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authorNao Pross <np@0hm.ch>2024-02-12 15:23:24 +0100
committerNao Pross <np@0hm.ch>2024-02-12 15:23:24 +0100
commitfbd6758fb4649b146176dbbc2dfe9384c69ef58d (patch)
tree0993d5c74a5cd1773ff9a572e4926d3102c0299f /src/EigenUnsupported/Polynomials
parentMove into version control (diff)
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Remove old stuff with Eigen
Diffstat (limited to 'src/EigenUnsupported/Polynomials')
-rw-r--r--src/EigenUnsupported/Polynomials137
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diff --git a/src/EigenUnsupported/Polynomials b/src/EigenUnsupported/Polynomials
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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 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$.
-
- \subsection poly_eval
- The function
- \code
- T poly_eval( const Polynomials& poly, const T& x )
- \endcode
- evaluates a polynomial at a given point using stabilized H&ouml;rner method.
-
- The following code: first computes the coefficients in the monomial basis of the monic polynomial that has the provided roots;
- then, it evaluates the computed polynomial, using a stabilized H&ouml;rner method.
-
- \include PolynomialUtils1.cpp
- Output: \verbinclude PolynomialUtils1.out
-
- \subsection Cauchy bounds
- The function
- \code
- Real cauchy_max_bound( const Polynomial& poly )
- \endcode
- provides a maximum bound (the Cauchy one: \f$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