Creation of the Polynomials module with the following features:

* convenient functions:
  - Horner and stabilized Horner evaluation
  - polynomial coefficients from a set of given roots
  - Cauchy bounds
* a QR based polynomial solver

4 files changed, 835 insertions, 0 deletions

diff --git a/unsupported/Eigen/src/Polynomials/CMakeLists.txt b/unsupported/Eigen/src/Polynomials/CMakeLists.txt
new file mode 100644
index 000000000..51f13f3cb
--- /dev/null
+++ b/unsupported/Eigen/src/Polynomials/CMakeLists.txt
@@ -0,0 +1,6 @@
+FILE(GLOB Eigen_Polynomials_SRCS "*.h")
+
+INSTALL(FILES
+  ${Eigen_Polynomials_SRCS}
+  DESTINATION ${INCLUDE_INSTALL_DIR}/unsupported/Eigen/src/Polynomials COMPONENT Devel
+  )
diff --git a/unsupported/Eigen/src/Polynomials/Companion.h b/unsupported/Eigen/src/Polynomials/Companion.h
new file mode 100644
index 000000000..7c9e9c518
--- /dev/null
+++ b/unsupported/Eigen/src/Polynomials/Companion.h
@@ -0,0 +1,281 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_COMPANION_H
+#define EIGEN_COMPANION_H
+
+// This file requires the user to include
+// * Eigen/Core
+// * Eigen/src/PolynomialSolver.h
+
+#ifndef EIGEN_PARSED_BY_DOXYGEN
+
+template <typename T>
+T ei_radix(){ return 2; }
+
+template <typename T>
+T ei_radix2(){ return ei_radix<T>()*ei_radix<T>(); }
+
+template<int Size>
+struct ei_decrement_if_fixed_size
+{
+  enum {
+    ret = (Size == Dynamic) ? Dynamic : Size-1 };
+};
+
+#endif
+
+template< typename _Scalar, int _Deg >
+class ei_companion
+{
+  public:
+    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
+
+    enum {
+      Deg = _Deg,
+      Deg_1=ei_decrement_if_fixed_size<Deg>::ret
+    };
+
+    typedef _Scalar                                Scalar;
+    typedef typename NumTraits<Scalar>::Real       RealScalar;
+    typedef Matrix<Scalar, Deg, 1>                 RightColumn;
+    //typedef DiagonalMatrix< Scalar, Deg_1, Deg_1 > BottomLeftDiagonal;
+    typedef Matrix<Scalar, Deg_1, 1>               BottomLeftDiagonal;
+
+    typedef Matrix<Scalar, Deg, Deg>               DenseCompanionMatrixType;
+    typedef Matrix< Scalar, _Deg, Deg_1 >          LeftBlock;
+    typedef Matrix< Scalar, Deg_1, Deg_1 >         BottomLeftBlock;
+    typedef Matrix< Scalar, 1, Deg_1 >             LeftBlockFirstRow;
+
+  public:
+    EIGEN_STRONG_INLINE const _Scalar operator()( int row, int col ) const
+    {
+      if( m_bl_diag.rows() > col )
+      {
+        if( 0 < row ){ return m_bl_diag[col]; }
+        else{ return 0; }
+      }
+      else{ return m_monic[row]; }
+    }
+
+  public:
+    template<typename VectorType>
+    void setPolynomial( const VectorType& poly )
+    {
+      const int deg = poly.size()-1;
+      m_monic = -1/poly[deg] * poly.head(deg);
+      //m_bl_diag.setIdentity( deg-1 );
+      m_bl_diag.setOnes(deg-1);
+    }
+
+    template<typename VectorType>
+    ei_companion( const VectorType& poly ){
+      setPolynomial( poly ); }
+
+  public:
+    DenseCompanionMatrixType denseMatrix() const
+    {
+      const int deg   = m_monic.size();
+      const int deg_1 = deg-1;
+      DenseCompanionMatrixType companion(deg,deg);
+      companion <<
+        ( LeftBlock(deg,deg_1)
+          << LeftBlockFirstRow::Zero(1,deg_1),
+          BottomLeftBlock::Identity(deg-1,deg-1)*m_bl_diag.asDiagonal() ).finished()
+        , m_monic;
+      return companion;
+    }
+
+
+
+  protected:
+    /** Helper function for the balancing algorithm.
