PolynomialUtils.h revision c981c48f5bc9aefeffc0bcb0cc3934c2fae179dd 
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
13namespace 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 */
26template <typename Polynomials, typename T>
27inline
28T 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$.
42 * \param[in] x : the value to evaluate the polynomial at.
43 */
44template <typename Polynomials, typename T>
45inline
46T poly_eval( const Polynomials& poly, const T& x )
47{
48  typedef typename NumTraits<T>::Real Real;
49
50  if( internal::abs2( x ) <= Real(1) ){
51    return poly_eval_horner( poly, x ); }
52  else
53  {
54    T val=poly[0];
55    T inv_x = T(1)/x;
56    for( DenseIndex i=1; i<poly.size(); ++i ){
57      val = val*inv_x + poly[i]; }
58
59    return std::pow(x,(T)(poly.size()-1)) * val;
60  }
61}
62
63/** \ingroup Polynomials_Module
64 * \returns a maximum bound for the absolute value of any root of the polynomial.
65 *
66 * \param[in] poly : the vector of coefficients of the polynomial ordered
67 *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
68 *  e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
69 *
70 *  <i><b>Precondition:</b></i>
71 *  <dd> the leading coefficient of the input polynomial poly must be non zero </dd>
72 */
73template <typename Polynomial>
74inline
75typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly )
76{
77  typedef typename Polynomial::Scalar Scalar;
78  typedef typename NumTraits<Scalar>::Real Real;
79
80  assert( Scalar(0) != poly[poly.size()-1] );
81  const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1];
82  Real cb(0);
83
84  for( DenseIndex i=0; i<poly.size()-1; ++i ){
85    cb += internal::abs(poly[i]*inv_leading_coeff); }
86  return cb + Real(1);
87}
88
89/** \ingroup Polynomials_Module
90 * \returns a minimum bound for the absolute value of any non zero root of the polynomial.
91 * \param[in] poly : the vector of coefficients of the polynomial ordered
92 *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
93 *  e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
94 */
95template <typename Polynomial>
96inline
97typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly )
98{
99  typedef typename Polynomial::Scalar Scalar;
100  typedef typename NumTraits<Scalar>::Real Real;
101
102  DenseIndex i=0;
103  while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; }
104  if( poly.size()-1 == i ){
105    return Real(1); }
106
107  const Scalar inv_min_coeff = Scalar(1)/poly[i];
108  Real cb(1);
109  for( DenseIndex j=i+1; j<poly.size(); ++j ){
110    cb += internal::abs(poly[j]*inv_min_coeff); }
111  return Real(1)/cb;
112}
113
114/** \ingroup Polynomials_Module
115 * Given the roots of a polynomial compute the coefficients in the
116 * monomial basis of the monic polynomial with same roots and minimal degree.
117 * If RootVector is a vector of complexes, Polynomial should also be a vector
118 * of complexes.
119 * \param[in] rv : a vector containing the roots of a polynomial.
120 * \param[out] poly : the vector of coefficients of the polynomial ordered
121 *  by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
122 *  e.g. \f$ 3 + x^2 \f$ is stored as a vector \f$ [ 3, 0, 1 ] \f$.
123 */
124template <typename RootVector, typename Polynomial>
125void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
126{
127
128  typedef typename Polynomial::Scalar Scalar;
129
130  poly.setZero( rv.size()+1 );
131  poly[0] = -rv[0]; poly[1] = Scalar(1);
132  for( DenseIndex i=1; i< rv.size(); ++i )
133  {
134    for( DenseIndex j=i+1; j>0; --j ){ poly[j] = poly[j-1] - rv[i]*poly[j]; }
135    poly[0] = -rv[i]*poly[0];
136  }
137}
138
139} // end namespace Eigen
140
141#endif // EIGEN_POLYNOMIAL_UTILS_H
142