Polynomial Class Reference

Polynomials of degree lower than or equal to 11. More...

#include <polynomial.h>

Public Member Functions
	Polynomial ()
	Creates a null polynomial.
	~Polynomial ()
	Empty.
	Polynomial (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a degree 9 polynomial.
	Polynomial (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a degree 8 polynomial.
	Polynomial (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a septic.
	Polynomial (const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a sextic.
	Polynomial (const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a quintic.
	Polynomial (const double &, const double &, const double &, const double &, const double &)
	Creates a quartic.
	Polynomial (const double &, const double &, const double &, const double &)
	Creates a cubic.
	Polynomial (const double &, const double &, const double &)
	Creates a quadric.
	Polynomial (const double &, const double &)
	Creates a linear polynomial.
	Polynomial (const double &)
	Creates a constant polynomial.
	Polynomial (const Linear &)
	Creates a polynomial given a linear.
	Polynomial (const Quadric &)
	Creates a polynomial given a quadric.
	Polynomial (const Cubic &)
	Creates a polynomial given a cubic.
	Polynomial (const Quartic &)
	Creates a polynomial given a quartic.
	Polynomial (const Quintic &)
	Creates a polynomial given a quintic.
	Polynomial (const Sextic &)
	Creates a polynomial given a sextic.
	Polynomial (const Septic &)
	Creates a polynomial given a septic.
	Polynomial (const Octic &)
	Creates a polynomial given an octic.
	Polynomial (const Nonic &)
	Creates a polynomial given a nonic.
constexpr double &	operator[] (int)
	Access to the cefficients of the polynomial.
constexpr double	operator[] (int) const
	Overloaded.
Polynomial	operator+ () const
	Overloaded.
Polynomial	operator- () const
	Unary.
Polynomial &	operator+= (const Polynomial &)
	Destructive sum of two polynomials.
Polynomial &	operator-= (const Polynomial &)
	Destructive difference of two polynomials.
Polynomial &	operator*= (const Polynomial &)
	Overloaded.
Polynomial &	operator*= (const double &)
	Scale a polynomial by a double value. Optimizations are provided if scalar is 0.0 or 1.0.
Polynomial &	operator/= (const double &)
	Scale a polynomial by a double value.
Polynomial	Prime () const
	Compute the derivative of the polynomial.
Polynomial	Compose (const Polynomial &) const
	Compose the polynomial by the argument polynomial.
Polynomial	Reversed () const
	Reverse the terms of the polynomial.
double	operator() (const double &) const
	Evaluates the polynomial.
int	Solve (double *) const
	Search the roots of a polynomial equation.
int	SturmSolve (double *) const
	Search the roots of a polynomial equation using the Sturm sequences.
int	Solve (double *, const double &, const double &) const
	Search the roots of a polynomial equation over a given interval.
int	Degree () const
	Returns the degree of the polynomial.
void	Check ()
	Check the degree of the polynomial.
void	Range (double &, double &, const double &=0.0, const double &=1.0) const
	This function computes the range of values taken by a polynomial over a given interval.
int	Bissection (double, double, double &, const double &) const
	Compute the roots of a polynomial equation using bissection.

Static Public Member Functions
static Polynomial	Compose (const Quartic &, const Quadric &)
	Compose the quartic by a quadric.

Static Public Attributes
static const Polynomial	Null
	Set the null polynomial member constant.

Protected Attributes
int	n = 0
	The degree of the polynomial.
double	c [MaxDegree+1] = { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 }
	Array of coefficients.

Static Protected Attributes
static constexpr int	MaxDegree = 11
	Maximum degree.

Friends
Polynomial	operator+ (const Polynomial &u, const Polynomial &v)
	Overloaded.
Polynomial	operator- (const Polynomial &u, const Polynomial &v)
	Overloaded.
Polynomial	operator* (const Polynomial &u, const Polynomial &v)
	Multiply two polynomials.
Polynomial	operator* (const Polynomial &u, const double &e)
	Multiply a polynomial by a scalar value. Optimizations are provided if scalar is 0.0 or 1.0.
Polynomial	operator* (const double &a, const Polynomial &p)
	Overloaded.
Polynomial	operator/ (const Polynomial &p, const double &a)
	Overloaded.
std::ostream &	operator<< (std::ostream &s, const Polynomial &p)
	Overloaded output-stream operator.

