text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2016 Rasmus Munk Larsen <rmlarsen@google.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_COMPLETEORTHOGONALDECOMPOSITION_H
  #define EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H
  
  namespace Eigen {
  
  namespace internal {
  template <typename _MatrixType>
  struct traits<CompleteOrthogonalDecomposition<_MatrixType> >
      : traits<_MatrixType> {
    enum { Flags = 0 };
  };
  
  }  // end namespace internal
  
  /** \ingroup QR_Module
    *
    * \class CompleteOrthogonalDecomposition
    *
    * \brief Complete orthogonal decomposition (COD) of a matrix.
    *
    * \param MatrixType the type of the matrix of which we are computing the COD.
    *
    * This class performs a rank-revealing complete orthogonal decomposition of a
    * matrix  \b A into matrices \b P, \b Q, \b T, and \b Z such that
    * \f[
    *  \mathbf{A} \, \mathbf{P} = \mathbf{Q} \,
    *                     \begin{bmatrix} \mathbf{T} &  \mathbf{0} \\
    *                                     \mathbf{0} & \mathbf{0} \end{bmatrix} \, \mathbf{Z}
    * \f]
    * by using Householder transformations. Here, \b P is a permutation matrix,
    * \b Q and \b Z are unitary matrices and \b T an upper triangular matrix of
    * size rank-by-rank. \b A may be rank deficient.
    *
    * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
    * 
    * \sa MatrixBase::completeOrthogonalDecomposition()
    */
  template <typename _MatrixType>
  class CompleteOrthogonalDecomposition {
   public:
    typedef _MatrixType MatrixType;
    enum {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::StorageIndex StorageIndex;
    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
    typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime>
        PermutationType;
    typedef typename internal::plain_row_type<MatrixType, Index>::type
        IntRowVectorType;
    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
    typedef typename internal::plain_row_type<MatrixType, RealScalar>::type
        RealRowVectorType;
    typedef HouseholderSequence<
        MatrixType, typename internal::remove_all<
                        typename HCoeffsType::ConjugateReturnType>::type>
        HouseholderSequenceType;
    typedef typename MatrixType::PlainObject PlainObject;
  
   private:
    typedef typename PermutationType::Index PermIndexType;
  
   public:
    /**
     * \brief Default Constructor.
     *
     * The default constructor is useful in cases in which the user intends to
     * perform decompositions via
     * \c CompleteOrthogonalDecomposition::compute(const* MatrixType&).
     */
    CompleteOrthogonalDecomposition() : m_cpqr(), m_zCoeffs(), m_temp() {}
  
    /** \brief Default Constructor with memory preallocation
     *
     * Like the default constructor but with preallocation of the internal data
     * according to the specified problem \a size.
     * \sa CompleteOrthogonalDecomposition()
     */
    CompleteOrthogonalDecomposition(Index rows, Index cols)
        : m_cpqr(rows, cols), m_zCoeffs((std::min)(rows, cols)), m_temp(cols) {}
  
    /** \brief Constructs a complete orthogonal decomposition from a given
     * matrix.
     *
     * This constructor computes the complete orthogonal decomposition of the
     * matrix \a matrix by calling the method compute(). The default
     * threshold for rank determination will be used. It is a short cut for:
     *
     * \code
     * CompleteOrthogonalDecomposition<MatrixType> cod(matrix.rows(),
     *                                                 matrix.cols());
     * cod.setThreshold(Default);
     * cod.compute(matrix);
     * \endcode
     *
     * \sa compute()
     */
    template <typename InputType>
    explicit CompleteOrthogonalDecomposition(const EigenBase<InputType>& matrix)
        : m_cpqr(matrix.rows(), matrix.cols()),
          m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
          m_temp(matrix.cols())
    {
      compute(matrix.derived());
    }
  
    /** \brief Constructs a complete orthogonal decomposition from a given matrix
      *
      * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
      *
      * \sa CompleteOrthogonalDecomposition(const EigenBase&)
      */
    template<typename InputType>
    explicit CompleteOrthogonalDecomposition(EigenBase<InputType>& matrix)
      : m_cpqr(matrix.derived()),
        m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_temp(matrix.cols())
    {
      computeInPlace();
    }
  
