unsupported/Eigen/src/Polynomials/PolynomialSolver.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// 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_POLYNOMIAL_SOLVER_H
#define EIGEN_POLYNOMIAL_SOLVER_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

/** \ingroup Polynomials_Module
 *  \class PolynomialSolverBase.
 *
 * \brief Defined to be inherited by polynomial solvers: it provides
 * convenient methods such as
 *  - real roots,
 *  - greatest, smallest complex roots,
 *  - real roots with greatest, smallest absolute real value,
 *  - greatest, smallest real roots.
 *
 * It stores the set of roots as a vector of complexes.
 *
 */
template <typename Scalar_, int Deg_>
class PolynomialSolverBase {
 public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_, Deg_ == Dynamic ? Dynamic : Deg_)

  typedef Scalar_ Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef std::complex<RealScalar> RootType;
  typedef Matrix<RootType, Deg_, 1> RootsType;

  typedef DenseIndex Index;

 protected:
  template <typename OtherPolynomial>
  inline void setPolynomial(const OtherPolynomial& poly) {
    m_roots.resize(poly.size() - 1);
  }

 public:
  template <typename OtherPolynomial>
  inline PolynomialSolverBase(const OtherPolynomial& poly) {
    setPolynomial(poly());
  }

  inline PolynomialSolverBase() {}

 public:
  /** \returns the complex roots of the polynomial */
  inline const RootsType& roots() const { return m_roots; }

 public:
  /** Clear and fills the back insertion sequence with the real roots of the polynomial
   * i.e. the real part of the complex roots that have an imaginary part which
   * absolute value is smaller than absImaginaryThreshold.
   * absImaginaryThreshold takes the dummy_precision associated
   * with the Scalar_ template parameter of the PolynomialSolver class as the default value.
   *
   * \param[out] bi_seq : the back insertion sequence (stl concept)
   * \param[in]  absImaginaryThreshold : the maximum bound of the imaginary part of a complex
   *  number that is considered as real.
   * */
  template <typename Stl_back_insertion_sequence>
  inline void realRoots(Stl_back_insertion_sequence& bi_seq,
                        const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    using std::abs;
    bi_seq.clear();
    for (Index i = 0; i < m_roots.size(); ++i) {
      if (abs(m_roots[i].imag()) < absImaginaryThreshold) {
        bi_seq.push_back(m_roots[i].real());
      }
    }
  }

 protected:
  template <typename squaredNormBinaryPredicate>
  inline const RootType& selectComplexRoot_withRespectToNorm(squaredNormBinaryPredicate& pred) const {
    Index res = 0;
    RealScalar norm2 = numext::abs2(m_roots[0]);
    for (Index i = 1; i < m_roots.size(); ++i) {
      const RealScalar currNorm2 = numext::abs2(m_roots[i]);
      if (pred(currNorm2, norm2)) {
        res = i;
        norm2 = currNorm2;
      }
    }
    return m_roots[res];
  }

 public:
  /**
   * \returns the complex root with greatest norm.
   */
  inline const RootType& greatestRoot() const {
    std::greater<RealScalar> greater;
    return selectComplexRoot_withRespectToNorm(greater);
  }

  /**
   * \returns the complex root with smallest norm.
   */
  inline const RootType& smallestRoot() const {
    std::less<RealScalar> less;
    return selectComplexRoot_withRespectToNorm(less);
  }

 protected:
  template <typename squaredRealPartBinaryPredicate>
  inline const RealScalar& selectRealRoot_withRespectToAbsRealPart(
      squaredRealPartBinaryPredicate& pred, bool& hasArealRoot,
      const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    using std::abs;
    hasArealRoot = false;
    Index res = 0;
    RealScalar abs2(0);

    for (Index i = 0; i < m_roots.size(); ++i) {
      if (abs(m_roots[i].imag()) <= absImaginaryThreshold) {
        if (!hasArealRoot) {
          hasArealRoot = true;
          res = i;
          abs2 = m_roots[i].real() * m_roots[i].real();
        } else {
          const RealScalar currAbs2 = m_roots[i].real() * m_roots[i].real();
          if (pred(currAbs2, abs2)) {
            abs2 = currAbs2;
            res = i;
          }
        }
      } else if (!hasArealRoot) {
        if (abs(m_roots[i].imag()) < abs(m_roots[res].imag())) {
          res = i;
        }
      }
    }
    return numext::real_ref(m_roots[res]);
  }