Detailed Description

Polynomials of degree lower than or equal to 11.

This class is defined as a fixed size data-structure to speed-up and facilitate memory management. It is more memory and computationnally efficient to use Linear, Quadric, Cubic, and Quartic classes whenever possible.

When using constructors, the coefficients are given in descending order, i.e. start from the highest degree coefficient to the lowest degree.

Example of how to code the quartic polynomial 3 x⁴-x+1:

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

See also

Septic, Sextic, Quintic, Quartic, Cubic, Quadric, Linear.

Constructor & Destructor Documentation

Polynomial() [1/9]

Polynomial::Polynomial ( const Linear & p )

inline

Creates a polynomial given a linear.

Parameters

p	Linear polynomial.

Polynomial() [2/9]

Polynomial::Polynomial ( const Quadric & p )

inline

Creates a polynomial given a quadric.

Parameters

p	Quadric polynomial.

Polynomial() [3/9]

Polynomial::Polynomial ( const Cubic & p )

inline

Creates a polynomial given a cubic.

Parameters

p	Cubic polynomial.

Polynomial() [4/9]

Polynomial::Polynomial ( const Quartic & p )

inline

Creates a polynomial given a quartic.

Parameters

p	Quartic polynomial.

Polynomial() [5/9]

Polynomial::Polynomial ( const Quintic & p )

inline

Creates a polynomial given a quintic.

Parameters

p	Quintic polynomial.

Polynomial() [6/9]

Polynomial::Polynomial ( const Sextic & p )

inline

Creates a polynomial given a sextic.

Parameters

p	Sextic polynomial.

Polynomial() [7/9]

Polynomial::Polynomial ( const Septic & p )

inline

Creates a polynomial given a septic.

Parameters

p	Septic polynomial.

Polynomial() [8/9]

Polynomial::Polynomial ( const Octic & p )

inline

Creates a polynomial given an octic.

Parameters

p	Octic polynomial.

Polynomial() [9/9]

Polynomial::Polynomial ( const Nonic & p )

inline

Creates a polynomial given a nonic.

Parameters

p	Nonic polynomial.

Member Function Documentation

Bissection()

int Polynomial::Bissection	(	double	a,
		double	b,
		double &	val,
		const double &	epsilon ) const

Compute the roots of a polynomial equation using bissection.

Parameters

a,b	Interval.
val	Value.
epsilon	Precision.

Check()

void Polynomial::Check

(

)

Check the degree of the polynomial.

The function decreases the degree if higher coefficients are null.

Compose() [1/2]

Polynomial Polynomial::Compose

(

const Polynomial &

v

)

const

Compose the polynomial by the argument polynomial.

Parameters

v	Argument polynomial.

Compose() [2/2]

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

static

Compose the quartic by a quadric.

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

Parameters

p	The quartic.
q	The quadric.

operator()()

double Polynomial::operator()

(

const double &

x

)

const

Evaluates the polynomial.

Parameters

x	Parameter.

Range()

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

This function computes the range of values taken by a polynomial over a given interval.

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

Parameters

a,b	Interval.
x,y	Returned range.

Solve() [1/2]

int Polynomial::Solve

(

double *

roots

)

const

Search the roots of a polynomial equation.

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

Otherwise, rely on a numeric root finder to isolate roots and converge using the Sturm sequences.

Parameters

roots

Array for storing the roots.

Returns

The number of roots.

Solve() [2/2]

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

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

Parameters

roots	The array of roots.
a,b	The search interval.

Returns

The number of roots.

SturmSolve()

int Polynomial::SturmSolve

(

double *

roots

)

const

Search the roots of a polynomial equation using the Sturm sequences.

Parameters

roots

Array for storing the roots.

Returns

The number of roots.

Friends And Related Symbol Documentation

operator<<

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

friend

Overloaded output-stream operator.

Parameters

s	Stream.
p	The polynomial.