polynomial_tools.h

// Description: Algorithms for polynomials
// Detail: differentiate, derivative, trim, divide, randomPolynomial, etc.

#ifndef PASTELMATH_POLYNOMIAL_TOOLS_H
#define PASTELMATH_POLYNOMIAL_TOOLS_H

#include "pastel/math/polynomial/polynomial.h"

#include <ostream>

namespace Pastel
{

    //! Differentiates a polynomial.
    template <typename Real>
    void differentiate(Polynomial<Real>& that);

    //! Returns the derivative of a polynomial.
    template <typename Real>
    Polynomial<Real> derivative(const Polynomial<Real>& that);

    //! Removes leading zero factors.
    template <typename Real>
    void trim(Polynomial<Real>& that);

    //! Divides a polynomial by another.
    template <typename Real>
    void divide(
        const Polynomial<Real>& that,
        const Polynomial<Real>& divider,
        Polynomial<Real>& quotient,
        Polynomial<Real>& remainder);

    //! Returns the quotient of dividing a polynomial by another.
    template <typename Real>
    Polynomial<Real> quotient(
        const Polynomial<Real>& that,
        const Polynomial<Real>& divider);

    //! Returns the remainder of dividing a polynomial by another.
    template <typename Real>
    Polynomial<Real> remainder(
        const Polynomial<Real>& that,
        const Polynomial<Real>& divider);

    //! Computes the Sturm-sequence of a polynomial.
    template <typename Real>
    void sturmSequence(
        const Polynomial<Real>& that,
        std::vector<Polynomial<Real> > sturm);

    //! Returns the number of roots before 'max'.
    /*!
   'sturm' must be a Sturm-sequence.
   */
    template <typename Real>
    integer roots(
        const std::vector<Polynomial<Real> >& sturm,
        const Real& max);

    //! Returns the number of roots in the interval [min, max].
    /*!
   'sturm' must be a Sturm-sequence.
   */
    template <typename Real>
    integer roots(
        const std::vector<Polynomial<Real> >& sturm,
        const Real& min, const Real& max);

    //! Returns a random polynomial of the given degree.
    /*!
   The coefficients of the polynomial will
   be chosen uniformly from the range [0, 1].
   */
    template <typename Real>
    Polynomial<Real> randomPolynomial(
        integer degree);

    //! Pretty-prints a polynomial to a stream.
    template <typename Real>
    std::ostream& operator<<(
        std::ostream& stream,
        const Polynomial<Real>& that);

}

#include "pastel/math/polynomial/polynomial_tools.hpp"

#endif