+     * \returns true if the row and the column, having colNorm and rowNorm
+     * as norms, are balanced, false otherwise.
+     * colB and rowB are repectively the multipliers for
+     * the column and the row in order to balance them.
+     * */
+    bool balanced( Scalar colNorm, Scalar rowNorm,
+        bool& isBalanced, Scalar& colB, Scalar& rowB );
+
+    /** Helper function for the balancing algorithm.
+     * \returns true if the row and the column, having colNorm and rowNorm
+     * as norms, are balanced, false otherwise.
+     * colB and rowB are repectively the multipliers for
+     * the column and the row in order to balance them.
+     * */
+    bool balancedR( Scalar colNorm, Scalar rowNorm,
+        bool& isBalanced, Scalar& colB, Scalar& rowB );
+
+  public:
+    /**
+     * Balancing algorithm from B. N. PARLETT and C. REINSCH (1969)
+     * "Balancing a matrix for calculation of eigenvalues and eigenvectors"
+     * adapted to the case of companion matrices.
+     * A matrix with non zero row and non zero column is balanced
+     * for a certain norm if the i-th row and the i-th column
+     * have same norm for all i.
+     */
+    void balance();
+
+  protected:
+      RightColumn                m_monic;
+      BottomLeftDiagonal         m_bl_diag;
+};
+
+
+
+template< typename _Scalar, int _Deg >
+inline
+bool ei_companion<_Scalar,_Deg>::balanced( Scalar colNorm, Scalar rowNorm,
+    bool& isBalanced, Scalar& colB, Scalar& rowB )
+{
+  if( Scalar(0) == colNorm || Scalar(0) == rowNorm ){ return true; }
+  else
+  {
+    //To find the balancing coefficients, if the radix is 2,
+    //one finds \f$ \sigma \f$ such that
+    // \f$ 2^{2\sigma-1} < rowNorm / colNorm \le 2^{2\sigma+1} \f$
+    // then the balancing coefficient for the row is \f$ 1/2^{\sigma} \f$
+    // and the balancing coefficient for the column is \f$ 2^{\sigma} \f$
+    rowB = rowNorm / ei_radix<Scalar>();
+    colB = Scalar(1);
+    const Scalar s = colNorm + rowNorm;
+
+    while (colNorm < rowB)
+    {
+      colB *= ei_radix<Scalar>();
+      colNorm *= ei_radix2<Scalar>();
+    }
+
+    rowB = rowNorm * ei_radix<Scalar>();
+
+    while (colNorm >= rowB)
+    {
+      colB /= ei_radix<Scalar>();
+      colNorm /= ei_radix2<Scalar>();
+    }
+
+    //This line is used to avoid insubstantial balancing
+    if ((rowNorm + colNorm) < Scalar(0.95) * s * colB)
+    {
+      isBalanced = false;
+      rowB = Scalar(1) / colB;
+      return false;
+    }
+    else{
+      return true; }
+  }
+}
+
+template< typename _Scalar, int _Deg >
+inline
+bool ei_companion<_Scalar,_Deg>::balancedR( Scalar colNorm, Scalar rowNorm,
+    bool& isBalanced, Scalar& colB, Scalar& rowB )
+{
+  if( Scalar(0) == colNorm || Scalar(0) == rowNorm ){ return true; }
+  else
+  {
+    /**
+     * Set the norm of the column and the row to the geometric mean
+     * of the row and column norm
+     */
+    const _Scalar q = colNorm/rowNorm;
+    if( !ei_isApprox( q, _Scalar(1) ) )
+    {
+      rowB = ei_sqrt( colNorm/rowNorm );
+      colB = Scalar(1)/rowB;
+
+      isBalanced = false;
+      return false;
+    }
+    else{
+      return true; }
+  }
+}
+
+
+template< typename _Scalar, int _Deg >
+void ei_companion<_Scalar,_Deg>::balance()
+{
+  EIGEN_STATIC_ASSERT( 1 < Deg, YOU_MADE_A_PROGRAMMING_MISTAKE );
+  const int deg   = m_monic.size();
+  const int deg_1 = deg-1;
+
+  bool hasConverged=false;
+  while( !hasConverged )
+  {
