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// 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 {
namespace internal {
template<typename Derived> struct traits<SVDBase<Derived> >
: traits<Derived>
{
typedef MatrixXpr XprKind;
typedef SolverStorage StorageKind;
typedef int StorageIndex;
enum { Flags = 0 };
};
}
/** \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 retrived 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;
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
{
_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$.
*
* If you want to come back to the default behavior, call setThreshold(Default_t)
*/
Derived& setThreshold(const RealScalar& threshold)
{
m_usePrescribedThreshold = true;
m_prescribedThreshold = threshold;
return derived();
}
/** Allows to come back to the default behavior, letting Eigen use its default formula for
* determining the threshold.
*
* You should pass the special object Eigen::Default as parameter here.
* \code svd.setThreshold(Eigen::Default); \endcode
*
* See the documentation of setThreshold(const RealScalar&).
*/
Derived& setThreshold(Default_t)
{
m_usePrescribedThreshold = false;
return derived();
}
/** Returns the threshold that will be used by certain methods such as rank().
*
* See the documentation of setThreshold(const RealScalar&).
*/
RealScalar threshold() const
{
eigen_assert(m_isInitialized || m_usePrescribedThreshold);
// this temporary is needed to workaround a MSVC issue
Index diagSize = (std::max<Index>)(1,m_diagSize);
return m_usePrescribedThreshold ? m_prescribedThreshold
: RealScalar(diagSize)*NumTraits<Scalar>::epsilon();
}
/** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
inline bool computeU() const { return m_computeFullU || m_computeThinU; }
/** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
inline bool computeV() const { return m_computeFullV || m_computeThinV; }
inline Index rows() const { return m_rows; }
inline Index cols() const { return m_cols; }
#ifdef EIGEN_PARSED_BY_DOXYGEN
/** \returns a (least squares) solution of \f$ 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:
static void check_template_parameters()
{
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, m_rows, m_cols, m_diagSize;
RealScalar m_prescribedThreshold;
/** \brief Default Constructor.
*
* Default constructor of SVDBase
*/
SVDBase()
: m_info(Success),
m_isInitialized(false),
m_isAllocated(false),
m_usePrescribedThreshold(false),
m_computeFullU(false),
m_computeThinU(false),
m_computeFullV(false),
m_computeThinV(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
{
// 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 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_info = Success;
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