1c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This file is part of Eigen, a lightweight C++ template library 2c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// for linear algebra. 3c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 4c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com> 5c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 6c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This Source Code Form is subject to the terms of the Mozilla 7c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Public License v. 2.0. If a copy of the MPL was not distributed 8c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 10c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#ifndef EIGEN_POLYNOMIAL_UTILS_H 11c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#define EIGEN_POLYNOMIAL_UTILS_H 12c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 13c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathnamespace Eigen { 14c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 15c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup Polynomials_Module 16c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns the evaluation of the polynomial at x using Horner algorithm. 17c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 18c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] poly : the vector of coefficients of the polynomial ordered 19c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial 20c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. 21c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] x : the value to evaluate the polynomial at. 22c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 23c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * <i><b>Note for stability:</b></i> 24c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * <dd> \f$ |x| \le 1 \f$ </dd> 25c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 26c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename Polynomials, typename T> 27c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline 28c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathT poly_eval_horner( const Polynomials& poly, const T& x ) 29c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 30c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath T val=poly[poly.size()-1]; 31c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(DenseIndex i=poly.size()-2; i>=0; --i ){ 32c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath val = val*x + poly[i]; } 33c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return val; 34c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 35c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 36c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup Polynomials_Module 37c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns the evaluation of the polynomial at x using stabilized Horner algorithm. 38c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 39c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] poly : the vector of coefficients of the polynomial ordered 40c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial 41c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. 42c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] x : the value to evaluate the polynomial at. 43c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 44c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename Polynomials, typename T> 45c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline 46c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathT poly_eval( const Polynomials& poly, const T& x ) 47c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 48c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename NumTraits<T>::Real Real; 49c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if( numext::abs2( x ) <= Real(1) ){ 51c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return poly_eval_horner( poly, x ); } 52c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else 53c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 54c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath T val=poly[0]; 55c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath T inv_x = T(1)/x; 56c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for( DenseIndex i=1; i<poly.size(); ++i ){ 57c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath val = val*inv_x + poly[i]; } 58c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 59c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return std::pow(x,(T)(poly.size()-1)) * val; 60c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 61c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 62c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 63c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup Polynomials_Module 64c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns a maximum bound for the absolute value of any root of the polynomial. 65c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 66c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] poly : the vector of coefficients of the polynomial ordered 67c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial 68c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. 69c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 70c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * <i><b>Precondition:</b></i> 71c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * <dd> the leading coefficient of the input polynomial poly must be non zero </dd> 72c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 73c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename Polynomial> 74c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline 75c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtypename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly ) 76c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 78c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Polynomial::Scalar Scalar; 79c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename NumTraits<Scalar>::Real Real; 80c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert( Scalar(0) != poly[poly.size()-1] ); 82c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1]; 83c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Real cb(0); 84c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 85c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for( DenseIndex i=0; i<poly.size()-1; ++i ){ 867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez cb += abs(poly[i]*inv_leading_coeff); } 87c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return cb + Real(1); 88c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 89c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 90c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup Polynomials_Module 91c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns a minimum bound for the absolute value of any non zero root of the polynomial. 92c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] poly : the vector of coefficients of the polynomial ordered 93c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial 94c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. 95c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 96c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename Polynomial> 97c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline 98c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtypename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly ) 99c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 1007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 101c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Polynomial::Scalar Scalar; 102c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename NumTraits<Scalar>::Real Real; 103c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 104c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath DenseIndex i=0; 105c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; } 106c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if( poly.size()-1 == i ){ 107c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return Real(1); } 108c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 109c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Scalar inv_min_coeff = Scalar(1)/poly[i]; 110c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Real cb(1); 111c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for( DenseIndex j=i+1; j<poly.size(); ++j ){ 1127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez cb += abs(poly[j]*inv_min_coeff); } 113c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return Real(1)/cb; 114c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 115c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 116c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup Polynomials_Module 117c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Given the roots of a polynomial compute the coefficients in the 118c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * monomial basis of the monic polynomial with same roots and minimal degree. 119c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * If RootVector is a vector of complexes, Polynomial should also be a vector 120c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * of complexes. 121c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] rv : a vector containing the roots of a polynomial. 122c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[out] poly : the vector of coefficients of the polynomial ordered 123c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial 124c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * e.g. \f$ 3 + x^2 \f$ is stored as a vector \f$ [ 3, 0, 1 ] \f$. 125c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 126c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename RootVector, typename Polynomial> 127c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly ) 128c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 129c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 130c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Polynomial::Scalar Scalar; 131c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 132c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath poly.setZero( rv.size()+1 ); 133c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath poly[0] = -rv[0]; poly[1] = Scalar(1); 134c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for( DenseIndex i=1; i< rv.size(); ++i ) 135c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 136c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for( DenseIndex j=i+1; j>0; --j ){ poly[j] = poly[j-1] - rv[i]*poly[j]; } 137c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath poly[0] = -rv[i]*poly[0]; 138c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 139c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 140c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 141c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} // end namespace Eigen 142c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 143c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#endif // EIGEN_POLYNOMIAL_UTILS_H 144