| // 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/. |
| |
| #ifndef EIGEN_POLYNOMIAL_UTILS_H |
| #define EIGEN_POLYNOMIAL_UTILS_H |
| |
| #include "./InternalHeaderCheck.h" |
| |
| 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 |
| |
| #endif // EIGEN_POLYNOMIAL_UTILS_H |