Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 1 | // This file is part of Eigen, a lightweight C++ template library |
| 2 | // for linear algebra. |
| 3 | // |
| 4 | // Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com> |
| 5 | // |
| 6 | // This Source Code Form is subject to the terms of the Mozilla |
| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed |
| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| 9 | |
| 10 | #ifndef EIGEN_POLYNOMIAL_UTILS_H |
| 11 | #define EIGEN_POLYNOMIAL_UTILS_H |
| 12 | |
| 13 | namespace Eigen { |
| 14 | |
| 15 | /** \ingroup Polynomials_Module |
| 16 | * \returns the evaluation of the polynomial at x using Horner algorithm. |
| 17 | * |
| 18 | * \param[in] poly : the vector of coefficients of the polynomial ordered |
| 19 | * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| 20 | * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| 21 | * \param[in] x : the value to evaluate the polynomial at. |
| 22 | * |
| 23 | * <i><b>Note for stability:</b></i> |
| 24 | * <dd> \f$ |x| \le 1 \f$ </dd> |
| 25 | */ |
| 26 | template <typename Polynomials, typename T> |
| 27 | inline |
| 28 | T poly_eval_horner( const Polynomials& poly, const T& x ) |
| 29 | { |
| 30 | T val=poly[poly.size()-1]; |
| 31 | for(DenseIndex i=poly.size()-2; i>=0; --i ){ |
| 32 | val = val*x + poly[i]; } |
| 33 | return val; |
| 34 | } |
| 35 | |
| 36 | /** \ingroup Polynomials_Module |
| 37 | * \returns the evaluation of the polynomial at x using stabilized Horner algorithm. |
| 38 | * |
| 39 | * \param[in] poly : the vector of coefficients of the polynomial ordered |
| 40 | * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| 41 | * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| 42 | * \param[in] x : the value to evaluate the polynomial at. |
| 43 | */ |
| 44 | template <typename Polynomials, typename T> |
| 45 | inline |
| 46 | T poly_eval( const Polynomials& poly, const T& x ) |
| 47 | { |
| 48 | typedef typename NumTraits<T>::Real Real; |
| 49 | |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 50 | if( numext::abs2( x ) <= Real(1) ){ |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 51 | return poly_eval_horner( poly, x ); } |
| 52 | else |
| 53 | { |
| 54 | T val=poly[0]; |
| 55 | T inv_x = T(1)/x; |
| 56 | for( DenseIndex i=1; i<poly.size(); ++i ){ |
| 57 | val = val*inv_x + poly[i]; } |
| 58 | |
| 59 | return std::pow(x,(T)(poly.size()-1)) * val; |
| 60 | } |
| 61 | } |
| 62 | |
| 63 | /** \ingroup Polynomials_Module |
| 64 | * \returns a maximum bound for the absolute value of any root of the polynomial. |
| 65 | * |
| 66 | * \param[in] poly : the vector of coefficients of the polynomial ordered |
| 67 | * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| 68 | * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| 69 | * |
| 70 | * <i><b>Precondition:</b></i> |
| 71 | * <dd> the leading coefficient of the input polynomial poly must be non zero </dd> |
| 72 | */ |
| 73 | template <typename Polynomial> |
| 74 | inline |
| 75 | typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly ) |
| 76 | { |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 77 | using std::abs; |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 78 | typedef typename Polynomial::Scalar Scalar; |
| 79 | typedef typename NumTraits<Scalar>::Real Real; |
| 80 | |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 81 | eigen_assert( Scalar(0) != poly[poly.size()-1] ); |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 82 | const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1]; |
| 83 | Real cb(0); |
| 84 | |
| 85 | for( DenseIndex i=0; i<poly.size()-1; ++i ){ |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 86 | cb += abs(poly[i]*inv_leading_coeff); } |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 87 | return cb + Real(1); |
| 88 | } |
| 89 | |
| 90 | /** \ingroup Polynomials_Module |
| 91 | * \returns a minimum bound for the absolute value of any non zero root of the polynomial. |
| 92 | * \param[in] poly : the vector of coefficients of the polynomial ordered |
| 93 | * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| 94 | * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| 95 | */ |
| 96 | template <typename Polynomial> |
| 97 | inline |
| 98 | typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly ) |
| 99 | { |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 100 | using std::abs; |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 101 | typedef typename Polynomial::Scalar Scalar; |
| 102 | typedef typename NumTraits<Scalar>::Real Real; |
| 103 | |
| 104 | DenseIndex i=0; |
| 105 | while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; } |
| 106 | if( poly.size()-1 == i ){ |
| 107 | return Real(1); } |
| 108 | |
| 109 | const Scalar inv_min_coeff = Scalar(1)/poly[i]; |
| 110 | Real cb(1); |
| 111 | for( DenseIndex j=i+1; j<poly.size(); ++j ){ |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 112 | cb += abs(poly[j]*inv_min_coeff); } |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 113 | return Real(1)/cb; |
| 114 | } |
| 115 | |
| 116 | /** \ingroup Polynomials_Module |
| 117 | * Given the roots of a polynomial compute the coefficients in the |
| 118 | * monomial basis of the monic polynomial with same roots and minimal degree. |
| 119 | * If RootVector is a vector of complexes, Polynomial should also be a vector |
| 120 | * of complexes. |
| 121 | * \param[in] rv : a vector containing the roots of a polynomial. |
| 122 | * \param[out] poly : the vector of coefficients of the polynomial ordered |
| 123 | * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| 124 | * e.g. \f$ 3 + x^2 \f$ is stored as a vector \f$ [ 3, 0, 1 ] \f$. |
| 125 | */ |
| 126 | template <typename RootVector, typename Polynomial> |
| 127 | void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly ) |
| 128 | { |
| 129 | |
| 130 | typedef typename Polynomial::Scalar Scalar; |
| 131 | |
| 132 | poly.setZero( rv.size()+1 ); |
| 133 | poly[0] = -rv[0]; poly[1] = Scalar(1); |
| 134 | for( DenseIndex i=1; i< rv.size(); ++i ) |
| 135 | { |
| 136 | for( DenseIndex j=i+1; j>0; --j ){ poly[j] = poly[j-1] - rv[i]*poly[j]; } |
| 137 | poly[0] = -rv[i]*poly[0]; |
| 138 | } |
| 139 | } |
| 140 | |
| 141 | } // end namespace Eigen |
| 142 | |
| 143 | #endif // EIGEN_POLYNOMIAL_UTILS_H |