+    hasConverged = true;
+    Scalar colNorm,rowNorm;
+    Scalar colB,rowB;
+
+    //First row, first column excluding the diagonal
+    //==============================================
+    colNorm = ei_abs(m_bl_diag[0]);
+    rowNorm = ei_abs(m_monic[0]);
+
+    //Compute balancing of the row and the column
+    if( !balanced( colNorm, rowNorm, hasConverged, colB, rowB ) )
+    {
+      m_bl_diag[0] *= colB;
+      m_monic[0] *= rowB;
+    }
+
+    //Middle rows and columns excluding the diagonal
+    //==============================================
+    for( int i=1; i<deg_1; ++i )
+    {
+      // column norm, excluding the diagonal
+      colNorm = ei_abs(m_bl_diag[i]);
+
+      // row norm, excluding the diagonal
+      rowNorm = ei_abs(m_bl_diag[i-1]) + ei_abs(m_monic[i]);
+
+      //Compute balancing of the row and the column
+      if( !balanced( colNorm, rowNorm, hasConverged, colB, rowB ) )
+      {
+        m_bl_diag[i]   *= colB;
+        m_bl_diag[i-1] *= rowB;
+        m_monic[i]     *= rowB;
+      }
+    }
+
+    //Last row, last column excluding the diagonal
+    //============================================
+    const int ebl = m_bl_diag.size()-1;
+    VectorBlock<RightColumn,Deg_1> headMonic( m_monic, 0, deg_1 );
+    colNorm = headMonic.array().abs().sum();
+    rowNorm = ei_abs( m_bl_diag[ebl] );
+
+    //Compute balancing of the row and the column
+    if( !balanced( colNorm, rowNorm, hasConverged, colB, rowB ) )
+    {
+      headMonic      *= colB;
+      m_bl_diag[ebl] *= rowB;
+    }
+  }
+}
+
+
+#endif // EIGEN_COMPANION_H
diff --git a/unsupported/Eigen/src/Polynomials/PolynomialSolver.h b/unsupported/Eigen/src/Polynomials/PolynomialSolver.h
new file mode 100644
index 000000000..49a5ffffa
--- /dev/null
+++ b/unsupported/Eigen/src/Polynomials/PolynomialSolver.h
@@ -0,0 +1,395 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_POLYNOMIAL_SOLVER_H
+#define EIGEN_POLYNOMIAL_SOLVER_H
+
+/** \ingroup Polynomials_Module
+ *  \class PolynomialSolverBase.
+ *
+ * \brief Defined to be inherited by polynomial solvers: it provides
+ * convenient methods such as
+ *  - real roots,
+ *  - greatest, smallest complex roots,
+ *  - real roots with greatest, smallest absolute real value,
+ *  - greatest, smallest real roots.
+ *
+ * It stores the set of roots as a vector of complexes.
+ *
+ */
+template< typename _Scalar, int _Deg >
+class PolynomialSolverBase
+{
+  public:
+    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
+
+    typedef _Scalar                             Scalar;
+    typedef typename NumTraits<Scalar>::Real    RealScalar;
+    typedef std::complex<RealScalar>            RootType;
+    typedef Matrix<RootType,_Deg,1>             RootsType;
+
+  protected:
+    template< typename OtherPolynomial >
+    inline void setPolynomial( const OtherPolynomial& poly ){
+      m_roots.resize(poly.size()); }
+
+  public:
+    template< typename OtherPolynomial >
+    inline PolynomialSolverBase( const OtherPolynomial& poly ){
+      setPolynomial( poly() ); }
+
+    inline PolynomialSolverBase(){}
+
+  public:
+    /** \returns the complex roots of the polynomial */
+    inline const RootsType& roots() const { return m_roots; }
+
+  public:
+    /** Clear and fills the back insertion sequence with the real roots of the polynomial
+     * i.e. the real part of the complex roots that have an imaginary part which
+     * absolute value is smaller than absImaginaryThreshold.
