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