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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
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//
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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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//
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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//
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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//
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_SVD_H
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#define EIGEN_SVD_H
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template<typename MatrixType, typename Rhs> struct ei_svd_solve_impl;
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/** \ingroup SVD_Module
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*
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*
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* \class SVD
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*
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* \brief Standard SVD decomposition of a matrix and associated features
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*
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* \param MatrixType the type of the matrix of which we are computing the SVD decomposition
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*
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* This class performs a standard SVD decomposition of a real matrix A of size \c M x \c N.
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*
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* Requires M >= N, in other words, at least as many rows as columns.
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*
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* \sa MatrixBase::SVD()
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*/
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template<typename _MatrixType> class SVD
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{
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public:
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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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typedef typename MatrixType::Index Index;
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enum {
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RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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PacketSize = ei_packet_traits<Scalar>::size,
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AlignmentMask = int(PacketSize)-1,
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MinSize = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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MatrixOptions = MatrixType::Options
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};
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typedef typename ei_plain_col_type<MatrixType>::type ColVector;
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typedef typename ei_plain_row_type<MatrixType>::type RowVector;
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typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixUType;
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typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> MatrixVType;
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typedef ColVector SingularValuesType;
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/**
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* \brief Default Constructor.
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*
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via SVD::compute(const MatrixType&).
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*/
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SVD() : m_matU(), m_matV(), m_sigma(), m_isInitialized(false) {}
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/** \brief Default Constructor with memory preallocation
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*
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* Like the default constructor but with preallocation of the internal data
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* according to the specified problem \a size.
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* \sa JacobiSVD()
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*/
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SVD(Index rows, Index cols) : m_matU(rows, rows),
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m_matV(cols,cols),
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m_sigma(std::min(rows, cols)),
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m_workMatrix(rows, cols),
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m_rv1(cols),
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m_isInitialized(false)
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{
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ei_assert(rows >= cols && "SVD is only defined if rows>=cols.");
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}
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SVD(const MatrixType& matrix) : m_matU(matrix.rows(), matrix.rows()),
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m_matV(matrix.cols(),matrix.cols()),
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m_sigma(std::min(matrix.rows(), matrix.cols())),
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m_workMatrix(matrix.rows(), matrix.cols()),
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m_rv1(matrix.cols()),
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m_isInitialized(false)
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{
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compute(matrix);
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}
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/** \returns a solution of \f$ A x = b \f$ using the current SVD decomposition of A.
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*
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* \param b the right-hand-side of the equation to solve.
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*
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* \note_about_checking_solutions
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*
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* \note_about_arbitrary_choice_of_solution
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*
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* \sa MatrixBase::svd(),
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*/
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template<typename Rhs>
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inline const ei_solve_retval<SVD, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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{
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ei_assert(m_isInitialized && "SVD is not initialized.");
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return ei_solve_retval<SVD, Rhs>(*this, b.derived());
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}
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const MatrixUType& matrixU() const
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{
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ei_assert(m_isInitialized && "SVD is not initialized.");
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return m_matU;
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}
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const SingularValuesType& singularValues() const
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{
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ei_assert(m_isInitialized && "SVD is not initialized.");
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return m_sigma;
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}
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const MatrixVType& matrixV() const
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{
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ei_assert(m_isInitialized && "SVD is not initialized.");
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return m_matV;
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}
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SVD& compute(const MatrixType& matrix);
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template<typename UnitaryType, typename PositiveType>
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void computeUnitaryPositive(UnitaryType *unitary, PositiveType *positive) const;
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template<typename PositiveType, typename UnitaryType>
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void computePositiveUnitary(PositiveType *positive, UnitaryType *unitary) const;
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template<typename RotationType, typename ScalingType>
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void computeRotationScaling(RotationType *unitary, ScalingType *positive) const;
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template<typename ScalingType, typename RotationType>
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void computeScalingRotation(ScalingType *positive, RotationType *unitary) const;
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inline Index rows() const
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{
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ei_assert(m_isInitialized && "SVD is not initialized.");
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return m_rows;
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}
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inline Index cols() const
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{
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ei_assert(m_isInitialized && "SVD is not initialized.");
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return m_cols;
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}
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protected:
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// Computes (a^2 + b^2)^(1/2) without destructive underflow or overflow.
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inline static Scalar pythag(Scalar a, Scalar b)
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{
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Scalar abs_a = ei_abs(a);
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Scalar abs_b = ei_abs(b);
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if (abs_a > abs_b)
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return abs_a*ei_sqrt(Scalar(1.0)+ei_abs2(abs_b/abs_a));
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else
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return (abs_b == Scalar(0.0) ? Scalar(0.0) : abs_b*ei_sqrt(Scalar(1.0)+ei_abs2(abs_a/abs_b)));
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}
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inline static Scalar sign(Scalar a, Scalar b)
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{
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return (b >= Scalar(0.0) ? ei_abs(a) : -ei_abs(a));
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}
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protected:
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/** \internal */
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MatrixUType m_matU;
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/** \internal */
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MatrixVType m_matV;
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/** \internal */
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SingularValuesType m_sigma;
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MatrixType m_workMatrix;
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RowVector m_rv1;
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bool m_isInitialized;
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Index m_rows, m_cols;
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};
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/** Computes / recomputes the SVD decomposition A = U S V^* of \a matrix
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*
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* \note this code has been adapted from Numerical Recipes, third edition.