  
    /** This method computes the minimum-norm solution X to a least squares
     * problem \f[\mathrm{minimize} \|A X - B\|, \f] where \b A is the matrix of
     * which \c *this is the complete orthogonal decomposition.
     *
     * \param b the right-hand sides of the problem to solve.
     *
     * \returns a solution.
     *
     */
    template <typename Rhs>
    inline const Solve<CompleteOrthogonalDecomposition, Rhs> solve(
        const MatrixBase<Rhs>& b) const {
      eigen_assert(m_cpqr.m_isInitialized &&
                   "CompleteOrthogonalDecomposition is not initialized.");
      return Solve<CompleteOrthogonalDecomposition, Rhs>(*this, b.derived());
    }
  
    HouseholderSequenceType householderQ(void) const;
    HouseholderSequenceType matrixQ(void) const { return m_cpqr.householderQ(); }
  
    /** \returns the matrix \b Z.
     */
    MatrixType matrixZ() const {
      MatrixType Z = MatrixType::Identity(m_cpqr.cols(), m_cpqr.cols());
      applyZAdjointOnTheLeftInPlace(Z);
      return Z.adjoint();
    }
  
    /** \returns a reference to the matrix where the complete orthogonal
     * decomposition is stored
     */
    const MatrixType& matrixQTZ() const { return m_cpqr.matrixQR(); }
  
    /** \returns a reference to the matrix where the complete orthogonal
     * decomposition is stored.
     * \warning The strict lower part and \code cols() - rank() \endcode right
     * columns of this matrix contains internal values.
     * Only the upper triangular part should be referenced. To get it, use
     * \code matrixT().template triangularView<Upper>() \endcode
     * For rank-deficient matrices, use
     * \code
     * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
     * \endcode
     */
    const MatrixType& matrixT() const { return m_cpqr.matrixQR(); }
  
    template <typename InputType>
    CompleteOrthogonalDecomposition& compute(const EigenBase<InputType>& matrix) {
      // Compute the column pivoted QR factorization A P = Q R.
      m_cpqr.compute(matrix);
      computeInPlace();
      return *this;
    }
  
    /** \returns a const reference to the column permutation matrix */
    const PermutationType& colsPermutation() const {
      return m_cpqr.colsPermutation();
    }
  
    /** \returns the absolute value of the determinant of the matrix of which
     * *this is the complete orthogonal decomposition. It has only linear
     * complexity (that is, O(n) where n is the dimension of the square matrix)
     * as the complete orthogonal decomposition has already been computed.
     *
     * \note This is only for square matrices.
     *
     * \warning a determinant can be very big or small, so for matrices
     * of large enough dimension, there is a risk of overflow/underflow.
     * One way to work around that is to use logAbsDeterminant() instead.
     *
     * \sa logAbsDeterminant(), MatrixBase::determinant()
     */
    typename MatrixType::RealScalar absDeterminant() const;
  
    /** \returns the natural log of the absolute value of the determinant of the
     * matrix of which *this is the complete orthogonal decomposition. It has
     * only linear complexity (that is, O(n) where n is the dimension of the
     * square matrix) as the complete orthogonal decomposition has already been
     * computed.
     *
     * \note This is only for square matrices.
     *
     * \note This method is useful to work around the risk of overflow/underflow
     * that's inherent to determinant computation.
     *
     * \sa absDeterminant(), MatrixBase::determinant()
     */
    typename MatrixType::RealScalar logAbsDeterminant() const;
  
    /** \returns the rank of the matrix of which *this is the complete orthogonal
     * decomposition.
     *
     * \note This method has to determine which pivots should be considered
     * nonzero. For that, it uses the threshold value that you can control by
     * calling setThreshold(const RealScalar&).
     */
    inline Index rank() const { return m_cpqr.rank(); }
  
    /** \returns the dimension of the kernel of the matrix of which *this is the
     * complete orthogonal decomposition.
     *
     * \note This method has to determine which pivots should be considered
     * nonzero. For that, it uses the threshold value that you can control by
     * calling setThreshold(const RealScalar&).
     */
    inline Index dimensionOfKernel() const { return m_cpqr.dimensionOfKernel(); }
  
