Eigen/src/SVD/SVDBase.h

// 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

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

namespace Eigen {

namespace internal {

enum OptionsMasks {
  QRPreconditionerBits = NoQRPreconditioner | HouseholderQRPreconditioner | ColPivHouseholderQRPreconditioner |
                         FullPivHouseholderQRPreconditioner,
  ComputationOptionsBits = ComputeThinU | ComputeFullU | ComputeThinV | ComputeFullV
};

constexpr int get_qr_preconditioner(int options) { return options & QRPreconditionerBits; }

constexpr int get_computation_options(int options) { return options & ComputationOptionsBits; }

constexpr bool should_svd_compute_thin_u(int options) { return (options & ComputeThinU) != 0; }
constexpr bool should_svd_compute_full_u(int options) { return (options & ComputeFullU) != 0; }
constexpr bool should_svd_compute_thin_v(int options) { return (options & ComputeThinV) != 0; }
constexpr bool should_svd_compute_full_v(int options) { return (options & ComputeFullV) != 0; }

template <typename MatrixType, int Options>
void check_svd_options_assertions(unsigned int computationOptions, Index rows, Index cols) {
  EIGEN_STATIC_ASSERT((Options & ComputationOptionsBits) == 0,
                      "SVDBase: Cannot request U or V using both static and runtime options, even if they match. "
                      "Requesting unitaries at runtime is DEPRECATED: "
                      "Prefer requesting unitaries statically, using the Options template parameter.");
  eigen_assert(
      !(should_svd_compute_thin_u(computationOptions) && cols < rows && MatrixType::RowsAtCompileTime != Dynamic) &&
      !(should_svd_compute_thin_v(computationOptions) && rows < cols && MatrixType::ColsAtCompileTime != Dynamic) &&
      "SVDBase: If thin U is requested at runtime, your matrix must have more rows than columns or a dynamic number of "
      "rows."
      "Similarly, if thin V is requested at runtime, you matrix must have more columns than rows or a dynamic number "
      "of columns.");
  (void)computationOptions;
  (void)rows;
  (void)cols;
}

template <typename Derived>
struct traits<SVDBase<Derived> > : traits<Derived> {
  typedef MatrixXpr XprKind;
  typedef SolverStorage StorageKind;
  typedef int StorageIndex;
  enum { Flags = 0 };
};

template <typename MatrixType, int Options_>
struct svd_traits : traits<MatrixType> {
  static constexpr int Options = Options_;
  static constexpr bool ShouldComputeFullU = internal::should_svd_compute_full_u(Options);
  static constexpr bool ShouldComputeThinU = internal::should_svd_compute_thin_u(Options);
  static constexpr bool ShouldComputeFullV = internal::should_svd_compute_full_v(Options);
  static constexpr bool ShouldComputeThinV = internal::should_svd_compute_thin_v(Options);
  enum {
    DiagSizeAtCompileTime =
        internal::min_size_prefer_dynamic(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime),
    MaxDiagSizeAtCompileTime =
        internal::min_size_prefer_dynamic(MatrixType::MaxRowsAtCompileTime, MatrixType::MaxColsAtCompileTime),
    MatrixUColsAtCompileTime = ShouldComputeThinU ? DiagSizeAtCompileTime : MatrixType::RowsAtCompileTime,
    MatrixVColsAtCompileTime = ShouldComputeThinV ? DiagSizeAtCompileTime : MatrixType::ColsAtCompileTime,
    MatrixUMaxColsAtCompileTime = ShouldComputeThinU ? MaxDiagSizeAtCompileTime : MatrixType::MaxRowsAtCompileTime,
    MatrixVMaxColsAtCompileTime = ShouldComputeThinV ? MaxDiagSizeAtCompileTime : MatrixType::MaxColsAtCompileTime
  };
};
}  // namespace internal

