unsupported/Eigen/src/Polynomials/Companion.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_COMPANION_H
#define EIGEN_COMPANION_H

// This file requires the user to include
// * Eigen/Core
// * Eigen/src/PolynomialSolver.h

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

namespace Eigen {

namespace internal {

#ifndef EIGEN_PARSED_BY_DOXYGEN

template <int Size>
struct decrement_if_fixed_size {
  enum { ret = (Size == Dynamic) ? Dynamic : Size - 1 };
};

#endif

template <typename Scalar_, int Deg_>
class companion {
 public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_, Deg_ == Dynamic ? Dynamic : Deg_)

  enum { Deg = Deg_, Deg_1 = decrement_if_fixed_size<Deg>::ret };

  typedef Scalar_ Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Matrix<Scalar, Deg, 1> RightColumn;
  // typedef DiagonalMatrix< Scalar, Deg_1, Deg_1 > BottomLeftDiagonal;
  typedef Matrix<Scalar, Deg_1, 1> BottomLeftDiagonal;

  typedef Matrix<Scalar, Deg, Deg> DenseCompanionMatrixType;
  typedef Matrix<Scalar, Deg_, Deg_1> LeftBlock;
  typedef Matrix<Scalar, Deg_1, Deg_1> BottomLeftBlock;
  typedef Matrix<Scalar, 1, Deg_1> LeftBlockFirstRow;

  typedef DenseIndex Index;

 public:
  EIGEN_STRONG_INLINE const Scalar_ operator()(Index row, Index col) const {
    if (m_bl_diag.rows() > col) {
      if (0 < row) {
        return m_bl_diag[col];
      } else {
        return 0;
      }
    } else {
      return m_monic[row];
    }
  }

 public:
  template <typename VectorType>
  void setPolynomial(const VectorType& poly) {
    const Index deg = poly.size() - 1;
    m_monic = -poly.head(deg) / poly[deg];
    m_bl_diag.setOnes(deg - 1);
  }

  template <typename VectorType>
  companion(const VectorType& poly) {
    setPolynomial(poly);
  }

 public:
  DenseCompanionMatrixType denseMatrix() const {
    const Index deg = m_monic.size();
    const Index deg_1 = deg - 1;
    DenseCompanionMatrixType companMat(deg, deg);
    companMat << (LeftBlock(deg, deg_1) << LeftBlockFirstRow::Zero(1, deg_1),
                  BottomLeftBlock::Identity(deg - 1, deg - 1) * m_bl_diag.asDiagonal())
                     .finished(),
        m_monic;
    return companMat;
  }

 protected:
  /** Helper function for the balancing algorithm.
   * \returns true if the row and the column, having colNorm and rowNorm
   * as norms, are balanced, false otherwise.
   * colB and rowB are respectively the multipliers for
   * the column and the row in order to balance them.
   * */
  bool balanced(RealScalar colNorm, RealScalar rowNorm, bool& isBalanced, RealScalar& colB, RealScalar& rowB);

  /** Helper function for the balancing algorithm.
   * \returns true if the row and the column, having colNorm and rowNorm
   * as norms, are balanced, false otherwise.
   * colB and rowB are respectively the multipliers for
   * the column and the row in order to balance them.
   * */
  bool balancedR(RealScalar colNorm, RealScalar rowNorm, bool& isBalanced, RealScalar& colB, RealScalar& rowB);

 public:
  /**
   * Balancing algorithm from B. N. PARLETT and C. REINSCH (1969)
   * "Balancing a matrix for calculation of eigenvalues and eigenvectors"
   * adapted to the case of companion matrices.
   * A matrix with non zero row and non zero column is balanced
   * for a certain norm if the i-th row and the i-th column
   * have same norm for all i.
   */
  void balance();

 protected:
  RightColumn m_monic;
  BottomLeftDiagonal m_bl_diag;
};

template <typename Scalar_, int Deg_>
inline bool companion<Scalar_, Deg_>::balanced(RealScalar colNorm, RealScalar rowNorm, bool& isBalanced,
                                               RealScalar& colB, RealScalar& rowB) {
  if (RealScalar(0) == colNorm || RealScalar(0) == rowNorm || !(numext::isfinite)(colNorm) ||
      !(numext::isfinite)(rowNorm)) {
    return true;
  } else {
    // To find the balancing coefficients, if the radix is 2,
    // one finds \f$ \sigma \f$ such that
    //  \f$ 2^{2\sigma-1} < rowNorm / colNorm \le 2^{2\sigma+1} \f$
    //  then the balancing coefficient for the row is \f$ 1/2^{\sigma} \f$
    //  and the balancing coefficient for the column is \f$ 2^{\sigma} \f$
    const RealScalar radix = RealScalar(2);
    const RealScalar radix2 = RealScalar(4);

    rowB = rowNorm / radix;
    colB = RealScalar(1);
    const RealScalar s = colNorm + rowNorm;