    /** \returns true if the matrix of which *this is the decomposition represents
     * an injective linear map, i.e. has trivial kernel; false otherwise.
     *
     * \note This method has to determine which pivots should be considered
     * nonzero. For that, it uses the threshold value that you can control by
     * calling setThreshold(const RealScalar&).
     */
    inline bool isInjective() const { return m_cpqr.isInjective(); }
  
    /** \returns true if the matrix of which *this is the decomposition represents
     * a surjective linear map; false otherwise.
     *
     * \note This method has to determine which pivots should be considered
     * nonzero. For that, it uses the threshold value that you can control by
     * calling setThreshold(const RealScalar&).
     */
    inline bool isSurjective() const { return m_cpqr.isSurjective(); }
  
    /** \returns true if the matrix of which *this is the complete orthogonal
     * decomposition is invertible.
     *
     * \note This method has to determine which pivots should be considered
     * nonzero. For that, it uses the threshold value that you can control by
     * calling setThreshold(const RealScalar&).
     */
    inline bool isInvertible() const { return m_cpqr.isInvertible(); }
  
    /** \returns the pseudo-inverse of the matrix of which *this is the complete
     * orthogonal decomposition.
     * \warning: Do not compute \c this->pseudoInverse()*rhs to solve a linear systems.
     * It is more efficient and numerically stable to call \c this->solve(rhs).
     */
    inline const Inverse<CompleteOrthogonalDecomposition> pseudoInverse() const
    {
      return Inverse<CompleteOrthogonalDecomposition>(*this);
    }
  
    inline Index rows() const { return m_cpqr.rows(); }
    inline Index cols() const { return m_cpqr.cols(); }
  
    /** \returns a const reference to the vector of Householder coefficients used
     * to represent the factor \c Q.
     *
     * For advanced uses only.
     */
    inline const HCoeffsType& hCoeffs() const { return m_cpqr.hCoeffs(); }
  
    /** \returns a const reference to the vector of Householder coefficients
     * used to represent the factor \c Z.
     *
     * For advanced uses only.
     */
    const HCoeffsType& zCoeffs() const { return m_zCoeffs; }
  
    /** Allows to prescribe a threshold to be used by certain methods, such as
     * rank(), who need to determine when pivots are to be considered nonzero.
     * Most be called before calling compute().
     *
     * When it needs to get the threshold value, Eigen calls threshold(). By
     * default, this uses a formula to automatically determine a reasonable
     * threshold. Once you have called the present method
     * setThreshold(const RealScalar&), your value is used instead.
     *
     * \param threshold The new value to use as the threshold.
     *
     * A pivot will be considered nonzero if its absolute value is strictly
     * greater than
     *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
     * where maxpivot is the biggest pivot.
     *
     * If you want to come back to the default behavior, call
     * setThreshold(Default_t)
     */
    CompleteOrthogonalDecomposition& setThreshold(const RealScalar& threshold) {
      m_cpqr.setThreshold(threshold);
      return *this;
    }
  
    /** Allows to come back to the default behavior, letting Eigen use its default
     * formula for determining the threshold.
     *
     * You should pass the special object Eigen::Default as parameter here.
     * \code qr.setThreshold(Eigen::Default); \endcode
     *
     * See the documentation of setThreshold(const RealScalar&).
     */
    CompleteOrthogonalDecomposition& setThreshold(Default_t) {
      m_cpqr.setThreshold(Default);
      return *this;
    }
  
    /** Returns the threshold that will be used by certain methods such as rank().
     *
     * See the documentation of setThreshold(const RealScalar&).
     */
    RealScalar threshold() const { return m_cpqr.threshold(); }
  
    /** \returns the number of nonzero pivots in the complete orthogonal
     * decomposition. Here nonzero is meant in the exact sense, not in a
     * fuzzy sense. So that notion isn't really intrinsically interesting,
     * but it is still useful when implementing algorithms.
     *
     * \sa rank()
     */
    inline Index nonzeroPivots() const { return m_cpqr.nonzeroPivots(); }
  
