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| -rw-r--r-- | src/EigenUnsupported/Polynomials | 137 |
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diff --git a/src/EigenUnsupported/Polynomials b/src/EigenUnsupported/Polynomials deleted file mode 100644 index 32ce2a2..0000000 --- a/src/EigenUnsupported/Polynomials +++ /dev/null @@ -1,137 +0,0 @@ -// 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 http://mozilla.org/MPL/2.0/. - -#ifndef EIGEN_POLYNOMIALS_MODULE_H -#define EIGEN_POLYNOMIALS_MODULE_H - -#include "../../Eigen/Core" - -#include "../../Eigen/Eigenvalues" - -#include "../../Eigen/src/Core/util/DisableStupidWarnings.h" - -// Note that EIGEN_HIDE_HEAVY_CODE has to be defined per module -#if (defined EIGEN_EXTERN_INSTANTIATIONS) && (EIGEN_EXTERN_INSTANTIATIONS>=2) - #ifndef EIGEN_HIDE_HEAVY_CODE - #define EIGEN_HIDE_HEAVY_CODE - #endif -#elif defined EIGEN_HIDE_HEAVY_CODE - #undef EIGEN_HIDE_HEAVY_CODE -#endif - -/** - * \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 - solver. - - \section polynomialUtils convenient functions to deal with polynomials - \subsection roots_to_monicPolynomial - The function - \code - void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly ) - \endcode - 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$. - - \subsection poly_eval - The function - \code - T poly_eval( const Polynomials& poly, const T& x ) - \endcode - evaluates a polynomial at a given point using stabilized Hörner method. - - The following code: first computes the coefficients in the monomial basis of the monic polynomial that has the provided roots; - then, it evaluates the computed polynomial, using a stabilized Hörner method. - - \include PolynomialUtils1.cpp - Output: \verbinclude PolynomialUtils1.out - - \subsection Cauchy bounds - The function - \code - Real cauchy_max_bound( const Polynomial& poly ) - \endcode - provides a maximum bound (the Cauchy one: \f$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$. - - - The function - \code - Real cauchy_min_bound( const Polynomial& poly ) - \endcode - provides a minimum bound (the Cauchy one: \f$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 - \f$ - \left [ - \begin{array}{cccc} - 0 & 0 & 0 & a_0 \\ - 1 & 0 & 0 & a_1 \\ - 0 & 1 & 0 & a_2 \\ - 0 & 0 & 1 & a_3 - \end{array} \right ] - \f$ - - 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" - -#endif // EIGEN_POLYNOMIALS_MODULE_H |