A polynomial solver. More...

#include <PolynomialSolver.h>

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

Public Types
typedef PolynomialSolverBase< Scalar_, Deg_ >	PS_Base

typedef Matrix< Scalar, Deg_, Deg_ >	CompanionMatrixType

typedef std::conditional_t< NumTraits< Scalar >::IsComplex, ComplexEigenSolver< CompanionMatrixType >, EigenSolver< CompanionMatrixType > >	EigenSolverType

typedef std::conditional_t< NumTraits< Scalar >::IsComplex, Scalar, std::complex< Scalar > >	ComplexScalar

Public Types inherited from Eigen::PolynomialSolverBase< Scalar_, Deg_ >
typedef Scalar_	Scalar

typedef NumTraits< Scalar >::Real	RealScalar

typedef std::complex< RealScalar >	RootType

typedef Matrix< RootType, Deg_, 1 >	RootsType

typedef DenseIndex	Index

Public Member Functions
template<typename OtherPolynomial >
void	compute (const OtherPolynomial &poly)

template<typename OtherPolynomial >
	PolynomialSolver (const OtherPolynomial &poly)

	PolynomialSolver ()

Public Member Functions inherited from Eigen::PolynomialSolverBase< Scalar_, Deg_ >
template<typename OtherPolynomial >
	PolynomialSolverBase (const OtherPolynomial &poly)

	PolynomialSolverBase ()

const RootsType &	roots () const

template<typename Stl_back_insertion_sequence >
void	realRoots (Stl_back_insertion_sequence &bi_seq, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) const

const RootType &	greatestRoot () const

const RootType &	smallestRoot () const

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 RealScalar &	smallestRealRoot (bool &hasArealRoot, const RealScalar &absImaginaryThreshold=NumTraits< Scalar >::dummy_precision()) 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 OtherPolynomial >
void	setPolynomial (const OtherPolynomial &poly)

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

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).

Member Typedef Documentation

◆ CompanionMatrixType

template<typename Scalar_ , int Deg_>

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

◆ ComplexScalar

template<typename Scalar_ , int Deg_>

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

◆ EigenSolverType

template<typename Scalar_ , int Deg_>

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

◆ PS_Base

template<typename Scalar_ , int Deg_>

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

Constructor & Destructor Documentation

◆ PolynomialSolver() [1/2]

template<typename Scalar_ , int Deg_>

template<typename OtherPolynomial >

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

inline

                                                        {
     compute(poly);
   }

References Eigen::PolynomialSolver< Scalar_, Deg_ >::compute().

◆ PolynomialSolver() [2/2]

template<typename Scalar_ , int Deg_>

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

inline

351 {}

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.

                                             {
     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());
       eigen_assert(m_eigenSolver.info() == Eigen::Success);
       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];
     }
   }