Nonic Class Reference

Nonic polynomials. More...

#include <nonic.h>

Public Member Functions
	Nonic ()
	Empty.
	Nonic (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a Nonic polynomial.
	~Nonic ()
	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 Nonic.
Nonic	operator- () const
	Unary.
Nonic &	operator+= (const Nonic &)
	Destructive sum of two Nonics.
Nonic &	operator-= (const Nonic &)
	Destructive difference of two Nonics.
Nonic &	operator*= (const double &)
	Scale a Nonics by a double value.
Nonic &	operator/= (const double &)
	Scale a Nonics by a double value.
int	Solve (double *) const
	Solve the octic equation over a given interval.
Octic	Prime () const
	Computes the first derivative of a septic, which is a quintic.
Septic	Second () const
	Computes the second derivative of a Nonic.
Sextic	Third () const
	Computes the third derivative of a Nonic.
constexpr double	operator() (const double &) const
	Evaluates the Nonic.
constexpr double	Derivative (const double &) const
	Computes the derivative of the sextic.

Static Public Member Functions
static Nonic	Compose (const Cubic &, const Cubic &)
	Compose the cubic by another one.
static double	Smooth (const double &)
	Compute the value of smooth C4 interpolating function over unit interval.
static double	SmoothStep (const double &, const double &, const double &)
	Compute a Nonic smooth step.
static double	Gaussian (const double &, const double &, const double &)
	Compute a compactly supported Gaussian-like pulse.

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 [10] = { 0.0 }
	Array of coefficients.

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

Detailed Description

Nonic polynomials.

Constructors should provide the coefficients in descending order. Example of how to code the Nonic 19x⁹-x+1:

Nonic p(19.0,0.0,0.0,0.0,0.0,0.0,0.0,-1.0,1.0);

Member Function Documentation

Compose()

Nonic Nonic::Compose ( const Cubic & p, const Cubic & q )

static

Compose the cubic by another one.

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

Parameters

q	The cubic.
p	The argument cubic.

Derivative()

double Nonic::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.

Gaussian()

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

inlinestatic

Compute a compactly supported Gaussian-like pulse.

The function has C⁴ continuity.

See also

Cubic::Gaussian(), Quintic::Gaussian() and Septic::Gaussian()

Parameters

c	Center.
r	Radius.
x	Value.

operator()()

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

inlineconstexpr

Evaluates the Nonic.

Parameters

x	Argument value of the function.

Smooth()

double Nonic::Smooth ( const double & x )

inlinestatic

Compute the value of smooth C⁴ interpolating function over unit interval.

The Nonic is defined as x⁴(-20 x³ + 70 x² -84 x + 35). Its first four derivatives at 0.0 and 1.0 are 0.0.

The Lipschitz constant of the smooth nonic over [0,1] is λ=315/128.

Parameters

x	Argument in [0,1].

See also

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

SmoothStep()

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

inlinestatic

Compute a Nonic smooth step.

The code is slightly more efficient than:

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

Parameters

x	Input value.
a,b	Interval values.

See also

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

Solve()

int Nonic::Solve

(

double *

roots

)

const

Solve the octic equation over a given interval.

Parameters

roots

Roots.

Friends And Related Symbol Documentation

operator<<

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

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The Nonic.