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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at
#include <Eigen/Core>
#include <Eigen/src/Core/util/DisableStupidWarnings.h>
#include <Eigen/Eigenvalues>
// Note that EIGEN_HIDE_HEAVY_CODE has to be defined per module
* \defgroup Polynomials_Module Polynomials module
* \brief This module provides a QR based polynomial solver.
* To use this module, add
* \code
* #include <unsupported/Eigen/Polynomials>
* \endcode
* at the start of your source file.
#include "src/Polynomials/PolynomialUtils.h"
#include "src/Polynomials/Companion.h"
#include "src/Polynomials/PolynomialSolver.h"
\page polynomials Polynomials defines functions for dealing with polynomials
and a QR based polynomial solver.
\ingroup Polynomials_Module
The remainder of the page documents first the functions for evaluating, computing
polynomials, computing estimates about polynomials and next the QR based polynomial
\section polynomialUtils convenient functions to deal with polynomials
\subsection roots_to_monicPolynomial
The function
void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
computes the coefficients \f$ a_i \f$ of
\f$ p(x) = a_0 + a_{1}x + ... + a_{n-1}x^{n-1} + x^n \f$
where \f$ p \f$ is known through its roots i.e. \f$ p(x) = (x-r_1)(x-r_2)...(x-r_n) \f$.
\subsection poly_eval
The function
T poly_eval( const Polynomials& poly, const T& x )
evaluates a polynomial at a given point using stabilized H&ouml;rner method.
The following code: first computes the coefficients in the monomial basis of the monic polynomial that has the provided roots;
then, it evaluates the computed polynomial, using a stabilized H&ouml;rner method.
\include PolynomialUtils1.cpp
Output: \verbinclude PolynomialUtils1.out
\subsection Cauchy bounds
The function
Real cauchy_max_bound( const Polynomial& poly )
provides a maximum bound (the Cauchy one: \f$C(p)\f$) for the absolute value of a root of the given polynomial i.e.
\f$ \forall r_i \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
\f$ |r_i| \le C(p) = \sum_{k=0}^{d} \left | \frac{a_k}{a_d} \right | \f$
The leading coefficient \f$ p \f$: should be non zero \f$a_d \neq 0\f$.
The function
Real cauchy_min_bound( const Polynomial& poly )
provides a minimum bound (the Cauchy one: \f$c(p)\f$) for the absolute value of a non zero root of the given polynomial i.e.
\f$ \forall r_i \neq 0 \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
\f$ |r_i| \ge c(p) = \left( \sum_{k=0}^{d} \left | \frac{a_k}{a_0} \right | \right)^{-1} \f$
\section QR polynomial solver class
Computes the complex roots of a polynomial by computing the eigenvalues of the associated companion matrix with the QR algorithm.
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
\left [
0 & 0 & 0 & a_0 \\
1 & 0 & 0 & a_1 \\
0 & 1 & 0 & a_2 \\
0 & 0 & 1 & a_3
\end{array} \right ]
However, the QR algorithm is not guaranteed to converge when there are several eigenvalues with same modulus.
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.
\f$ \forall i,j \in [1;d],~ \| r_i \| \neq \| r_j \| \f$.
With 32bit (float) floating types this problem shows up frequently.
However, almost always, correct accuracy is reached even in these cases for 64bit
(double) floating types and small polynomial degree (<20).
\include PolynomialSolver1.cpp
In the above example:
-# a simple use of the polynomial solver is shown;
-# the accuracy problem with the QR algorithm is presented: a polynomial with almost conjugate roots is provided to the solver.
Those roots have almost same module therefore the QR algorithm failed to converge: the accuracy
of the last root is bad;
-# a simple way to circumvent the problem is shown: use doubles instead of floats.
Output: \verbinclude PolynomialSolver1.out
#include <Eigen/src/Core/util/ReenableStupidWarnings.h>
/* vim: set filetype=cpp et sw=2 ts=2 ai: */