    /** \returns the absolute value of the biggest pivot, i.e. the biggest
     *          diagonal coefficient of R.
     */
    inline RealScalar maxPivot() const { return m_cpqr.maxPivot(); }
  
    /** \brief Reports whether the complete orthogonal decomposition was
     * succesful.
     *
     * \note This function always returns \c Success. It is provided for
     * compatibility
     * with other factorization routines.
     * \returns \c Success
     */
    ComputationInfo info() const {
      eigen_assert(m_cpqr.m_isInitialized && "Decomposition is not initialized.");
      return Success;
    }
  
  #ifndef EIGEN_PARSED_BY_DOXYGEN
    template <typename RhsType, typename DstType>
    EIGEN_DEVICE_FUNC void _solve_impl(const RhsType& rhs, DstType& dst) const;
  #endif
  
   protected:
    static void check_template_parameters() {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }
  
    void computeInPlace();
  
    /** Overwrites \b rhs with \f$ \mathbf{Z}^* * \mathbf{rhs} \f$.
     */
    template <typename Rhs>
    void applyZAdjointOnTheLeftInPlace(Rhs& rhs) const;
  
    ColPivHouseholderQR<MatrixType> m_cpqr;
    HCoeffsType m_zCoeffs;
    RowVectorType m_temp;
  };
  
  template <typename MatrixType>
  typename MatrixType::RealScalar
  CompleteOrthogonalDecomposition<MatrixType>::absDeterminant() const {
    return m_cpqr.absDeterminant();
  }
  
  template <typename MatrixType>
  typename MatrixType::RealScalar
  CompleteOrthogonalDecomposition<MatrixType>::logAbsDeterminant() const {
    return m_cpqr.logAbsDeterminant();
  }
  
  /** Performs the complete orthogonal decomposition of the given matrix \a
   * matrix. The result of the factorization is stored into \c *this, and a
   * reference to \c *this is returned.
   *
   * \sa class CompleteOrthogonalDecomposition,
   * CompleteOrthogonalDecomposition(const MatrixType&)
   */
  template <typename MatrixType>
  void CompleteOrthogonalDecomposition<MatrixType>::computeInPlace()
  {
    check_template_parameters();
  
    // the column permutation is stored as int indices, so just to be sure:
    eigen_assert(m_cpqr.cols() <= NumTraits<int>::highest());
  
    const Index rank = m_cpqr.rank();
    const Index cols = m_cpqr.cols();
    const Index rows = m_cpqr.rows();
    m_zCoeffs.resize((std::min)(rows, cols));
    m_temp.resize(cols);
  
    if (rank < cols) {
      // We have reduced the (permuted) matrix to the form
      //   [R11 R12]
      //   [ 0  R22]
      // where R11 is r-by-r (r = rank) upper triangular, R12 is
      // r-by-(n-r), and R22 is empty or the norm of R22 is negligible.
      // We now compute the complete orthogonal decomposition by applying
      // Householder transformations from the right to the upper trapezoidal
      // matrix X = [R11 R12] to zero out R12 and obtain the factorization
      // [R11 R12] = [T11 0] * Z, where T11 is r-by-r upper triangular and
      // Z = Z(0) * Z(1) ... Z(r-1) is an n-by-n orthogonal matrix.
      // We store the data representing Z in R12 and m_zCoeffs.
      for (Index k = rank - 1; k >= 0; --k) {
        if (k != rank - 1) {
          // Given the API for Householder reflectors, it is more convenient if
          // we swap the leading parts of columns k and r-1 (zero-based) to form
          // the matrix X_k = [X(0:k, k), X(0:k, r:n)]
          m_cpqr.m_qr.col(k).head(k + 1).swap(
              m_cpqr.m_qr.col(rank - 1).head(k + 1));
        }
        // Construct Householder reflector Z(k) to zero out the last row of X_k,
        // i.e. choose Z(k) such that
        // [X(k, k), X(k, r:n)] * Z(k) = [beta, 0, .., 0].
        RealScalar beta;
        m_cpqr.m_qr.row(k)
            .tail(cols - rank + 1)
            .makeHouseholderInPlace(m_zCoeffs(k), beta);
        m_cpqr.m_qr(k, rank - 1) = beta;
        if (k > 0) {
          // Apply Z(k) to the first k rows of X_k
          m_cpqr.m_qr.topRightCorner(k, cols - rank + 1)
              .applyHouseholderOnTheRight(
                  m_cpqr.m_qr.row(k).tail(cols - rank).transpose(), m_zCoeffs(k),
                  &m_temp(0));
        }
        if (k != rank - 1) {
          // Swap X(0:k,k) back to its proper location.
          m_cpqr.m_qr.col(k).head(k + 1).swap(
              m_cpqr.m_qr.col(rank - 1).head(k + 1));
        }
      }
    }
  }
  
