text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
  // Copyright (C) 2014 Gael Guennebaud <gael.guennebaud@inria.fr>
  //
  // Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
  // Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
  // Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
  // Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
  //
  // 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_SVDBASE_H
  #define EIGEN_SVDBASE_H
  
  namespace Eigen {
  /** \ingroup SVD_Module
   *
   *
   * \class SVDBase
   *
   * \brief Base class of SVD algorithms
   *
   * \tparam Derived the type of the actual SVD decomposition
   *
   * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
   *   \f[ A = U S V^* \f]
   * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
   * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
   * and right \em singular \em vectors of \a A respectively.
   *
   * Singular values are always sorted in decreasing order.
   *
   * 
   * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
   * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
   * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
   * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
   *  
   * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
   * terminate in finite (and reasonable) time.
   * \sa class BDCSVD, class JacobiSVD
   */
      template<typename Derived>
      class SVDBase {
  
      public:
          typedef typename internal::traits<Derived>::MatrixType MatrixType;
          typedef typename MatrixType::Scalar Scalar;
          typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
          typedef typename MatrixType::StorageIndex StorageIndex;
          typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
          enum {
              RowsAtCompileTime = MatrixType::RowsAtCompileTime,
              ColsAtCompileTime = MatrixType::ColsAtCompileTime,
              DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
              MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
              MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
              MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
              MatrixOptions = MatrixType::Options
          };
  
          typedef Matrix <Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixUType;
          typedef Matrix <Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> MatrixVType;
          typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
  
          Derived &derived() { return *static_cast<Derived *>(this); }
  
          const Derived &derived() const { return *static_cast<const Derived *>(this); }
  
          /** \returns the \a U matrix.
           *
           * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
           * the U matrix is n-by-n if you asked for \link Eigen::ComputeFullU ComputeFullU \endlink, and is n-by-m if you asked for \link Eigen::ComputeThinU ComputeThinU \endlink.
           *
           * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed.
           *
           * This method asserts that you asked for \a U to be computed.
           */
          const MatrixUType &matrixU() const {
              eigen_assert(m_isInitialized && "SVD is not initialized.");
              eigen_assert(computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
              return m_matrixU;
          }
  
          /** \returns the \a V matrix.
           *
           * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
           * the V matrix is p-by-p if you asked for \link Eigen::ComputeFullV ComputeFullV \endlink, and is p-by-m if you asked for \link Eigen::ComputeThinV ComputeThinV \endlink.
           *
           * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed.
           *
           * This method asserts that you asked for \a V to be computed.
           */
          const MatrixVType &matrixV() const {
              eigen_assert(m_isInitialized && "SVD is not initialized.");
              eigen_assert(computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
              return m_matrixV;
          }
  
          /** \returns the vector of singular values.
           *
           * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the
           * returned vector has size \a m.  Singular values are always sorted in decreasing order.
           */
          const SingularValuesType &singularValues() const {
              eigen_assert(m_isInitialized && "SVD is not initialized.");
              return m_singularValues;
          }
  
          /** \returns the number of singular values that are not exactly 0 */
          Index nonzeroSingularValues() const {
              eigen_assert(m_isInitialized && "SVD is not initialized.");
              return m_nonzeroSingularValues;
          }
  
          /** \returns the rank of the matrix of which \c *this is the SVD.
            *
            * \note This method has to determine which singular values should be considered nonzero.
            *       For that, it uses the threshold value that you can control by calling
            *       setThreshold(const RealScalar&).
            */
          inline Index rank() const {
              using std::abs;
              eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
              if (m_singularValues.size() == 0) return 0;
              RealScalar premultiplied_threshold = numext::maxi<RealScalar>(m_singularValues.coeff(0) * threshold(),
                                                                            (std::numeric_limits<RealScalar>::min)());
              Index i = m_nonzeroSingularValues - 1;
              while (i >= 0 && m_singularValues.coeff(i) < premultiplied_threshold) --i;
              return i + 1;
          }
  
          /** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(),
            * which need to determine when singular values are to be considered nonzero.
            * This is not used for the SVD decomposition itself.
            *
            * When it needs to get the threshold value, Eigen calls threshold().
            * The default is \c NumTraits<Scalar>::epsilon()
            *
            * \param threshold The new value to use as the threshold.
            *
            * A singular value will be considered nonzero if its value is strictly greater than
            *  \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$.
            *
            * If you want to come back to the default behavior, call setThreshold(Default_t)
            */
          Derived &setThreshold(const RealScalar &threshold) {
              m_usePrescribedThreshold = true;
              m_prescribedThreshold = threshold;
              return derived();
          }
  
