Quartic Class Reference

Quartic polynomials. More...

#include <quartic.h>

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

Static Public Member Functions
static Quartic	Compose (const Quadric &, const Quadric &)
	Compose the quadric by another one. More...

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

Protected Member Functions
bool	Analyze ()
	Check if any coefficient of the quartic is more than 1012 times larger than the smallest. More...

Protected Attributes
double	c [5] = { 0.0,0.0,0.0,0.0,0.0 }
	Array of coefficients.

Friends
Quartic	operator+ (const Quartic &, const Quartic &)
	Overloaded.
Quartic	operator- (const Quartic &, const Quartic &)
	Overloaded.
Quartic	operator* (const Quartic &, const double &)
	Overloaded.
Quartic	operator* (const double &, const Quartic &)
	Overloaded.
Quartic	operator/ (const Quartic &, const double &)
	Overloaded.
Quartic	operator* (const Quadric &, const Quadric &)
	Multiply two quadrics which generates a quartic. More...
std::ostream &	operator<< (std::ostream &, const Quartic &)
	Overloaded output-stream operator. More...

Detailed Description

Quartic polynomials.

Closed form expression of roots exist for quartics.

Constructors should provide the coefficients in descending order. Example of how to code the quartic x⁴+x³-x+1:

Quartic p(3.0,1.0,0.0,-1.0,1.0);

Constructor & Destructor Documentation

Quartic()

Quartic::Quartic ( const double &  a, const double &  b, const double &  c, const double &  d, const double &  e  )

inlineexplicit

Empty.

Creates a quartic.

Member Function Documentation

Analyze()

bool Quartic::Analyze ( )

protected

Check if any coefficient of the quartic is more than 10¹² times larger than the smallest.

Small coefficients are set to 0.

Compose()

Quartic Quartic::Compose ( const Quadric &  p, const Quadric &  q  )

static

Compose the quadric by another one.

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

Parameters

q	The quadric.
p	The argument quadric.

Derivative()

constexpr double Quartic::Derivative ( const double &  x ) const

inlineconstexpr

Computes the derivative value at a given point.

If only one evaluation is to be made, this function is more efficient than Prime() and then evaluating the derivative for a given input value.

Parameters

x

Real.

See also

Quartic::Prime

K()

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

Compute the Lipschitz constant of the quartic.

Parameters

a,b

Interval.

operator()()

constexpr double Quartic::operator() ( const double &  x ) const

inlineconstexpr

Evaluates the quartic.

Parameters

x	Argument value of the function.

operator[]()

constexpr double & Quartic::operator[] ( int  i )

inlineconstexpr

Empty.

Access to the coefficients of the quartic.

Range()

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

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

Compute the roots of the first derivative, and evaluates the cubic at the roots if they are within the interval bounds.

Parameters

a,b	Interval.
x,y	Returned range.

Solve() [1/2]

int Quartic::Solve

(

double *

results

)

Search the roots of a quartic equation.

The roots are not sorted.

This function calls other root solving functions, invoking Cubic::Solve() or Quadric::Solve() as appropriate if the degree is lower than four (which means that closed form expressions of the roots exist).

Otherwise, solve the quartic using the method of Francois Vieta (Circa 1735).

This function checks if the higher coefficients are nul, possibly calling Quadric::Solve() and Cubic::Solve() if need be.

Parameters

results

Array for storing the roots.

Solve() [2/2]

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

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

Parameters

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

Friends And Related Function Documentation

operator*

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

friend

Multiply two quadrics which generates a quartic.

Parameters

u,v	Argument quadrics.

operator<<

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

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The quartic.