Eigen::PolynomialSolver< Scalar_, Deg_ > Class Template Reference

A polynomial solver. More...

Inheritance diagram for Eigen::PolynomialSolver< Scalar_, Deg_ >:

Public Types
typedef Matrix< Scalar, Deg_, Deg_ >	CompanionMatrixType
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, Scalar, std::complex< Scalar > >	ComplexScalar
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, ComplexEigenSolver< CompanionMatrixType >, EigenSolver< CompanionMatrixType > >	EigenSolverType
typedef PolynomialSolverBase< Scalar_, Deg_ >	PS_Base
Public Types inherited from Eigen::PolynomialSolverBase< Scalar_, Deg_ >
typedef DenseIndex	Index
typedef NumTraits< Scalar >::Real	RealScalar
typedef Matrix< RootType, Deg_, 1 >	RootsType
typedef std::complex< RealScalar >	RootType
typedef Scalar_	Scalar

Public Member Functions
template<typename OtherPolynomial >
void	compute (const OtherPolynomial &poly)
	PolynomialSolver ()
template<typename OtherPolynomial >
	PolynomialSolver (const OtherPolynomial &poly)
Public Member Functions inherited from Eigen::PolynomialSolverBase< Scalar_, Deg_ >
const RealScalar &	absGreatestRealRoot (bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
const RealScalar &	absSmallestRealRoot (bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
const RealScalar &	greatestRealRoot (bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
const RootType &	greatestRoot () const
	PolynomialSolverBase ()
template<typename OtherPolynomial >
	PolynomialSolverBase (const OtherPolynomial &poly)
template<typename Stl_back_insertion_sequence >
void	realRoots (Stl_back_insertion_sequence &bi_seq, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
const RootsType &	roots () const
const RealScalar &	smallestRealRoot (bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
const RootType &	smallestRoot () const

Protected Attributes
EigenSolverType	m_eigenSolver
RootsType	m_roots
Protected Attributes inherited from Eigen::PolynomialSolverBase< Scalar_, Deg_ >
RootsType	m_roots

Additional Inherited Members
Protected Member Functions inherited from Eigen::PolynomialSolverBase< Scalar_, Deg_ >
template<typename squaredNormBinaryPredicate >
const RootType &	selectComplexRoot_withRespectToNorm (squaredNormBinaryPredicate &pred) const
template<typename squaredRealPartBinaryPredicate >
const RealScalar &	selectRealRoot_withRespectToAbsRealPart (squaredRealPartBinaryPredicate &pred, bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
template<typename RealPartBinaryPredicate >
const RealScalar &	selectRealRoot_withRespectToRealPart (RealPartBinaryPredicate &pred, bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const
template<typename OtherPolynomial >
void	setPolynomial (const OtherPolynomial &poly)

Detailed Description

template<typename Scalar_, int Deg_>
class Eigen::PolynomialSolver< Scalar_, Deg_ >

A polynomial solver.

Computes the complex roots of a real polynomial.

Parameters

Scalar_	the scalar type, i.e., the type of the polynomial coefficients
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 unsupported 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).

Definition at line 333 of file PolynomialSolver.h.

Member Typedef Documentation

CompanionMatrixType

template<typename Scalar_ , int Deg_>

typedef Matrix<Scalar,Deg_,Deg_> Eigen::PolynomialSolver< Scalar_, Deg_ >::CompanionMatrixType

Definition at line 341 of file PolynomialSolver.h.

ComplexScalar

template<typename Scalar_ , int Deg_>

typedef std::conditional_t<NumTraits<Scalar>::IsComplex, Scalar, std::complex<Scalar> > Eigen::PolynomialSolver< Scalar_, Deg_ >::ComplexScalar

Definition at line 345 of file PolynomialSolver.h.

EigenSolverType

template<typename Scalar_ , int Deg_>

typedef std::conditional_t<NumTraits<Scalar>::IsComplex, ComplexEigenSolver<CompanionMatrixType>, EigenSolver<CompanionMatrixType> > Eigen::PolynomialSolver< Scalar_, Deg_ >::EigenSolverType

Definition at line 344 of file PolynomialSolver.h.

PS_Base

template<typename Scalar_ , int Deg_>

typedef PolynomialSolverBase<Scalar_,Deg_> Eigen::PolynomialSolver< Scalar_, Deg_ >::PS_Base

Definition at line 338 of file PolynomialSolver.h.

Constructor & Destructor Documentation

PolynomialSolver() [1/2]

template<typename Scalar_ , int Deg_>

template<typename OtherPolynomial >

Eigen::PolynomialSolver< Scalar_, Deg_ >::PolynomialSolver ( const OtherPolynomial &  poly )

inline

Definition at line 389 of file PolynomialSolver.h.

                                                          {
      compute( poly ); }

PolynomialSolver() [2/2]

template<typename Scalar_ , int Deg_>

Eigen::PolynomialSolver< Scalar_, Deg_ >::PolynomialSolver ( )

inline

Definition at line 392 of file PolynomialSolver.h.

{}

Member Function Documentation

compute()

template<typename Scalar_ , int Deg_>

template<typename OtherPolynomial >

void Eigen::PolynomialSolver< Scalar_, Deg_ >::compute ( const OtherPolynomial &  poly )

inline

Computes the complex roots of a new polynomial.

Definition at line 350 of file PolynomialSolver.h.

    {
      eigen_assert( Scalar(0) != poly[poly.size()-1] );
      eigen_assert( poly.size() > 1 );
      if(poly.size() >  2 )
      {
        internal::companion<Scalar,Deg_> companion( poly );
        companion.balance();
        m_eigenSolver.compute( companion.denseMatrix() );
        m_roots = m_eigenSolver.eigenvalues();
        // cleanup noise in imaginary part of real roots:
        // if the imaginary part is rather small compared to the real part
        // and that cancelling the imaginary part yield a smaller evaluation,
        // then it's safe to keep the real part only.
        RealScalar coarse_prec = RealScalar(std::pow(4,poly.size()+1))*NumTraits<RealScalar>::epsilon();
        for(Index i = 0; i<m_roots.size(); ++i)
        {
          if( internal::isMuchSmallerThan(numext::abs(numext::imag(m_roots[i])),
                                          numext::abs(numext::real(m_roots[i])),
                                          coarse_prec) )
          {
            ComplexScalar as_real_root = ComplexScalar(numext::real(m_roots[i]));
            if(    numext::abs(poly_eval(poly, as_real_root))
                <= numext::abs(poly_eval(poly, m_roots[i])))
            {
              m_roots[i] = as_real_root;
            }
          }
        }
      }
      else if(poly.size () == 2)
      {
        m_roots.resize(1);
        m_roots[0] = -poly[0]/poly[1];
      }
    }

Member Data Documentation

m_eigenSolver

template<typename Scalar_ , int Deg_>

EigenSolverType Eigen::PolynomialSolver< Scalar_, Deg_ >::m_eigenSolver

protected

Definition at line 396 of file PolynomialSolver.h.

m_roots

template<typename Scalar_ , int Deg_>

RootsType Eigen::PolynomialSolverBase< Scalar_, Deg_ >::m_roots

protected

Definition at line 292 of file PolynomialSolver.h.

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The documentation for this class was generated from the following file:

• PolynomialSolver.h