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