Septic Class Reference

Septic (heptic) polynomials. More...

#include <septic.h>

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

Static Public Member Functions
static double	Smooth (const double &)
	Compute the value of smooth \(C^3\) interpolating function over unit interval.
static double	SmoothStep (const double &, const double &, const double &)
	Compute a septic smooth step.
static double	Gaussian (const double &, const double &, const double &)
	Compute a compactly supported Gaussian-like pulse.
static Septic	Hermite (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates an Hermite septic polynomial on interval [0,1].

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 [8] = { 0.0 }
	Array of coefficients.

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

Detailed Description

Septic (heptic) polynomials.

Constructors should provide the coefficients in descending order. Example of how to code the septic \(12x^7 - x + 1\):

Septic p(12.0,0.0,0.0,0.0,0.0,-1.0,1.0);

Constructor & Destructor Documentation

Septic() [1/5]

Septic::Septic ( const Sextic & p )

inline

Creates a septic given a sextic.

Simply set higher coefficients to 0.

Parameters

p

Quintic.

Septic() [2/5]

Septic::Septic ( const Quintic & p )

inline

Creates a septic given a quintic.

Simply set higher coefficients to 0.

Parameters

p

Quintic.

Septic() [3/5]

Septic::Septic ( const Quartic & p )

inline

Creates a septic given a quartic.

Parameters

p

Quartic.

Septic() [4/5]

Septic::Septic ( const Cubic & p )

inline

Creates a septic given a cubic.

Simply set higher coefficients to 0.

Parameters

p

Cubic.

Septic() [5/5]

Septic::Septic ( const Quadric & p )

inline

Creates a septic given a quadric.

Simply set higher coefficients to 0.

Parameters

p

Quadric.

Member Function Documentation

Derivative()

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

inlineconstexpr

Computes the derivative of the septic.

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

Parameters

x

Real.

Gaussian()

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

inlinestatic

Compute a compactly supported Gaussian-like pulse.

The function has \(C^3\) continuity.

See also

Cubic::Gaussian() and Quintic::Gaussian()

Parameters

c	Center.
r	Radius.
x	Value.

Hermite()

Septic Septic::Hermite ( const double & a, const double & b, const double & da, const double & db, const double & sa, const double & sb, const double & ta, const double & tb )

static

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

Author

Loïs Paulin

Parameters

a,b	Values at t=0 and t=1.
da,db	Derivatives.
sa,sb	Second derivatives.
ta,tb	Third derivatives.

K()

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

Compute the Lipschitz constant of the septic.

Parameters

a,b

Interval.

operator()()

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

inlineconstexpr

Evaluates the septic.

Parameters

x	Argument value of the function.

Range()

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

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

Parameters

a,b	Interval.
x,y	Returned range.

Smooth()

double Septic::Smooth ( const double & x )

inlinestatic

Compute the value of smooth \(C^3\) interpolating function over unit interval.

The septic is defined as \(x^4 (-20 x^3 + 70 x^2 - 84 x + 35)\). Its first, second and third derivatives at 0.0 and 1.0 are 0.0.

The Lipschitz constant of the smooth septic over [0,1] is \(\lambda = 35/16\).

Parameters

x	Argument in [0,1].

See also

Quintic::Smooth(), Cubic::Smooth()

SmoothStep()

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

inlinestatic

Compute a septic smooth step.

The code is slightly more efficient than:

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

Parameters

x	Input value.
a,b	Interval values.

See also

Cubic::SmoothStep(), Quintic::SmoothStep()

Solve() [1/2]

int Septic::Solve

(

double *

roots

)

const

Solve the septics 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 Septic::Solve	(	double *	roots,
		const double &	a,
		const double &	b ) const

Search the roots of a septic 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 Septic & p )

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The septic.