/** \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.
 *
 * The status of the computation can be retrieved using the \a info() method. Unless \a info() returns \a Success, the
 * results should be not considered well defined.
 *
 * If the input matrix has inf or nan coefficients, the result of the computation is undefined, and \a info() will
 * return \a InvalidInput, but the computation is guaranteed to terminate in finite (and reasonable) time. \sa class
 * BDCSVD, class JacobiSVD
 */
template <typename Derived>
class SVDBase : public SolverBase<SVDBase<Derived> > {
 public:
  template <typename Derived_>
  friend struct internal::solve_assertion;

  typedef typename internal::traits<Derived>::MatrixType MatrixType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
  typedef typename Eigen::internal::traits<SVDBase>::StorageIndex StorageIndex;

  static constexpr bool ShouldComputeFullU = internal::traits<Derived>::ShouldComputeFullU;
  static constexpr bool ShouldComputeThinU = internal::traits<Derived>::ShouldComputeThinU;
  static constexpr bool ShouldComputeFullV = internal::traits<Derived>::ShouldComputeFullV;
  static constexpr bool ShouldComputeThinV = internal::traits<Derived>::ShouldComputeThinV;

  enum {
    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
    DiagSizeAtCompileTime = internal::min_size_prefer_dynamic(RowsAtCompileTime, ColsAtCompileTime),
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
    MaxDiagSizeAtCompileTime = internal::min_size_prefer_fixed(MaxRowsAtCompileTime, MaxColsAtCompileTime),
    MatrixOptions = internal::traits<MatrixType>::Options,
    MatrixUColsAtCompileTime = internal::traits<Derived>::MatrixUColsAtCompileTime,
    MatrixVColsAtCompileTime = internal::traits<Derived>::MatrixVColsAtCompileTime,
    MatrixUMaxColsAtCompileTime = internal::traits<Derived>::MatrixUMaxColsAtCompileTime,
    MatrixVMaxColsAtCompileTime = internal::traits<Derived>::MatrixVMaxColsAtCompileTime
  };

  EIGEN_STATIC_ASSERT(!(ShouldComputeFullU && ShouldComputeThinU), "SVDBase: Cannot request both full and thin U")
  EIGEN_STATIC_ASSERT(!(ShouldComputeFullV && ShouldComputeThinV), "SVDBase: Cannot request both full and thin V")

  typedef
      typename internal::make_proper_matrix_type<Scalar, RowsAtCompileTime, MatrixUColsAtCompileTime, MatrixOptions,
                                                 MaxRowsAtCompileTime, MatrixUMaxColsAtCompileTime>::type MatrixUType;
  typedef
      typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, MatrixVColsAtCompileTime, MatrixOptions,
                                                 MaxColsAtCompileTime, MatrixVMaxColsAtCompileTime>::type 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 {
    _check_compute_assertions();
    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 {
    _check_compute_assertions();
    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 {
    _check_compute_assertions();
    return m_singularValues;
  }

  /** \returns the number of singular values that are not exactly 0 */
  Index nonzeroSingularValues() const {
    _check_compute_assertions();
    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;
    _check_compute_assertions();
    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.value(); }
  inline Index cols() const { return m_cols.value(); }
  inline Index diagSize() const { return m_diagSize.value(); }

#ifdef EIGEN_PARSED_BY_DOXYGEN
  /** \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;
#endif

  /** \brief Reports whether previous computation was successful.
   *
   * \returns \c Success if computation was successful.
   */
  EIGEN_DEVICE_FUNC ComputationInfo info() const {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    return m_info;
  }

#ifndef EIGEN_PARSED_BY_DOXYGEN
  template <typename RhsType, typename DstType>
  void _solve_impl(const RhsType& rhs, DstType& dst) const;

  template <bool Conjugate, typename RhsType, typename DstType>
  void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

 protected:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  void _check_compute_assertions() const { eigen_assert(m_isInitialized && "SVD is not initialized."); }

  template <bool Transpose_, typename Rhs>
  void _check_solve_assertion(const Rhs& b) const {
    EIGEN_ONLY_USED_FOR_DEBUG(b);
    _check_compute_assertions();
    eigen_assert(computeU() && computeV() &&
                 "SVDBase::solve(): Both unitaries U and V are required to be computed (thin unitaries suffice).");
    eigen_assert((Transpose_ ? cols() : rows()) == b.rows() &&
                 "SVDBase::solve(): invalid number of rows of the right hand side matrix b");
  }

