Quintic Class Reference

Quintic polynomials. More...

#include <quintic.h>

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

Static Public Member Functions
static Quintic	Hermite (const double &, const double &, const double &, const double &, const double &, const double &)
	Creates an Hermite quintic polynomial on interval [0,1].
static double	Smooth (const double &)
	Compute the value of smooth \(C^2\) interpolating function over unit interval.
static double	SmoothCompact (const double &, const double &)
	Compactly supported smooth interpolating function.
static double	SmoothStep (const double &, const double &, const double &)
	Compute a quintic smooth step.
static double	Gaussian (const double &, const double &, const double &)
	Compute a compactly supported Gaussian-like pulse.
static double	GaussianThick (const double &, const double &, const double &, const double &)
	Compute a compactly supported Gaussian-like pulse with a thick plateau.
static Quintic	FromRoots (const QVector< double > &)
	Create a quintic from a set of roots.

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

Protected Attributes
double	c [6] = { 0.0,0.0,0.0,0.0,0.0,0.0 }
	Array of coefficients.

Friends
Quintic	operator+ (const Quintic &u, const Quintic &v)
	Overloaded.
Quintic	operator- (const Quintic &v, const Quintic &u)
	Overloaded.
Quintic	operator* (const Quintic &u, const double &e)
	Overloaded.
Quintic	operator* (const double &a, const Quintic &p)
	Overloaded.
Quintic	operator/ (const Quintic &p, const double &a)
	Overloaded.
std::ostream &	operator<< (std::ostream &s, const Quintic &p)
	Overloaded output-stream operator.

Detailed Description

Quintic polynomials.

Closed form expression of roots does not exist for polynomials with a degree greater than 5.

Constructors should provide the coefficients in descending order. Example of how to code the quintic \(x^5 - x + 1\):

Quintic p(1.0,0.0,0.0,-1.0,1.0);

Quintic implements some inline static member function such as:

inline double Quintic::Smooth(const double& x)
{
return x * x * x * (x * (x * 6 - 15) + 10);
}

Constructor & Destructor Documentation

Quintic() [1/3]

Quintic::Quintic ( const Quadric & p )

inline

Creates a quintic from a quadric.

Simply set higher coefficients to 0.

Parameters

p

Quadric.

Quintic() [2/3]

Quintic::Quintic ( const Cubic & p )

inline

Creates a quintic from a cubic.

Simply set higher coefficients to 0.

Parameters

p

Cubic.

Quintic() [3/3]

Quintic::Quintic ( const Quartic & p )

inline

Creates a quintic from a quartic.

Parameters

p

Quartic.

Member Function Documentation

Derivative()

double Quintic::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

Quintic::Prime

FromRoots()

Quintic Quintic::FromRoots ( const QVector< double > & r )

static

Create a quintic from a set of roots.

Parameters

r	Set of roots.

Example of how to code the quintic \((x-1)^3 (x+1) (x+2)\):

Quintic p=Quintic::FromRoots({1.0,1.0,1.0,-1.0,-2.0});

Gaussian()

double Quintic::Gaussian ( const double & c, const double & r, const double & x )

inlinestatic

Compute a compactly supported Gaussian-like pulse.

The function has \(C^2\) continuity.

See also

Cubic::Gaussian()

Parameters

c	Center.
r	Radius.
x	Value.

GaussianThick()

double Quintic::GaussianThick ( const double & c, const double & t, const double & r, const double & x )

inlinestatic

Compute a compactly supported Gaussian-like pulse with a thick plateau.

See also

Cubic::GaussianThick()

Quintic::Gaussian()

Parameters

c	Center.
t,r	Thickness (plateau) and radius.
x	Value.

Hermite()

Quintic Quintic::Hermite ( const double & a, const double & b, const double & ta, const double & tb, const double & na, const double & nb )

static

Creates an Hermite quintic polynomial on interval [0,1].

Parameters

a,b	Values at t=0 and t=1.
ta,tb	Derivatives at t=0 and t=1.
na,nb	Second derivatives at t=0 and t=1.

K()

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

Compute the Lipschitz constant of the quintic.

Parameters

a,b

Interval.

operator()()

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

inlineconstexpr

Evaluates the quintic.

Parameters

x	Argument value of the function.

Range()

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

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

Parameters

a,b	Interval.
x,y	Returned range.

Smooth()

double Quintic::Smooth ( const double & x )

inlinestatic

Compute the value of smooth \(C^2\) interpolating function over unit interval.

The quintic is defined as \(x^3 (6 x^2 - 15 x + 10)\). Its first and second derivatives at 0.0 and 1.0 are 0.0.

The Lipschitz constant of the smooth quintic over [0,1] is \(\lambda = 15/8\).

Parameters

x	Argument in [0,1].

See also

Septic::Smooth(), Cubic::Smooth()

SmoothCompact()

double Quintic::SmoothCompact ( const double & x, const double & r )

inlinestatic

Compactly supported smooth interpolating function.

See also

Cubic::SmoothCompact(const double&, const double&)

Parameters

x	Squared distance.
r	Squared radius.

SmoothStep()

double Quintic::SmoothStep ( const double & x, const double & a, const double & b )

inlinestatic

Compute a quintic smooth step.

The code is slightly more efficient than:

double y=Quintic::Smooth(Linear::Step(x,a,b));

Parameters

x	Input value.
a,b	Interval values.

See also

Cubic::SmoothStep, Septic::SmoothStep()

Solve() [1/2]

int Quintic::Solve

(

double *

roots

)

const

Solve the quintic 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 Quintic::Solve	(	double *	roots,
		const double &	a,
		const double &	b ) const

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

Parameters

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

Friends And Related Symbol Documentation

operator<<

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

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The quintic.