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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
This commit is contained in:
Manuel Yguel
395
unsupported/Eigen/src/Polynomials/PolynomialSolver.h
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395
unsupported/Eigen/src/Polynomials/PolynomialSolver.h
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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
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//
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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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//
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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//
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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//
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_POLYNOMIAL_SOLVER_H
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#define EIGEN_POLYNOMIAL_SOLVER_H
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/** \ingroup Polynomials_Module
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* \class PolynomialSolverBase.
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*
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* \brief Defined to be inherited by polynomial solvers: it provides
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* convenient methods such as
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* - real roots,
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* - greatest, smallest complex roots,
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* - real roots with greatest, smallest absolute real value,
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* - greatest, smallest real roots.
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*
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* It stores the set of roots as a vector of complexes.
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*
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*/
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template< typename _Scalar, int _Deg >
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class PolynomialSolverBase
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{
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public:
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EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
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typedef _Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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typedef std::complex<RealScalar> RootType;
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typedef Matrix<RootType,_Deg,1> RootsType;
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protected:
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template< typename OtherPolynomial >
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inline void setPolynomial( const OtherPolynomial& poly ){
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m_roots.resize(poly.size()); }
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public:
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template< typename OtherPolynomial >
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inline PolynomialSolverBase( const OtherPolynomial& poly ){
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setPolynomial( poly() ); }
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inline PolynomialSolverBase(){}
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public:
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/** \returns the complex roots of the polynomial */
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inline const RootsType& roots() const { return m_roots; }
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public:
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/** Clear and fills the back insertion sequence with the real roots of the polynomial
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* i.e. the real part of the complex roots that have an imaginary part which
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* absolute value is smaller than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the _Scalar template parameter of the PolynomialSolver class as the default value.
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*
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* \param[out] bi_seq : the back insertion sequence (stl concept)
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* \param[in] absImaginaryThreshold : the maximum bound of the imaginary part of a complex
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* number that is considered as real.
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* */
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template<typename Stl_back_insertion_sequence>
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inline void realRoots( Stl_back_insertion_sequence& bi_seq,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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bi_seq.clear();
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for( int i=0; i<m_roots.size(); ++i )
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{
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if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold ){
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bi_seq.push_back( m_roots[i].real() ); }
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}
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}
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protected:
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template<typename squaredNormBinaryPredicate>
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inline const RootType& selectComplexRoot_withRespectToNorm( squaredNormBinaryPredicate& pred ) const
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{
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int res=0;
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RealScalar norm2 = ei_abs2( m_roots[0] );
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for( int i=1; i<m_roots.size(); ++i )
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{
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const RealScalar currNorm2 = ei_abs2( m_roots[i] );
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if( pred( currNorm2, norm2 ) ){
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res=i; norm2=currNorm2; }
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}
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return m_roots[res];
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}
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public:
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/**
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* \returns the complex root with greatest norm.
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*/
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inline const RootType& greatestRoot() const
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{
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std::greater<Scalar> greater;
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return selectComplexRoot_withRespectToNorm( greater );
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}
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/**
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* \returns the complex root with smallest norm.
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*/
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inline const RootType& smallestRoot() const
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{
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std::less<Scalar> less;
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return selectComplexRoot_withRespectToNorm( less );
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}
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protected:
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template<typename squaredRealPartBinaryPredicate>
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inline const RealScalar& selectRealRoot_withRespectToAbsRealPart(
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squaredRealPartBinaryPredicate& pred,
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bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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hasArealRoot = false;
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int res=0;
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RealScalar abs2;
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for( int i=0; i<m_roots.size(); ++i )
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{
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if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold )
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{
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if( !hasArealRoot )
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{
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hasArealRoot = true;
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res = i;
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abs2 = m_roots[i].real() * m_roots[i].real();
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}
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else
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{
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const RealScalar currAbs2 = m_roots[i].real() * m_roots[i].real();
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if( pred( currAbs2, abs2 ) )
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{
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abs2 = currAbs2;
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res = i;
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}
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}
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}
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else
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{
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if( ei_abs( m_roots[i].imag() ) < ei_abs( m_roots[res].imag() ) ){
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res = i; }
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}
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}
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return m_roots[res].real();
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}
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template<typename RealPartBinaryPredicate>
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inline const RealScalar& selectRealRoot_withRespectToRealPart(
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RealPartBinaryPredicate& pred,
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bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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hasArealRoot = false;
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int res=0;
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RealScalar val;
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for( int i=0; i<m_roots.size(); ++i )
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{
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if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold )
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{
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if( !hasArealRoot )
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{
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hasArealRoot = true;
