| // This file is part of Eigen, a lightweight C++ template library | |
| // for linear algebra. | |
| // | |
| // Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com> | |
| // | |
| // This Source Code Form is subject to the terms of the Mozilla | |
| // Public License v. 2.0. If a copy of the MPL was not distributed | |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. | |
| namespace Eigen { | |
| /** \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$. | |
| * \param[in] x : the value to evaluate the polynomial at. | |
| * | |
| * \note for stability: | |
| * \f$ |x| \le 1 \f$ | |
| */ | |
| template <typename Polynomials, typename T> | |
| inline | |
| T poly_eval_horner( const Polynomials& poly, const T& x ) | |
| { | |
| T val=poly[poly.size()-1]; | |
| for(DenseIndex 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$. | |
| * \param[in] x : the value to evaluate the polynomial at. | |
| */ | |
| template <typename Polynomials, typename T> | |
| inline | |
| T poly_eval( const Polynomials& poly, const T& x ) | |
| { | |
| typedef typename NumTraits<T>::Real Real; | |
| if( numext::abs2( x ) <= Real(1) ){ | |
| return poly_eval_horner( poly, x ); } | |
| else | |
| { | |
| T val=poly[0]; | |
| T inv_x = T(1)/x; | |
| for( DenseIndex i=1; i<poly.size(); ++i ){ | |
| val = val*inv_x + poly[i]; } | |
| return numext::pow(x,(T)(poly.size()-1)) * val; | |
| } | |
| } | |
| /** \ingroup Polynomials_Module | |
| * \returns a maximum bound for the absolute value of any root of the polynomial. | |
| * | |
| * \param[in] poly : the vector of coefficients of the polynomial ordered | |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial | |
| * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. | |
| * | |
| * \pre | |
| * the leading coefficient of the input polynomial poly must be non zero | |
| */ | |
| template <typename Polynomial> | |
| inline | |
| typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly ) | |
| { | |
| using std::abs; | |
| typedef typename Polynomial::Scalar Scalar; | |
| typedef typename NumTraits<Scalar>::Real Real; | |
| eigen_assert( Scalar(0) != poly[poly.size()-1] ); | |
| const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1]; | |
| Real cb(0); | |
| for( DenseIndex i=0; i<poly.size()-1; ++i ){ | |
| cb += abs(poly[i]*inv_leading_coeff); } | |
| return cb + Real(1); | |
| } | |
| /** \ingroup Polynomials_Module | |
| * \returns a minimum bound for the absolute value of any non zero root of the polynomial. | |
| * \param[in] poly : the vector of coefficients of the polynomial ordered | |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial | |
| * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. | |
| */ | |
| template <typename Polynomial> | |
| inline | |
| typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly ) | |
| { | |
| using std::abs; | |
| typedef typename Polynomial::Scalar Scalar; | |
| typedef typename NumTraits<Scalar>::Real Real; | |
| DenseIndex i=0; | |
| while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; } | |
| if( poly.size()-1 == i ){ | |
| return Real(1); } | |
| const Scalar inv_min_coeff = Scalar(1)/poly[i]; | |
| Real cb(1); | |
| for( DenseIndex j=i+1; j<poly.size(); ++j ){ | |
| cb += abs(poly[j]*inv_min_coeff); } | |
| return Real(1)/cb; | |
| } | |
| /** \ingroup Polynomials_Module | |
| * Given the roots of a polynomial compute the coefficients in the | |
| * monomial basis of the monic polynomial with same roots and minimal degree. | |
| * If RootVector is a vector of complexes, Polynomial should also be a vector | |
| * of complexes. | |
| * \param[in] rv : a vector containing the roots of a polynomial. | |
| * \param[out] poly : the vector of coefficients of the polynomial ordered | |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial | |
| * e.g. \f$ 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( DenseIndex i=1; i< rv.size(); ++i ) | |
| { | |
| for( DenseIndex j=i+1; j>0; --j ){ poly[j] = poly[j-1] - rv[i]*poly[j]; } | |
| poly[0] = -rv[i]*poly[0]; | |
| } | |
| } | |
| } // end namespace Eigen | |