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388 lines
14 KiB
C++
388 lines
14 KiB
C++
// 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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// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
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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_POLYNOMIAL_SOLVER_H
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#define EIGEN_POLYNOMIAL_SOLVER_H
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// IWYU pragma: private
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#include "./InternalHeaderCheck.h"
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namespace Eigen {
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/** \ingroup Polynomials_Module
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* \class PolynomialSolverBase.
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*
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* \brief Defined to be inherited by polynomial solvers: it provides
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* convenient methods such as
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* - real roots,
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* - greatest, smallest complex roots,
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* - real roots with greatest, smallest absolute real value,
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* - greatest, smallest real roots.
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*
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* It stores the set of roots as a vector of complexes.
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*
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*/
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template <typename Scalar_, int Deg_>
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class PolynomialSolverBase {
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public:
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EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_, Deg_ == Dynamic ? Dynamic : Deg_)
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typedef Scalar_ Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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typedef internal::make_complex_t<Scalar> RootType;
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typedef Matrix<RootType, Deg_, 1> RootsType;
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typedef DenseIndex Index;
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protected:
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template <typename OtherPolynomial>
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inline void setPolynomial(const OtherPolynomial& poly) {
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m_roots.resize(poly.size() - 1);
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}
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public:
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template <typename OtherPolynomial>
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inline PolynomialSolverBase(const OtherPolynomial& poly) {
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setPolynomial(poly());
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}
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inline PolynomialSolverBase() {}
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public:
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/** \returns the complex roots of the polynomial */
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inline const RootsType& roots() const { return m_roots; }
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public:
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/** Clear and fills the back insertion sequence with the real roots of the polynomial
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* i.e. the real part of the complex roots that have an imaginary part which
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* absolute value is smaller than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the Scalar_ template parameter of the PolynomialSolver class as the default value.
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*
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* \param[out] bi_seq : the back insertion sequence (stl concept)
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* \param[in] absImaginaryThreshold : the maximum bound of the imaginary part of a complex
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* number that is considered as real.
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* */
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template <typename Stl_back_insertion_sequence>
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inline void realRoots(Stl_back_insertion_sequence& bi_seq,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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using std::abs;
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bi_seq.clear();
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for (Index i = 0; i < m_roots.size(); ++i) {
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if (abs(m_roots[i].imag()) < absImaginaryThreshold) {
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bi_seq.push_back(m_roots[i].real());
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}
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}
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}
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protected:
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template <typename Predicate>
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inline const RootType& selectComplexRoot_withRespectToNorm(Predicate& pred) const {
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Index res = 0;
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RealScalar norm2 = numext::abs2(m_roots[0]);
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for (Index i = 1; i < m_roots.size(); ++i) {
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const RealScalar currNorm2 = numext::abs2(m_roots[i]);
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if (pred(currNorm2, norm2)) {
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res = i;
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norm2 = currNorm2;
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}
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}
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return m_roots[res];
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}
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public:
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/**
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* \returns the complex root with greatest norm.
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*/
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inline const RootType& greatestRoot() const {
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std::greater<RealScalar> greater;
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return selectComplexRoot_withRespectToNorm(greater);
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}
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/**
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* \returns the complex root with smallest norm.
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*/
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inline const RootType& smallestRoot() const {
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std::less<RealScalar> less;
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return selectComplexRoot_withRespectToNorm(less);
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}
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protected:
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template <typename Predicate>
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inline const RealScalar& selectRealRoot_withRespectToAbsRealPart(
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Predicate& pred, bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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using std::abs;
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hasArealRoot = false;
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Index res = 0;
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RealScalar val(0);
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for (Index i = 0; i < m_roots.size(); ++i) {
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if (abs(m_roots[i].imag()) <= absImaginaryThreshold) {
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if (!hasArealRoot) {
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hasArealRoot = true;
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res = i;
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val = abs(m_roots[i].real());
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} else {
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const RealScalar curr = abs(m_roots[i].real());
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if (pred(curr, val)) {
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val = curr;
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res = i;
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}
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}
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} else if (!hasArealRoot) {
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if (abs(m_roots[i].imag()) < abs(m_roots[res].imag())) {
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res = i;
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}
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}
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}
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return numext::real_ref(m_roots[res]);
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}
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template <typename Predicate>
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inline const RealScalar& selectRealRoot_withRespectToRealPart(
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Predicate& pred, bool& hasArealRoot,
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const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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using std::abs;
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hasArealRoot = false;
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Index res = 0;
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RealScalar val(0);
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for (Index i = 0; i < m_roots.size(); ++i) {
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if (abs(m_roots[i].imag()) <= absImaginaryThreshold) {
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if (!hasArealRoot) {
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hasArealRoot = true;
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res = i;
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val = m_roots[i].real();
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} else {
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const RealScalar curr = m_roots[i].real();
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if (pred(curr, val)) {
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val = curr;
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res = i;
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}
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}
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} else {
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if (abs(m_roots[i].imag()) < abs(m_roots[res].imag())) {
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res = i;
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}
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}
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}
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return numext::real_ref(m_roots[res]);
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}
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public:
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/**
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* \returns a real root with greatest absolute magnitude.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the Scalar_ template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& absGreatestRealRoot(
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bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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std::greater<RealScalar> greater;
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return selectRealRoot_withRespectToAbsRealPart(greater, hasArealRoot, absImaginaryThreshold);
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}
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/**
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* \returns a real root with smallest absolute magnitude.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the Scalar_ template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& absSmallestRealRoot(
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bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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std::less<RealScalar> less;
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return selectRealRoot_withRespectToAbsRealPart(less, hasArealRoot, absImaginaryThreshold);
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}
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/**
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* \returns the real root with greatest value.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the Scalar_ template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& greatestRealRoot(
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bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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std::greater<RealScalar> greater;
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return selectRealRoot_withRespectToRealPart(greater, hasArealRoot, absImaginaryThreshold);
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}
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/**
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* \returns the real root with smallest value.