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res = i;
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val = m_roots[i].real();
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}
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else
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{
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const RealScalar curr = m_roots[i].real();
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if( pred( curr, val ) )
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{
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val = curr;
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res = i;
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}
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}
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}
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else
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{
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if( ei_abs( m_roots[i].imag() ) < ei_abs( m_roots[res].imag() ) ){
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res = i; }
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}
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}
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return m_roots[res].real();
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}
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public:
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/**
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* \returns a real root with greatest absolute magnitude.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the _Scalar template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& absGreatestRealRoot(
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bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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std::greater<Scalar> greater;
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return selectRealRoot_withRespectToAbsRealPart( greater, hasArealRoot, absImaginaryThreshold );
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}
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/**
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* \returns a real root with smallest absolute magnitude.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the _Scalar template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& absSmallestRealRoot(
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bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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std::less<Scalar> less;
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return selectRealRoot_withRespectToAbsRealPart( less, hasArealRoot, absImaginaryThreshold );
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}
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/**
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* \returns the real root with greatest value.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the _Scalar template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& greatestRealRoot(
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bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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std::greater<Scalar> greater;
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return selectRealRoot_withRespectToRealPart( greater, hasArealRoot, absImaginaryThreshold );
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}
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/**
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* \returns the real root with smallest value.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the _Scalar template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& smallestRealRoot(
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bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
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{
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std::less<Scalar> less;
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return selectRealRoot_withRespectToRealPart( less, hasArealRoot, absImaginaryThreshold );
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}
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protected:
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RootsType m_roots;
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};
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#define EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( BASE ) \
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typedef typename BASE::Scalar Scalar; \
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typedef typename BASE::RealScalar RealScalar; \
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typedef typename BASE::RootType RootType; \
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typedef typename BASE::RootsType RootsType;
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/** \ingroup Polynomials_Module
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*
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* \class PolynomialSolver
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*
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* \brief A polynomial solver
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*
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* Computes the complex roots of a real polynomial.
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*
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* \param _Scalar the scalar type, i.e., the type of the polynomial coefficients
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* \param _Deg the degree of the polynomial, can be a compile time value or Dynamic.
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* Notice that the number of polynomial coefficients is _Deg+1.
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*
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* This class implements a polynomial solver and provides convenient methods such as
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* - real roots,
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* - greatest, smallest complex roots,
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* - real roots with greatest, smallest absolute real value.
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* - greatest, smallest real roots.
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*
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* WARNING: this polynomial solver is experimental, part of the unsuported Eigen modules.
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*
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*
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* Currently a QR algorithm is used to compute the eigenvalues of the companion matrix of
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* the polynomial to compute its roots.
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* This supposes that the complex moduli of the roots are all distinct: e.g. there should
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* be no multiple roots or conjugate roots for instance.
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* With 32bit (float) floating types this problem shows up frequently.
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* However, almost always, correct accuracy is reached even in these cases for 64bit
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* (double) floating types and small polynomial degree (<20).
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*/
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template< typename _Scalar, int _Deg >
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class PolynomialSolver : public PolynomialSolverBase<_Scalar,_Deg>
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{
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public:
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EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
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typedef PolynomialSolverBase<_Scalar,_Deg> PS_Base;
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EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( PS_Base )
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typedef Matrix<Scalar,_Deg,_Deg> CompanionMatrixType;
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typedef EigenSolver<CompanionMatrixType> EigenSolverType;
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public:
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/** Computes the complex roots of a new polynomial. */
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template< typename OtherPolynomial >
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void compute( const OtherPolynomial& poly )
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{
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assert( Scalar(0) != poly[poly.size()-1] );
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ei_companion<Scalar,_Deg> companion( poly );
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companion.balance();
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m_eigenSolver.compute( companion.denseMatrix() );
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m_roots = m_eigenSolver.eigenvalues();
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}
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public:
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template< typename OtherPolynomial >
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inline PolynomialSolver( const OtherPolynomial& poly ){
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compute( poly ); }
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inline PolynomialSolver(){}
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protected:
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using PS_Base::m_roots;
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EigenSolverType m_eigenSolver;
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};
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template< typename _Scalar >
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class PolynomialSolver<_Scalar,1> : public PolynomialSolverBase<_Scalar,1>
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{
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public:
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typedef PolynomialSolverBase<_Scalar,1> PS_Base;
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EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( PS_Base )
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public:
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/** Computes the complex roots of a new polynomial. */
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template< typename OtherPolynomial >
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void compute( const OtherPolynomial& poly )
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{
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assert( Scalar(0) != poly[poly.size()-1] );
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m_roots[0] = -poly[0]/poly[poly.size()-1];
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}
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protected:
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using PS_Base::m_roots;
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};
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#endif // EIGEN_POLYNOMIAL_SOLVER_H
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