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.
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 Quintic	Bezier (const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a quintic Bezier polynomial.
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 const 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 of degree five or greater.

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

Quintic p(1.0,0.0,0.0,0.0,-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

Bezier()

Quintic Quintic::Bezier ( const double & a, const double & b, const double & c, const double & d, const double & e, const double & f )

static

Creates a quintic Bezier polynomial.

Interval parameterization is unit.

Parameters

a,b,c,d,e,f

Control values.

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});

Warning

The vector must contain exactly 5 elements. No bounds checking is performed; passing a vector with fewer than 5 elements causes undefined behaviour.

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.

This function stores the roots in an array and returns the number of roots. The roots are not guaranteed to be sorted, and when Polynomial::Solve() uses the Sturm sequences, as the degree is larger than four, only the roots in the interval [-1e7,1e7] are found. See (double*, const double&, const double&) to select the roots in a given interval.

Calls Polynomial::Solve() when the leading coefficient is non-zero. Otherwise cascades to lower-degree solvers (Quartic, Cubic, Quadric) by dropping zero leading coefficients.

Parameters

roots

The array of roots.

Returns

The number 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.