    // Find sigma s.t. rowNorm / 2 <= 2^(2*sigma) * colNorm
    RealScalar scout = colNorm;
    while (scout < rowB) {
      colB *= radix;
      scout *= radix2;
    }

    // We now have an upper-bound for sigma, try to lower it.
    // Find sigma s.t. 2^(2*sigma) * colNorm / 2 < rowNorm
    scout = colNorm * (colB / radix) * colB;  // Avoid overflow.
    while (scout >= rowNorm) {
      colB /= radix;
      scout /= radix2;
    }

    // This line is used to avoid insubstantial balancing.
    if ((rowNorm + radix * scout) < RealScalar(0.95) * s * colB) {
      isBalanced = false;
      rowB = RealScalar(1) / colB;
      return false;
    } else {
      return true;
    }
  }
}

template <typename Scalar_, int Deg_>
inline bool companion<Scalar_, Deg_>::balancedR(RealScalar colNorm, RealScalar rowNorm, bool& isBalanced,
                                                RealScalar& colB, RealScalar& rowB) {
  if (RealScalar(0) == colNorm || RealScalar(0) == rowNorm) {
    return true;
  } else {
    /**
     * Set the norm of the column and the row to the geometric mean
     * of the row and column norm
     */
    const RealScalar q = colNorm / rowNorm;
    if (!isApprox(q, Scalar_(1))) {
      rowB = sqrt(colNorm / rowNorm);
      colB = RealScalar(1) / rowB;

      isBalanced = false;
      return false;
    } else {
      return true;
    }
  }
}

template <typename Scalar_, int Deg_>
void companion<Scalar_, Deg_>::balance() {
  using std::abs;
  EIGEN_STATIC_ASSERT(Deg == Dynamic || 1 < Deg, YOU_MADE_A_PROGRAMMING_MISTAKE);
  const Index deg = m_monic.size();
  const Index deg_1 = deg - 1;

  bool hasConverged = false;
  while (!hasConverged) {
    hasConverged = true;
    RealScalar colNorm, rowNorm;
    RealScalar colB, rowB;

    // First row, first column excluding the diagonal
    //==============================================
    colNorm = abs(m_bl_diag[0]);
    rowNorm = abs(m_monic[0]);

    // Compute balancing of the row and the column
    if (!balanced(colNorm, rowNorm, hasConverged, colB, rowB)) {
      m_bl_diag[0] *= colB;
      m_monic[0] *= rowB;
    }

    // Middle rows and columns excluding the diagonal
    //==============================================
    for (Index i = 1; i < deg_1; ++i) {
      // column norm, excluding the diagonal
      colNorm = abs(m_bl_diag[i]);

      // row norm, excluding the diagonal
      rowNorm = abs(m_bl_diag[i - 1]) + abs(m_monic[i]);

      // Compute balancing of the row and the column
      if (!balanced(colNorm, rowNorm, hasConverged, colB, rowB)) {
        m_bl_diag[i] *= colB;
        m_bl_diag[i - 1] *= rowB;
        m_monic[i] *= rowB;
      }
    }

    // Last row, last column excluding the diagonal
    //============================================
    const Index ebl = m_bl_diag.size() - 1;
    VectorBlock<RightColumn, Deg_1> headMonic(m_monic, 0, deg_1);
    colNorm = headMonic.array().abs().sum();
    rowNorm = abs(m_bl_diag[ebl]);

    // Compute balancing of the row and the column
    if (!balanced(colNorm, rowNorm, hasConverged, colB, rowB)) {
      headMonic *= colB;
      m_bl_diag[ebl] *= rowB;
    }
  }
}

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_COMPANION_H