  template <typename RealPartBinaryPredicate>
  inline const RealScalar& selectRealRoot_withRespectToRealPart(
      RealPartBinaryPredicate& pred, bool& hasArealRoot,
      const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    using std::abs;
    hasArealRoot = false;
    Index res = 0;
    RealScalar val(0);

    for (Index i = 0; i < m_roots.size(); ++i) {
      if (abs(m_roots[i].imag()) <= absImaginaryThreshold) {
        if (!hasArealRoot) {
          hasArealRoot = true;
          res = i;
          val = m_roots[i].real();
        } else {
          const RealScalar curr = m_roots[i].real();
          if (pred(curr, val)) {
            val = curr;
            res = i;
          }
        }
      } else {
        if (abs(m_roots[i].imag()) < abs(m_roots[res].imag())) {
          res = i;
        }
      }
    }
    return numext::real_ref(m_roots[res]);
  }

 public:
  /**
   * \returns a real root with greatest absolute magnitude.
   * A real root is defined as the real part of a complex root with absolute imaginary
   * part smallest than absImaginaryThreshold.
   * absImaginaryThreshold takes the dummy_precision associated
   * with the Scalar_ template parameter of the PolynomialSolver class as the default value.
   * If no real root is found the boolean hasArealRoot is set to false and the real part of
   * the root with smallest absolute imaginary part is returned instead.
   *
   * \param[out] hasArealRoot : boolean true if a real root is found according to the
   *  absImaginaryThreshold criterion, false otherwise.
   * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
   *  whether or not a root is real.
   */
  inline const RealScalar& absGreatestRealRoot(
      bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    std::greater<RealScalar> greater;
    return selectRealRoot_withRespectToAbsRealPart(greater, hasArealRoot, absImaginaryThreshold);
  }

  /**
   * \returns a real root with smallest absolute magnitude.
   * A real root is defined as the real part of a complex root with absolute imaginary
   * part smallest than absImaginaryThreshold.
   * absImaginaryThreshold takes the dummy_precision associated
   * with the Scalar_ template parameter of the PolynomialSolver class as the default value.
   * If no real root is found the boolean hasArealRoot is set to false and the real part of
   * the root with smallest absolute imaginary part is returned instead.
   *
   * \param[out] hasArealRoot : boolean true if a real root is found according to the
   *  absImaginaryThreshold criterion, false otherwise.
   * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
   *  whether or not a root is real.
   */
  inline const RealScalar& absSmallestRealRoot(
      bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    std::less<RealScalar> less;
    return selectRealRoot_withRespectToAbsRealPart(less, hasArealRoot, absImaginaryThreshold);
  }

  /**
   * \returns the real root with greatest value.
   * A real root is defined as the real part of a complex root with absolute imaginary
   * part smallest than absImaginaryThreshold.
   * absImaginaryThreshold takes the dummy_precision associated
   * with the Scalar_ template parameter of the PolynomialSolver class as the default value.
   * If no real root is found the boolean hasArealRoot is set to false and the real part of
   * the root with smallest absolute imaginary part is returned instead.
   *
   * \param[out] hasArealRoot : boolean true if a real root is found according to the
   *  absImaginaryThreshold criterion, false otherwise.
   * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
   *  whether or not a root is real.
   */
  inline const RealScalar& greatestRealRoot(
      bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    std::greater<RealScalar> greater;
    return selectRealRoot_withRespectToRealPart(greater, hasArealRoot, absImaginaryThreshold);
  }

  /**
   * \returns the real root with smallest value.
   * A real root is defined as the real part of a complex root with absolute imaginary
   * part smallest than absImaginaryThreshold.
   * absImaginaryThreshold takes the dummy_precision associated
   * with the Scalar_ template parameter of the PolynomialSolver class as the default value.
   * If no real root is found the boolean hasArealRoot is set to false and the real part of
   * the root with smallest absolute imaginary part is returned instead.
   *
   * \param[out] hasArealRoot : boolean true if a real root is found according to the
   *  absImaginaryThreshold criterion, false otherwise.
   * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
   *  whether or not a root is real.
   */
  inline const RealScalar& smallestRealRoot(
      bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const {
    std::less<RealScalar> less;
    return selectRealRoot_withRespectToRealPart(less, hasArealRoot, absImaginaryThreshold);
  }

 protected:
  RootsType m_roots;
};

#define EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(BASE) \
  typedef typename BASE::Scalar Scalar;                    \
  typedef typename BASE::RealScalar RealScalar;            \
  typedef typename BASE::RootType RootType;                \
  typedef typename BASE::RootsType RootsType;

