &v)
	Multiply two cubics which generates a sextic.
std::ostream &	operator<< (std::ostream &s, const Sextic &p)
	Overloaded output-stream operator.

Detailed Description

Sextic polynomials, also known as hextics.

Constructors should provide the coefficients in descending order. Example of how to code the sextic 4x⁶-x+1:

Sextic p(4.0,0.0,0.0,0.0,-1.0,1.0);

Constructor & Destructor Documentation

Sextic() [1/4]

Sextic::Sextic ( const Quintic & p )

inline

Creates a sextic given a quintic.

Simply set higher coefficients to 0.

Parameters

p

Quintic.

Sextic() [2/4]

Sextic::Sextic ( const Quadric & p )

inline

Creates a sextic given a quadric.

Simply set higher coefficients to 0.

Parameters

p

Quadric.

Sextic() [3/4]

Sextic::Sextic ( const Cubic & p )

inline

Creates a sextic given a cubic.

Simply set higher coefficients to 0.

Parameters

p

Cubic.

Sextic() [4/4]

Sextic::Sextic ( const Quartic & p )

inline

Creates a sextic given a quartic.

Simply set highest coefficient to 0.

Parameters

p

Quartic.

Member Function Documentation

Compose() [1/2]

Sextic Sextic::Compose ( const Cubic & p, const Quadric & q )

static

Compose the cubic by a quadric.

This function computes p ( q(x) ), where p denotes the cubic and q the quadric.

Parameters

p	The cubic.
q	The quadric.

Compose() [2/2]

Sextic Sextic::Compose ( const Quadric & p, const Cubic & q )

static

Compose the quadric by a cubic.

This function computes p ( q(x) ), where p denotes the current quadric and q the argument cubic.

Parameters

p	The quadric.
q	The cubic.

Derivative()

double Sextic::Derivative ( const double & x ) const

inlineconstexpr

Computes the derivative of the sextic.

This function is more efficient than using Prime() and then evaluating the derivative for a given input value.

Parameters

x

Real.

K()

double Sextic::K	(	const double &	a,
		const double &	b ) const

Compute the Lipschitz constant of the sextic.

Parameters

a,b

Interval.

operator()()

double Sextic::operator() ( const double & x ) const

inlineconstexpr

Evaluates the sextic.

Parameters

x	Argument value of the function.

Range()

void Sextic::Range	(	double &	x,
		double &	y,
		const double &	a = 0.0,
		const double &	b = 1.0 ) const

Compute the range of values taken by a sextic over a given interval.

Parameters

a,b	Interval.
x,y	Returned range.

Solve() [1/2]

int Sextic::Solve

(

double *

roots

)

const

Solve the sextic equation over a given interval.

This function stores the sorted roots in an array and returns the number of roots.

This function calls Polynomial::Solve() if the highest coefficient is not nul, otherwise it calls Quartic::Solve() and other lower degree polynomial solvers which are more efficient.

Parameters

roots

The array of roots.

Solve() [2/2]

int Sextic::Solve	(	double *	roots,
		const double &	a,
		const double &	b ) const

Search the roots of a sextic equation over a given interval.

Parameters

roots	Array for storing the roots.
a,b	Interval range.

Friends And Related Symbol Documentation

operator* [1/2]

Sextic operator* ( const Cubic & u, const Cubic & v )

friend

Multiply two cubics which generates a sextic.

Parameters

u,v	Arguments.

operator* [2/2]

Sextic operator* ( const Quartic & u, const Quadric & v )

friend

Multiply a quartic by a quadric.

Parameters

u,v	Arguments.

operator<<

std::ostream & operator<< ( std::ostream & s, const Sextic & p )

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The sextic.