// 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/.
/** \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:
/** \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$. * * If you want to come back to the default behavior, call setThreshold(Default_t)
*/
Derived& setThreshold(const RealScalar& threshold)
{
m_usePrescribedThreshold = true;
m_prescribedThreshold = threshold; return derived();
}
/** Allows to come back to the default behavior, letting Eigen use its default formula for * determining the threshold. * * You should pass the special object Eigen::Default as parameter here. * \code svd.setThreshold(Eigen::Default); \endcode * * See the documentation of setThreshold(const RealScalar&).
*/
Derived& setThreshold(Default_t)
{
m_usePrescribedThreshold = false; return derived();
}
/** Returns the threshold that will be used by certain methods such as rank(). * * See the documentation of setThreshold(const RealScalar&).
*/
RealScalar threshold() const
{
eigen_assert(m_isInitialized || m_usePrescribedThreshold); // this temporary is needed to workaround a MSVC issue
Index diagSize = (std::max<Index>)(1,m_diagSize); return m_usePrescribedThreshold ? m_prescribedThreshold
: RealScalar(diagSize)*NumTraits<Scalar>::epsilon();
}
/** \returns true if \a U (full or thin) is asked for in this SVD decomposition */ inlinebool computeU() const { return m_computeFullU || m_computeThinU; } /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */ inlinebool computeV() const { return m_computeFullV || m_computeThinV; }
inline Index rows() const { return m_rows; } inline Index cols() const { return m_cols; }
#ifdef EIGEN_PARSED_BY_DOXYGEN /** \returns a (least squares) solution of \f$ 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> inlineconst 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;
}
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, unsignedint computationOptions) ;
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.");
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