+     * absImaginaryThreshold takes the dummy_precision associated
+     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
+     *
+     * \param[out] bi_seq : the back insertion sequence (stl concept)
+     * \param[in]  absImaginaryThreshold : the maximum bound of the imaginary part of a complex
+     *  number that is considered as real.
+     * */
+    template<typename Stl_back_insertion_sequence>
+    inline void realRoots( Stl_back_insertion_sequence& bi_seq,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      bi_seq.clear();
+      for( int i=0; i<m_roots.size(); ++i )
+      {
+        if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold ){
+          bi_seq.push_back( m_roots[i].real() ); }
+      }
+    }
+
+  protected:
+    template<typename squaredNormBinaryPredicate>
+    inline const RootType& selectComplexRoot_withRespectToNorm( squaredNormBinaryPredicate& pred ) const
+    {
+      int res=0;
+      RealScalar norm2 = ei_abs2( m_roots[0] );
+      for( int i=1; i<m_roots.size(); ++i )
+      {
+        const RealScalar currNorm2 = ei_abs2( m_roots[i] );
+        if( pred( currNorm2, norm2 ) ){
+          res=i; norm2=currNorm2; }
+      }
+      return m_roots[res];
+    }
+
+  public:
+    /**
+     * \returns the complex root with greatest norm.
+     */
+    inline const RootType& greatestRoot() const
+    {
+      std::greater<Scalar> greater;
+      return selectComplexRoot_withRespectToNorm( greater );
+    }
+
+    /**
+     * \returns the complex root with smallest norm.
+     */
+    inline const RootType& smallestRoot() const
+    {
+      std::less<Scalar> less;
+      return selectComplexRoot_withRespectToNorm( less );
+    }
+
+  protected:
+    template<typename squaredRealPartBinaryPredicate>
+    inline const RealScalar& selectRealRoot_withRespectToAbsRealPart(
+        squaredRealPartBinaryPredicate& pred,
+        bool& hasArealRoot,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      hasArealRoot = false;
+      int res=0;
+      RealScalar abs2;
+
+      for( int i=0; i<m_roots.size(); ++i )
+      {
+        if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold )
+        {
+          if( !hasArealRoot )
+          {
+            hasArealRoot = true;
+            res = i;
+            abs2 = m_roots[i].real() * m_roots[i].real();
+          }
+          else
+          {
+            const RealScalar currAbs2 = m_roots[i].real() * m_roots[i].real();
+            if( pred( currAbs2, abs2 ) )
+            {
+              abs2 = currAbs2;
+              res = i;
+            }
+          }
+        }
+        else
+        {
+          if( ei_abs( m_roots[i].imag() ) < ei_abs( m_roots[res].imag() ) ){
+            res = i; }
+        }
+      }
+      return m_roots[res].real();
+    }
+
+
+    template<typename RealPartBinaryPredicate>
+    inline const RealScalar& selectRealRoot_withRespectToRealPart(
+        RealPartBinaryPredicate& pred,
+        bool& hasArealRoot,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      hasArealRoot = false;
+      int res=0;
+      RealScalar val;
+
+      for( int i=0; i<m_roots.size(); ++i )
+      {
+        if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold )
+        {
+          if( !hasArealRoot )
+          {
+            hasArealRoot = true;
+            res = i;
+            val = m_roots[i].real();
+          }
+          else
+          {
+            const RealScalar curr = m_roots[i].real();
+            if( pred( curr, val ) )
+            {
+              val = curr;
+              res = i;
+            }
+          }
+        }
+        else
+        {
+          if( ei_abs( m_roots[i].imag() ) < ei_abs( m_roots[res].imag() ) ){
+            res = i; }
+        }
+      }
+      return m_roots[res].real();
+    }
+
+  public:
+    /**
+     * \returns a real root with greatest absolute magnitude.
