1// This file is part of Eigen, a lightweight C++ template library
5// This Source Code Form is subject to the terms of the Mozilla
6// Public License v. 2.0. If a copy of the MPL was not distributed
7// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9#ifndef EIGEN_POLYNOMIALS_MODULE_H
10#define EIGEN_POLYNOMIALS_MODULE_H
12#include "../../Eigen/Core"
14#include "../../Eigen/Eigenvalues"
16#include "../../Eigen/src/Core/util/DisableStupidWarnings.h"
18// Note that EIGEN_HIDE_HEAVY_CODE has to be defined per module
19#if (defined EIGEN_EXTERN_INSTANTIATIONS) && (EIGEN_EXTERN_INSTANTIATIONS>=2)
20 #ifndef EIGEN_HIDE_HEAVY_CODE
21 #define EIGEN_HIDE_HEAVY_CODE
23#elif defined EIGEN_HIDE_HEAVY_CODE
24 #undef EIGEN_HIDE_HEAVY_CODE
28 * \defgroup Polynomials_Module Polynomials module
29 * \brief This module provides a QR based polynomial solver.
31 * To use this module, add
33 * #include <unsupported/Eigen/Polynomials>
35 * at the start of your source file.
38#include "src/Polynomials/PolynomialUtils.h"
39#include "src/Polynomials/Companion.h"
40#include "src/Polynomials/PolynomialSolver.h"
43 \page polynomials Polynomials defines functions for dealing with polynomials
44 and a QR based polynomial solver.
45 \ingroup Polynomials_Module
47 The remainder of the page documents first the functions for evaluating, computing
48 polynomials, computing estimates about polynomials and next the QR based polynomial
51 \section polynomialUtils convenient functions to deal with polynomials
52 \subsection roots_to_monicPolynomial
55 void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
57 computes the coefficients \f$ a_i \f$ of
59 \f$ p(x) = a_0 + a_{1}x + ... + a_{n-1}x^{n-1} + x^n \f$
61 where \f$ p \f$ is known through its roots i.e. \f$ p(x) = (x-r_1)(x-r_2)...(x-r_n) \f$.
66 T poly_eval( const Polynomials& poly, const T& x )
68 evaluates a polynomial at a given point using stabilized Hörner method.
70 The following code: first computes the coefficients in the monomial basis of the monic polynomial that has the provided roots;
71 then, it evaluates the computed polynomial, using a stabilized Hörner method.
73 \include PolynomialUtils1.cpp
74 Output: \verbinclude PolynomialUtils1.out
76 \subsection Cauchy bounds
79 Real cauchy_max_bound( const Polynomial& poly )
81 provides a maximum bound (the Cauchy one: \f$C(p)\f$) for the absolute value of a root of the given polynomial i.e.
82 \f$ \forall r_i \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
83 \f$ |r_i| \le C(p) = \sum_{k=0}^{d} \left | \frac{a_k}{a_d} \right | \f$
84 The leading coefficient \f$ p \f$: should be non zero \f$a_d \neq 0\f$.
89 Real cauchy_min_bound( const Polynomial& poly )
91 provides a minimum bound (the Cauchy one: \f$c(p)\f$) for the absolute value of a non zero root of the given polynomial i.e.
92 \f$ \forall r_i \neq 0 \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
93 \f$ |r_i| \ge c(p) = \left( \sum_{k=0}^{d} \left | \frac{a_k}{a_0} \right | \right)^{-1} \f$
98 \section QR polynomial solver class
99 Computes the complex roots of a polynomial by computing the eigenvalues of the associated companion matrix with the QR algorithm.
101 The roots of \f$ p(x) = a_0 + a_1 x + a_2 x^2 + a_{3} x^3 + x^4 \f$ are the eigenvalues of
112 However, the QR algorithm is not guaranteed to converge when there are several eigenvalues with same modulus.
114 Therefore the current polynomial solver is guaranteed to provide a correct result only when the complex roots \f$r_1,r_2,...,r_d\f$ have distinct moduli i.e.
116 \f$ \forall i,j \in [1;d],~ \| r_i \| \neq \| r_j \| \f$.
118 With 32bit (float) floating types this problem shows up frequently.
119 However, almost always, correct accuracy is reached even in these cases for 64bit
120 (double) floating types and small polynomial degree (<20).
122 \include PolynomialSolver1.cpp
124 In the above example:
126 -# a simple use of the polynomial solver is shown;
127 -# the accuracy problem with the QR algorithm is presented: a polynomial with almost conjugate roots is provided to the solver.
128 Those roots have almost same module therefore the QR algorithm failed to converge: the accuracy
129 of the last root is bad;
130 -# a simple way to circumvent the problem is shown: use doubles instead of floats.
132 Output: \verbinclude PolynomialSolver1.out
135#include "../../Eigen/src/Core/util/ReenableStupidWarnings.h"
137#endif // EIGEN_POLYNOMIALS_MODULE_H