Polynomials

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#ifndef EIGEN_POLYNOMIALS_MODULE_H
#define EIGEN_POLYNOMIALS_MODULE_H

#include <Eigen/Core>

#include <Eigen/src/Core/util/DisableStupidWarnings.h>

#include <Eigen/Eigenvalues>

// Note that EIGEN_HIDE_HEAVY_CODE has to be defined per module
#if (defined EIGEN_EXTERN_INSTANTIATIONS) && (EIGEN_EXTERN_INSTANTIATIONS>=2)
  #ifndef EIGEN_HIDE_HEAVY_CODE
  #define EIGEN_HIDE_HEAVY_CODE
  #endif
#elif defined EIGEN_HIDE_HEAVY_CODE
  #undef EIGEN_HIDE_HEAVY_CODE
#endif

namespace Eigen {

/** \ingroup Unsupported_modules
  * \defgroup Polynomials_Module Polynomials module  
 * \ingroup eigen_grp
 * \ingroup eigen_grp
  *
  *
  *
  * \brief This module provides a QR based polynomial solver.
        *
  * To use this module, add
  * \code
  * #include <unsupported/Eigen/Polynomials>
  * \endcode
        * at the start of your source file.
  */

#include "src/Polynomials/PolynomialUtils.h"
#include "src/Polynomials/Companion.h"
#include "src/Polynomials/PolynomialSolver.h"

/**
        \page polynomials Polynomials defines functions for dealing with polynomials
        and a QR based polynomial solver.
        \ingroup Polynomials_Module

        The remainder of the page documents first the functions for evaluating, computing
        polynomials, computing estimates about polynomials and next the QR based polynomial
        solver.

        \section polynomialUtils convenient functions to deal with polynomials
        \subsection roots_to_monicPolynomial
        The function
        \code
        void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
        \endcode
        computes the coefficients \f$ a_i \f$ of

        \f$ p(x) = a_0 + a_{1}x + ... + a_{n-1}x^{n-1} + x^n \f$

        where \f$ p \f$ is known through its roots i.e. \f$ p(x) = (x-r_1)(x-r_2)...(x-r_n) \f$.

        \subsection poly_eval
        The function
        \code
        T poly_eval( const Polynomials& poly, const T& x )
        \endcode
        evaluates a polynomial at a given point using stabilized H&ouml;rner method.

        The following code: first computes the coefficients in the monomial basis of the monic polynomial that has the provided roots;
        then, it evaluates the computed polynomial, using a stabilized H&ouml;rner method.

        \include PolynomialUtils1.cpp
  Output: \verbinclude PolynomialUtils1.out

        \subsection Cauchy bounds
        The function
        \code
        Real cauchy_max_bound( const Polynomial& poly )
        \endcode
        provides a maximum bound (the Cauchy one: \f$C(p)\f$) for the absolute value of a root of the given polynomial i.e.
        \f$ \forall r_i \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
        \f$ |r_i| \le C(p) = \sum_{k=0}^{d} \left | \frac{a_k}{a_d} \right | \f$
        The leading coefficient \f$ p \f$: should be non zero \f$a_d \neq 0\f$.


        The function
        \code
        Real cauchy_min_bound( const Polynomial& poly )
        \endcode
        provides a minimum bound (the Cauchy one: \f$c(p)\f$) for the absolute value of a non zero root of the given polynomial i.e.
        \f$ \forall r_i \neq 0 \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
        \f$ |r_i| \ge c(p) = \left( \sum_{k=0}^{d} \left | \frac{a_k}{a_0} \right | \right)^{-1} \f$




        \section QR polynomial solver class
        Computes the complex roots of a polynomial by computing the eigenvalues of the associated companion matrix with the QR algorithm.
        
        The roots of \f$ p(x) = a_0 + a_1 x + a_2 x^2 + a_{3} x^3 + x^4 \f$ are the eigenvalues of
        \f$
        \left [
        \begin{array}{cccc}
        0 & 0 &  0 & a_0 \\
        1 & 0 &  0 & a_1 \\
        0 & 1 &  0 & a_2 \\
        0 & 0 &  1 & a_3
        \end{array} \right ]
        \f$

        However, the QR algorithm is not guaranteed to converge when there are several eigenvalues with same modulus.

        Therefore the current polynomial solver is guaranteed to provide a correct result only when the complex roots \f$r_1,r_2,...,r_d\f$ have distinct moduli i.e.
        
        \f$ \forall i,j \in [1;d],~ \| r_i \| \neq \| r_j \| \f$.

        With 32bit (float) floating types this problem shows up frequently.
  However, almost always, correct accuracy is reached even in these cases for 64bit
  (double) floating types and small polynomial degree (<20).

        \include PolynomialSolver1.cpp
        
        In the above example:
        
        -# a simple use of the polynomial solver is shown;
        -# the accuracy problem with the QR algorithm is presented: a polynomial with almost conjugate roots is provided to the solver.
        Those roots have almost same module therefore the QR algorithm failed to converge: the accuracy
        of the last root is bad;
        -# a simple way to circumvent the problem is shown: use doubles instead of floats.

  Output: \verbinclude PolynomialSolver1.out
*/

} // namespace Eigen

#include <Eigen/src/Core/util/ReenableStupidWarnings.h>

#endif // EIGEN_POLYNOMIALS_MODULE_H
/* vim: set filetype=cpp et sw=2 ts=2 ai: */