+     * A real root is defined as the real part of a complex root with absolute imaginary
+     * part smallest than absImaginaryThreshold.
+     * absImaginaryThreshold takes the dummy_precision associated
+     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
+     * If no real root is found the boolean hasArealRoot is set to false and the real part of
+     * the root with smallest absolute imaginary part is returned instead.
+     *
+     * \param[out] hasArealRoot : boolean true if a real root is found according to the
+     *  absImaginaryThreshold criterion, false otherwise.
+     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
+     *  whether or not a root is real.
+     */
+    inline const RealScalar& absGreatestRealRoot(
+        bool& hasArealRoot,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      std::greater<Scalar> greater;
+      return selectRealRoot_withRespectToAbsRealPart( greater, hasArealRoot, absImaginaryThreshold );
+    }
+
+
+    /**
+     * \returns a real root with smallest absolute magnitude.
+     * A real root is defined as the real part of a complex root with absolute imaginary
+     * part smallest than absImaginaryThreshold.
+     * absImaginaryThreshold takes the dummy_precision associated
+     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
+     * If no real root is found the boolean hasArealRoot is set to false and the real part of
+     * the root with smallest absolute imaginary part is returned instead.
+     *
+     * \param[out] hasArealRoot : boolean true if a real root is found according to the
+     *  absImaginaryThreshold criterion, false otherwise.
+     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
+     *  whether or not a root is real.
+     */
+    inline const RealScalar& absSmallestRealRoot(
+        bool& hasArealRoot,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      std::less<Scalar> less;
+      return selectRealRoot_withRespectToAbsRealPart( less, hasArealRoot, absImaginaryThreshold );
+    }
+
+
+    /**
+     * \returns the real root with greatest value.
+     * A real root is defined as the real part of a complex root with absolute imaginary
+     * part smallest than absImaginaryThreshold.
+     * absImaginaryThreshold takes the dummy_precision associated
+     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
+     * If no real root is found the boolean hasArealRoot is set to false and the real part of
+     * the root with smallest absolute imaginary part is returned instead.
+     *
+     * \param[out] hasArealRoot : boolean true if a real root is found according to the
+     *  absImaginaryThreshold criterion, false otherwise.
+     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
+     *  whether or not a root is real.
+     */
+    inline const RealScalar& greatestRealRoot(
+        bool& hasArealRoot,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      std::greater<Scalar> greater;
+      return selectRealRoot_withRespectToRealPart( greater, hasArealRoot, absImaginaryThreshold );
+    }
+
+
+    /**
+     * \returns the real root with smallest value.
+     * A real root is defined as the real part of a complex root with absolute imaginary
+     * part smallest than absImaginaryThreshold.
+     * absImaginaryThreshold takes the dummy_precision associated
+     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
+     * If no real root is found the boolean hasArealRoot is set to false and the real part of
+     * the root with smallest absolute imaginary part is returned instead.
+     *
+     * \param[out] hasArealRoot : boolean true if a real root is found according to the
+     *  absImaginaryThreshold criterion, false otherwise.
+     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
+     *  whether or not a root is real.
+     */
+    inline const RealScalar& smallestRealRoot(
+        bool& hasArealRoot,
+        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
+    {
+      std::less<Scalar> less;
+      return selectRealRoot_withRespectToRealPart( less, hasArealRoot, absImaginaryThreshold );
+    }
+
+  protected:
+    RootsType               m_roots;
+};
+
+#define EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( BASE )  \
+  typedef typename BASE::Scalar                 Scalar;       \
+  typedef typename BASE::RealScalar             RealScalar;   \
+  typedef typename BASE::RootType               RootType;     \
+  typedef typename BASE::RootsType              RootsType;
+
+
+
+/** \ingroup Polynomials_Module
+  *
+  * \class PolynomialSolver
+  *
+  * \brief A polynomial solver
+  *
+  * Computes the complex roots of a real polynomial.
