[136] | 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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