/** \ingroup Polynomials_Module
 *
 * \class PolynomialSolver
 *
 * \brief A polynomial solver
 *
 * Computes the complex roots of a real polynomial.
 *
 * \param Scalar_ the scalar type, i.e., the type of the polynomial coefficients
 * \param Deg_ the degree of the polynomial, can be a compile time value or Dynamic.
 *             Notice that the number of polynomial coefficients is Deg_+1.
 *
 * This class implements a polynomial solver and provides convenient methods such as
 * - real roots,
 * - greatest, smallest complex roots,
 * - real roots with greatest, smallest absolute real value.
 * - greatest, smallest real roots.
 *
 * WARNING: this polynomial solver is experimental, part of the unsupported Eigen modules.
 *
 *
 * Currently a QR algorithm is used to compute the eigenvalues of the companion matrix of
 * the polynomial to compute its roots.
 * This supposes that the complex moduli of the roots are all distinct: e.g. there should
 * be no multiple roots or conjugate roots for instance.
 * 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).
 */
template <typename Scalar_, int Deg_>
class PolynomialSolver : public PolynomialSolverBase<Scalar_, Deg_> {
 public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_, Deg_ == Dynamic ? Dynamic : Deg_)

  typedef PolynomialSolverBase<Scalar_, Deg_> PS_Base;
  EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(PS_Base)

  typedef Matrix<Scalar, Deg_, Deg_> CompanionMatrixType;
  typedef std::conditional_t<NumTraits<Scalar>::IsComplex, ComplexEigenSolver<CompanionMatrixType>,
                             EigenSolver<CompanionMatrixType> >
      EigenSolverType;
  typedef std::conditional_t<NumTraits<Scalar>::IsComplex, Scalar, std::complex<Scalar> > ComplexScalar;

 public:
  /** Computes the complex roots of a new polynomial. */
  template <typename OtherPolynomial>
  void compute(const OtherPolynomial& poly) {
    eigen_assert(Scalar(0) != poly[poly.size() - 1]);
    eigen_assert(poly.size() > 1);
    if (poly.size() > 2) {
      internal::companion<Scalar, Deg_> companion(poly);
      companion.balance();
      m_eigenSolver.compute(companion.denseMatrix());
      eigen_assert(m_eigenSolver.info() == Eigen::Success);
      m_roots = m_eigenSolver.eigenvalues();
      // cleanup noise in imaginary part of real roots:
      // if the imaginary part is rather small compared to the real part
      // and that cancelling the imaginary part yield a smaller evaluation,
      // then it's safe to keep the real part only.
      RealScalar coarse_prec = RealScalar(std::pow(4, poly.size() + 1)) * NumTraits<RealScalar>::epsilon();
      for (Index i = 0; i < m_roots.size(); ++i) {
        if (internal::isMuchSmallerThan(numext::abs(numext::imag(m_roots[i])), numext::abs(numext::real(m_roots[i])),
                                        coarse_prec)) {
          ComplexScalar as_real_root = ComplexScalar(numext::real(m_roots[i]));
          if (numext::abs(poly_eval(poly, as_real_root)) <= numext::abs(poly_eval(poly, m_roots[i]))) {
            m_roots[i] = as_real_root;
          }
        }
      }
    } else if (poly.size() == 2) {
      m_roots.resize(1);
      m_roots[0] = -poly[0] / poly[1];
    }
  }

 public:
  template <typename OtherPolynomial>
  inline PolynomialSolver(const OtherPolynomial& poly) {
    compute(poly);
  }

  inline PolynomialSolver() {}

 protected:
  using PS_Base::m_roots;
  EigenSolverType m_eigenSolver;
};

template <typename Scalar_>
class PolynomialSolver<Scalar_, 1> : public PolynomialSolverBase<Scalar_, 1> {
 public:
  typedef PolynomialSolverBase<Scalar_, 1> PS_Base;
  EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(PS_Base)

 public:
  /** Computes the complex roots of a new polynomial. */
  template <typename OtherPolynomial>
  void compute(const OtherPolynomial& poly) {
    eigen_assert(poly.size() == 2);
    eigen_assert(Scalar(0) != poly[1]);
    m_roots[0] = -poly[0] / poly[1];
  }

 public:
  template <typename OtherPolynomial>
  inline PolynomialSolver(const OtherPolynomial& poly) {
    compute(poly);
  }

  inline PolynomialSolver() {}

 protected:
  using PS_Base::m_roots;
};

}  // end namespace Eigen

#endif  // EIGEN_POLYNOMIAL_SOLVER_H