          /** 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 svd.setThreshold(Eigen::Default); \endcode
            *
            * See the documentation of setThreshold(const RealScalar&).
            */
          Derived &setThreshold(Default_t) {
              m_usePrescribedThreshold = false;
              return derived();
          }
  
          /** Returns the threshold that will be used by certain methods such as rank().
            *
            * See the documentation of setThreshold(const RealScalar&).
            */
          RealScalar threshold() const {
              eigen_assert(m_isInitialized || m_usePrescribedThreshold);
              // this temporary is needed to workaround a MSVC issue
              Index diagSize = (std::max<Index>)(1, m_diagSize);
              return m_usePrescribedThreshold ? m_prescribedThreshold
                                              : RealScalar(diagSize) * NumTraits<Scalar>::epsilon();
          }
  
          /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
          inline bool computeU() const { return m_computeFullU || m_computeThinU; }
  
          /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
          inline bool computeV() const { return m_computeFullV || m_computeThinV; }
  
          inline Index rows() const { return m_rows; }
  
          inline Index cols() const { return m_cols; }
  
          /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
            *
            * \param b the right-hand-side of the equation to solve.
            *
            * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
            *
            * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
            * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
            */
          template<typename Rhs>
          inline const Solve <Derived, Rhs>
          solve(const MatrixBase <Rhs> &b) const {
              eigen_assert(m_isInitialized && "SVD is not initialized.");
              eigen_assert(computeU() && computeV() &&
                           "SVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
              return Solve<Derived, Rhs>(derived(), b.derived());
          }
  
  #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);
          }
  
          // return true if already allocated
          bool allocate(Index rows, Index cols, unsigned int computationOptions);
  
          MatrixUType m_matrixU;
          MatrixVType m_matrixV;
          SingularValuesType m_singularValues;
          bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold;
          bool m_computeFullU, m_computeThinU;
          bool m_computeFullV, m_computeThinV;
          unsigned int m_computationOptions;
          Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
          RealScalar m_prescribedThreshold;
  
          /** \brief Default Constructor.
           *
           * Default constructor of SVDBase
           */
          SVDBase()
                  : m_isInitialized(false),
                    m_isAllocated(false),
                    m_usePrescribedThreshold(false),
                    m_computationOptions(0),
                    m_rows(-1), m_cols(-1), m_diagSize(0) {
              check_template_parameters();
          }
  
  
      };
  
  #ifndef EIGEN_PARSED_BY_DOXYGEN
  
      template<typename Derived>
      template<typename RhsType, typename DstType>
      void SVDBase<Derived>::_solve_impl(const RhsType &rhs, DstType &dst) const {
          eigen_assert(rhs.rows() == rows());
  
          // A = U S V^*
          // So A^{-1} = V S^{-1} U^*
  
          Matrix<Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime, RhsType::MaxColsAtCompileTime> tmp;
          Index l_rank = rank();
          tmp.noalias() = m_matrixU.leftCols(l_rank).adjoint() * rhs;
          tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
          dst = m_matrixV.leftCols(l_rank) * tmp;
      }
  
  #endif
  
      template<typename MatrixType>
      bool SVDBase<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions) {
          eigen_assert(rows >= 0 && cols >= 0);
  
          if (m_isAllocated &&
              rows == m_rows &&
              cols == m_cols &&
              computationOptions == m_computationOptions) {
              return true;
          }
  
          m_rows = rows;
          m_cols = cols;
          m_isInitialized = false;
          m_isAllocated = true;
          m_computationOptions = computationOptions;
          m_computeFullU = (computationOptions & ComputeFullU) != 0;
          m_computeThinU = (computationOptions & ComputeThinU) != 0;
          m_computeFullV = (computationOptions & ComputeFullV) != 0;
          m_computeThinV = (computationOptions & ComputeThinV) != 0;
          eigen_assert(!(m_computeFullU && m_computeThinU) && "SVDBase: you can't ask for both full and thin U");
          eigen_assert(!(m_computeFullV && m_computeThinV) && "SVDBase: you can't ask for both full and thin V");
          eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime == Dynamic) &&
                       "SVDBase: thin U and V are only available when your matrix has a dynamic number of columns.");
  
          m_diagSize = (std::min)(m_rows, m_cols);
          m_singularValues.resize(m_diagSize);
          if (RowsAtCompileTime == Dynamic)
              m_matrixU.resize(m_rows, m_computeFullU ? m_rows : m_computeThinU ? m_diagSize : 0);
          if (ColsAtCompileTime == Dynamic)
              m_matrixV.resize(m_cols, m_computeFullV ? m_cols : m_computeThinV ? m_diagSize : 0);
  
          return false;
      }
  
  }// end namespace
  
  #endif // EIGEN_SVDBASE_H
  