  template <typename MatrixType>
  template <typename Rhs>
  void CompleteOrthogonalDecomposition<MatrixType>::applyZAdjointOnTheLeftInPlace(
      Rhs& rhs) const {
    const Index cols = this->cols();
    const Index nrhs = rhs.cols();
    const Index rank = this->rank();
    Matrix<typename MatrixType::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
    for (Index k = 0; k < rank; ++k) {
      if (k != rank - 1) {
        rhs.row(k).swap(rhs.row(rank - 1));
      }
      rhs.middleRows(rank - 1, cols - rank + 1)
          .applyHouseholderOnTheLeft(
              matrixQTZ().row(k).tail(cols - rank).adjoint(), zCoeffs()(k),
              &temp(0));
      if (k != rank - 1) {
        rhs.row(k).swap(rhs.row(rank - 1));
      }
    }
  }
  
  #ifndef EIGEN_PARSED_BY_DOXYGEN
  template <typename _MatrixType>
  template <typename RhsType, typename DstType>
  void CompleteOrthogonalDecomposition<_MatrixType>::_solve_impl(
      const RhsType& rhs, DstType& dst) const {
    eigen_assert(rhs.rows() == this->rows());
  
    const Index rank = this->rank();
    if (rank == 0) {
      dst.setZero();
      return;
    }
  
    // Compute c = Q^* * rhs
    // Note that the matrix Q = H_0^* H_1^*... so its inverse is
    // Q^* = (H_0 H_1 ...)^T
    typename RhsType::PlainObject c(rhs);
    c.applyOnTheLeft(
        householderSequence(matrixQTZ(), hCoeffs()).setLength(rank).transpose());
  
    // Solve T z = c(1:rank, :)
    dst.topRows(rank) = matrixT()
                            .topLeftCorner(rank, rank)
                            .template triangularView<Upper>()
                            .solve(c.topRows(rank));
  
    const Index cols = this->cols();
    if (rank < cols) {
      // Compute y = Z^* * [ z ]
      //                   [ 0 ]
      dst.bottomRows(cols - rank).setZero();
      applyZAdjointOnTheLeftInPlace(dst);
    }
  
    // Undo permutation to get x = P^{-1} * y.
    dst = colsPermutation() * dst;
  }
  #endif
  
  namespace internal {
  
  template<typename DstXprType, typename MatrixType>
  struct Assignment<DstXprType, Inverse<CompleteOrthogonalDecomposition<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename CompleteOrthogonalDecomposition<MatrixType>::Scalar>, Dense2Dense>
  {
    typedef CompleteOrthogonalDecomposition<MatrixType> CodType;
    typedef Inverse<CodType> SrcXprType;
    static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename CodType::Scalar> &)
    {
      dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.rows()));
    }
  };
  
  } // end namespace internal
  
  /** \returns the matrix Q as a sequence of householder transformations */
  template <typename MatrixType>
  typename CompleteOrthogonalDecomposition<MatrixType>::HouseholderSequenceType
  CompleteOrthogonalDecomposition<MatrixType>::householderQ() const {
    return m_cpqr.householderQ();
  }
  
  /** \return the complete orthogonal decomposition of \c *this.
    *
    * \sa class CompleteOrthogonalDecomposition
    */
  template <typename Derived>
  const CompleteOrthogonalDecomposition<typename MatrixBase<Derived>::PlainObject>
  MatrixBase<Derived>::completeOrthogonalDecomposition() const {
    return CompleteOrthogonalDecomposition<PlainObject>(eval());
  }
  
  }  // end namespace Eigen
  
  #endif  // EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H
  