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