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* A real root is defined as the real part of a complex root with absolute imaginary
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* part smallest than absImaginaryThreshold.
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* absImaginaryThreshold takes the dummy_precision associated
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* with the Scalar_ template parameter of the PolynomialSolver class as the default value.
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* If no real root is found the boolean hasArealRoot is set to false and the real part of
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* the root with smallest absolute imaginary part is returned instead.
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*
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* \param[out] hasArealRoot : boolean true if a real root is found according to the
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* absImaginaryThreshold criterion, false otherwise.
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* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
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* whether or not a root is real.
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*/
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inline const RealScalar& smallestRealRoot(
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bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
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std::less<RealScalar> less;
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return selectRealRoot_withRespectToRealPart(less, hasArealRoot, absImaginaryThreshold);
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}
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protected:
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RootsType m_roots;
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};
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#define EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(BASE) \
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typedef typename BASE::Scalar Scalar; \
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typedef typename BASE::RealScalar RealScalar; \
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typedef typename BASE::RootType RootType; \
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typedef typename BASE::RootsType RootsType;
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/** \ingroup Polynomials_Module
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*
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* \class PolynomialSolver
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*
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* \brief A polynomial solver
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*
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* Computes the complex roots of a real polynomial.
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*
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* \param Scalar_ the scalar type, i.e., the type of the polynomial coefficients
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* \param Deg_ the degree of the polynomial, can be a compile time value or Dynamic.
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* Notice that the number of polynomial coefficients is Deg_+1.
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*
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* This class implements a polynomial solver and provides convenient methods such as
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* - real roots,
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* - greatest, smallest complex roots,
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* - real roots with greatest, smallest absolute real value.
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* - greatest, smallest real roots.
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*
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* WARNING: this polynomial solver is experimental, part of the unsupported Eigen modules.
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*
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*
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* Currently a QR algorithm is used to compute the eigenvalues of the companion matrix of
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* the polynomial to compute its roots.
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* This supposes that the complex moduli of the roots are all distinct: e.g. there should
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* be no multiple roots or conjugate roots for instance.
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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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*/
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template <typename Scalar_, int Deg_>
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class PolynomialSolver : public PolynomialSolverBase<Scalar_, Deg_> {
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public:
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EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_, Deg_ == Dynamic ? Dynamic : Deg_)
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typedef PolynomialSolverBase<Scalar_, Deg_> PS_Base;
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EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(PS_Base)
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typedef Matrix<Scalar, Deg_, Deg_> CompanionMatrixType;
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typedef std::conditional_t<NumTraits<Scalar>::IsComplex, ComplexEigenSolver<CompanionMatrixType>,
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EigenSolver<CompanionMatrixType> >
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EigenSolverType;
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typedef internal::make_complex_t<Scalar_> ComplexScalar;
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public:
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/** Computes the complex roots of a new polynomial. */
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template <typename OtherPolynomial>
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void compute(const OtherPolynomial& poly) {
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eigen_assert(Scalar(0) != poly[poly.size() - 1]);
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eigen_assert(poly.size() > 1);
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if (poly.size() > 2) {
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internal::companion<Scalar, Deg_> companion(poly);
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companion.balance();
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m_eigenSolver.compute(companion.denseMatrix());
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eigen_assert(m_eigenSolver.info() == Eigen::Success);
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m_roots = m_eigenSolver.eigenvalues();
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// cleanup noise in imaginary part of real roots:
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// if the imaginary part is rather small compared to the real part
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// and that cancelling the imaginary part yield a smaller evaluation,
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// then it's safe to keep the real part only.
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RealScalar coarse_prec = RealScalar(std::pow(4, poly.size() + 1)) * NumTraits<RealScalar>::epsilon();
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for (Index i = 0; i < m_roots.size(); ++i) {
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if (internal::isMuchSmallerThan(numext::abs(numext::imag(m_roots[i])), numext::abs(numext::real(m_roots[i])),
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coarse_prec)) {
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ComplexScalar as_real_root = ComplexScalar(numext::real(m_roots[i]));
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if (numext::abs(poly_eval(poly, as_real_root)) <= numext::abs(poly_eval(poly, m_roots[i]))) {
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m_roots[i] = as_real_root;
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}
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}
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}
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} else if (poly.size() == 2) {
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m_roots.resize(1);
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m_roots[0] = -poly[0] / poly[1];
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}
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}
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public:
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template <typename OtherPolynomial>
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inline PolynomialSolver(const OtherPolynomial& poly) {
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compute(poly);
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}
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inline PolynomialSolver() {}
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protected:
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using PS_Base::m_roots;
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EigenSolverType m_eigenSolver;
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};
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template <typename Scalar_>
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class PolynomialSolver<Scalar_, 1> : public PolynomialSolverBase<Scalar_, 1> {
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public:
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typedef PolynomialSolverBase<Scalar_, 1> PS_Base;
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EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(PS_Base)
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public:
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/** Computes the complex roots of a new polynomial. */
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template <typename OtherPolynomial>
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void compute(const OtherPolynomial& poly) {
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eigen_assert(poly.size() == 2);
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eigen_assert(Scalar(0) != poly[1]);
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m_roots[0] = -poly[0] / poly[1];
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}
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public:
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template <typename OtherPolynomial>
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inline PolynomialSolver(const OtherPolynomial& poly) {
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compute(poly);
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}
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inline PolynomialSolver() {}
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protected:
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using PS_Base::m_roots;
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};
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} // end namespace Eigen
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#endif // EIGEN_POLYNOMIAL_SOLVER_H
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