+  *
+  * \param _Scalar the scalar type, i.e., the type of the polynomial coefficients
+  * \param _Deg the degree of the polynomial, can be a compile time value or Dynamic.
+  *             Notice that the number of polynomial coefficients is _Deg+1.
+  *
+  * This class implements a polynomial solver and provides convenient methods such as
+  * - real roots,
+  * - greatest, smallest complex roots,
+  * - real roots with greatest, smallest absolute real value.
+  * - greatest, smallest real roots.
+  *
+  * WARNING: this polynomial solver is experimental, part of the unsuported Eigen modules.
+  *
+  *
+  * Currently a QR algorithm is used to compute the eigenvalues of the companion matrix of
+  * the polynomial to compute its roots.
+  * This supposes that the complex moduli of the roots are all distinct: e.g. there should
+  * be no multiple roots or conjugate roots for instance.
+  * 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).
+  */
+template< typename _Scalar, int _Deg >
+class PolynomialSolver : public PolynomialSolverBase<_Scalar,_Deg>
+{
+  public:
+    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
+
+    typedef PolynomialSolverBase<_Scalar,_Deg>    PS_Base;
+    EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( PS_Base )
+
+    typedef Matrix<Scalar,_Deg,_Deg>                 CompanionMatrixType;
+    typedef EigenSolver<CompanionMatrixType>         EigenSolverType;
+
+  public:
+    /** Computes the complex roots of a new polynomial. */
+    template< typename OtherPolynomial >
+    void compute( const OtherPolynomial& poly )
+    {
+      assert( Scalar(0) != poly[poly.size()-1] );
+      ei_companion<Scalar,_Deg> companion( poly );
+      companion.balance();
+      m_eigenSolver.compute( companion.denseMatrix() );
+      m_roots = m_eigenSolver.eigenvalues();
+    }
+
+  public:
+    template< typename OtherPolynomial >
+    inline PolynomialSolver( const OtherPolynomial& poly ){
+      compute( poly ); }
+
+    inline PolynomialSolver(){}
+
+  protected:
+    using                   PS_Base::m_roots;
+    EigenSolverType         m_eigenSolver;
+};
+
+
+template< typename _Scalar >
+class PolynomialSolver<_Scalar,1> : public PolynomialSolverBase<_Scalar,1>
+{
+  public:
+    typedef PolynomialSolverBase<_Scalar,1>    PS_Base;
+    EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( PS_Base )
+
+  public:
+    /** Computes the complex roots of a new polynomial. */
+    template< typename OtherPolynomial >
+    void compute( const OtherPolynomial& poly )
+    {
+      assert( Scalar(0) != poly[poly.size()-1] );
+      m_roots[0] = -poly[0]/poly[poly.size()-1];
+    }
+
+  protected:
+    using                   PS_Base::m_roots;
+};
+
+#endif // EIGEN_POLYNOMIAL_SOLVER_H
diff --git a/unsupported/Eigen/src/Polynomials/PolynomialUtils.h b/unsupported/Eigen/src/Polynomials/PolynomialUtils.h
new file mode 100644
index 000000000..d78821f90
--- /dev/null
+++ b/unsupported/Eigen/src/Polynomials/PolynomialUtils.h
@@ -0,0 +1,153 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_POLYNOMIAL_UTILS_H
+#define EIGEN_POLYNOMIAL_UTILS_H
+
+/** \ingroup Polynomials_Module
+ * \returns the evaluation of the polynomial at x using Horner algorithm.
+ *
+ * \param[in] poly : the vector of coefficients of the polynomial ordered
+ *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
+ *  e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
+ * \param[in] x : the value to evaluate the polynomial at.