  // return true if already allocated
  bool allocate(Index rows, Index cols, unsigned int computationOptions);

  MatrixUType m_matrixU;
  MatrixVType m_matrixV;
  SingularValuesType m_singularValues;
  ComputationInfo m_info;
  bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold;
  bool m_computeFullU, m_computeThinU;
  bool m_computeFullV, m_computeThinV;
  unsigned int m_computationOptions;
  Index m_nonzeroSingularValues;
  internal::variable_if_dynamic<Index, RowsAtCompileTime> m_rows;
  internal::variable_if_dynamic<Index, ColsAtCompileTime> m_cols;
  internal::variable_if_dynamic<Index, DiagSizeAtCompileTime> m_diagSize;
  RealScalar m_prescribedThreshold;

  /** \brief Default Constructor.
   *
   * Default constructor of SVDBase
   */
  SVDBase()
      : m_matrixU(MatrixUType()),
        m_matrixV(MatrixVType()),
        m_singularValues(SingularValuesType()),
        m_info(Success),
        m_isInitialized(false),
        m_isAllocated(false),
        m_usePrescribedThreshold(false),
        m_computeFullU(ShouldComputeFullU),
        m_computeThinU(ShouldComputeThinU),
        m_computeFullV(ShouldComputeFullV),
        m_computeThinV(ShouldComputeThinV),
        m_computationOptions(internal::traits<Derived>::Options),
        m_nonzeroSingularValues(0),
        m_rows(RowsAtCompileTime),
        m_cols(ColsAtCompileTime),
        m_diagSize(DiagSizeAtCompileTime),
        m_prescribedThreshold(0) {}
};

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename Derived>
template <typename RhsType, typename DstType>
void SVDBase<Derived>::_solve_impl(const RhsType& rhs, DstType& dst) const {
  // A = U S V^*
  // So A^{-1} = V S^{-1} U^*

  Matrix<typename RhsType::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;
}

template <typename Derived>
template <bool Conjugate, typename RhsType, typename DstType>
void SVDBase<Derived>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const {
  // A = U S V^*
  // So  A^{-*} = U S^{-1} V^*
  // And A^{-T} = U_conj S^{-1} V^T
  Matrix<typename RhsType::Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime,
         RhsType::MaxColsAtCompileTime>
      tmp;
  Index l_rank = rank();

  tmp.noalias() = m_matrixV.leftCols(l_rank).transpose().template conjugateIf<Conjugate>() * rhs;
  tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
  dst = m_matrixU.template conjugateIf<!Conjugate>().leftCols(l_rank) * tmp;
}
#endif

template <typename Derived>
bool SVDBase<Derived>::allocate(Index rows, Index cols, unsigned int computationOptions) {
  eigen_assert(rows >= 0 && cols >= 0);

  if (m_isAllocated && rows == m_rows.value() && cols == m_cols.value() && computationOptions == m_computationOptions) {
    return true;
  }

  m_rows.setValue(rows);
  m_cols.setValue(cols);
  m_info = Success;
  m_isInitialized = false;
  m_isAllocated = true;
  m_computationOptions = computationOptions;
  m_computeFullU = ShouldComputeFullU || internal::should_svd_compute_full_u(computationOptions);
  m_computeThinU = ShouldComputeThinU || internal::should_svd_compute_thin_u(computationOptions);
  m_computeFullV = ShouldComputeFullV || internal::should_svd_compute_full_v(computationOptions);
  m_computeThinV = ShouldComputeThinV || internal::should_svd_compute_thin_v(computationOptions);

  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");

  m_diagSize.setValue(numext::mini(m_rows.value(), m_cols.value()));
  m_singularValues.resize(m_diagSize.value());
  if (RowsAtCompileTime == Dynamic)
    m_matrixU.resize(m_rows.value(), m_computeFullU ? m_rows.value() : m_computeThinU ? m_diagSize.value() : 0);
  if (ColsAtCompileTime == Dynamic)
    m_matrixV.resize(m_cols.value(), m_computeFullV ? m_cols.value() : m_computeThinV ? m_diagSize.value() : 0);

  return false;
}

}  // namespace Eigen

#endif  // EIGEN_SVDBASE_H