+ *
+ * <i><b>Note for stability:</b></i>
+ *  <dd> \f$ |x| \le 1 \f$ </dd>
+ */
+template <typename Polynomials, typename T>
+inline
+T poly_eval_horner( const Polynomials& poly, const T& x )
+{
+  T val=poly[poly.size()-1];
+  for( int i=poly.size()-2; i>=0; --i ){
+    val = val*x + poly[i]; }
+  return val;
+}
+
+/** \ingroup Polynomials_Module
+ * \returns the evaluation of the polynomial at x using stabilized Horner algorithm.
+ *
+ * \param[in] poly : the vector of coefficients of the polynomial ordered
+ *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
+ *  e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
+ * \param[in] x : the value to evaluate the polynomial at.
+ */
+template <typename Polynomials, typename T>
+inline
+T poly_eval( const Polynomials& poly, const T& x )
+{
+  typedef typename NumTraits<T>::Real Real;
+
+  if( ei_abs2( x ) <= Real(1) ){
+    return poly_eval_horner( poly, x ); }
+  else
+  {
+    T val=poly[0];
+    T inv_x = T(1)/x;
+    for( int i=1; i<poly.size(); ++i ){
+      val = val*inv_x + poly[i]; }
+
+    return std::pow(x,(T)(poly.size()-1)) * val;
+  }
+}
+
+/** \ingroup Polynomials_Module
+ * \returns a maximum bound for the absolute value of any root of the polynomial.
+ *
+ * \param[in] poly : the vector of coefficients of the polynomial ordered
+ *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
+ *  e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
+ *
+ *  <i><b>Precondition:</b></i>
+ *  <dd> the leading coefficient of the input polynomial poly must be non zero </dd>
+ */
+template <typename Polynomial>
+inline
+typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly )
+{
+  typedef typename Polynomial::Scalar Scalar;
+  typedef typename NumTraits<Scalar>::Real Real;
+
+  assert( Scalar(0) != poly[poly.size()-1] );
+  const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1];
+  Real cb(0);
+
+  for( int i=0; i<poly.size()-1; ++i ){
+    cb += ei_abs(poly[i]*inv_leading_coeff); }
+  return cb + Real(1);
+}
+
+/** \ingroup Polynomials_Module
+ * \returns a minimum bound for the absolute value of any non zero root of the polynomial.
+ * \param[in] poly : the vector of coefficients of the polynomial ordered
+ *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
+ *  e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
+ */
+template <typename Polynomial>
+inline
+typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly )
+{
+  typedef typename Polynomial::Scalar Scalar;
+  typedef typename NumTraits<Scalar>::Real Real;
+
+  int i=0;
+  while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; }
+  if( poly.size()-1 == i ){
+    return Real(1); }
+
+  const Scalar inv_min_coeff = Scalar(1)/poly[i];
+  Real cb(1);
+  for( int j=i+1; j<poly.size(); ++j ){
+    cb += ei_abs(poly[j]*inv_min_coeff); }
+  return Real(1)/cb;
+}
+
+/** \ingroup Polynomials_Module
+ * Given the roots of a polynomial compute the coefficients in the
+ * monomial basis of the monic polynomial with same roots and minimal degree.
+ * If RootVector is a vector of complexes, Polynomial should also be a vector
+ * of complexes.
+ * \param[in] rv : a vector containing the roots of a polynomial.
+ * \param[out] poly : the vector of coefficients of the polynomial ordered
+ *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
+ *  e.g. \f$ 3 + x^2 \f$ is stored as a vector \f$ [ 3, 0, 1 ] \f$.
+ */
+template <typename RootVector, typename Polynomial>
+void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
+{
+
+  typedef typename Polynomial::Scalar Scalar;
+
+  poly.setZero( rv.size()+1 );
+  poly[0] = -rv[0]; poly[1] = Scalar(1);
+  for( int i=1; i<(int)rv.size(); ++i )
+  {
+    for( int j=i+1; j>0; --j ){ poly[j] = poly[j-1] - rv[i]*poly[j]; }
+    poly[0] = -rv[i]*poly[0];
+  }
+}
+
+
+#endif // EIGEN_POLYNOMIAL_UTILS_H