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*
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* \returns a reference to *this
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*/
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template<typename MatrixType>
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SVD<MatrixType>& SVD<MatrixType>::compute(const MatrixType& matrix)
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{
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const Index m = m_rows = matrix.rows();
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const Index n = m_cols = matrix.cols();
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m_matU.resize(m, m);
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m_matU.setZero();
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m_sigma.resize(n);
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m_matV.resize(n,n);
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m_workMatrix = matrix;
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Index max_iters = 30;
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MatrixVType& V = m_matV;
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MatrixType& A = m_workMatrix;
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SingularValuesType& W = m_sigma;
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bool flag;
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Index i=0,its=0,j=0,k=0,l=0,nm=0;
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Scalar anorm, c, f, g, h, s, scale, x, y, z;
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bool convergence = true;
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Scalar eps = NumTraits<Scalar>::dummy_precision();
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m_rv1.resize(n);
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g = scale = anorm = 0;
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// Householder reduction to bidiagonal form.
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for (i=0; i<n; i++)
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{
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l = i+2;
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m_rv1[i] = scale*g;
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g = s = scale = 0.0;
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if (i < m)
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{
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scale = A.col(i).tail(m-i).cwiseAbs().sum();
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if (scale != Scalar(0))
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{
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for (k=i; k<m; k++)
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{
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A(k, i) /= scale;
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s += A(k, i)*A(k, i);
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}
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f = A(i, i);
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g = -sign( ei_sqrt(s), f );
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h = f*g - s;
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A(i, i)=f-g;
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for (j=l-1; j<n; j++)
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{
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s = A.col(j).tail(m-i).dot(A.col(i).tail(m-i));
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f = s/h;
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A.col(j).tail(m-i) += f*A.col(i).tail(m-i);
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}
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A.col(i).tail(m-i) *= scale;
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}
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}
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W[i] = scale * g;
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g = s = scale = 0.0;
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if (i+1 <= m && i+1 != n)
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{
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scale = A.row(i).tail(n-l+1).cwiseAbs().sum();
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if (scale != Scalar(0))
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{
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for (k=l-1; k<n; k++)
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{
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A(i, k) /= scale;
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s += A(i, k)*A(i, k);
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}
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f = A(i,l-1);
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g = -sign(ei_sqrt(s),f);
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h = f*g - s;
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A(i,l-1) = f-g;
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m_rv1.tail(n-l+1) = A.row(i).tail(n-l+1)/h;
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for (j=l-1; j<m; j++)
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{
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s = A.row(i).tail(n-l+1).dot(A.row(j).tail(n-l+1));
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A.row(j).tail(n-l+1) += s*m_rv1.tail(n-l+1).transpose();
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}
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A.row(i).tail(n-l+1) *= scale;
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}
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}
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anorm = std::max( anorm, (ei_abs(W[i])+ei_abs(m_rv1[i])) );
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}
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// Accumulation of right-hand transformations.
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for (i=n-1; i>=0; i--)
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{
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//Accumulation of right-hand transformations.
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if (i < n-1)
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{
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if (g != Scalar(0.0))
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{
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for (j=l; j<n; j++) //Double division to avoid possible underflow.
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V(j, i) = (A(i, j)/A(i, l))/g;
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for (j=l; j<n; j++)
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{
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s = V.col(j).tail(n-l).dot(A.row(i).tail(n-l));
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V.col(j).tail(n-l) += s * V.col(i).tail(n-l);
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}
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}
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V.row(i).tail(n-l).setZero();
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V.col(i).tail(n-l).setZero();
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}
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V(i, i) = 1.0;
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g = m_rv1[i];
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l = i;
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}
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// Accumulation of left-hand transformations.
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for (i=std::min(m,n)-1; i>=0; i--)
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{
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l = i+1;
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g = W[i];
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if (n-l>0)
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A.row(i).tail(n-l).setZero();
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if (g != Scalar(0.0))
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{
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g = Scalar(1.0)/g;
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if (m-l)
|
|
|
|
|
{
|
|
|
|
|
for (j=l; j<n; j++)
|
|
|
|
|
{
|
|
|
|
|
s = A.col(j).tail(m-l).dot(A.col(i).tail(m-l));
|
|
|
|
|
f = (s/A(i,i))*g;
|
|
|
|
|
A.col(j).tail(m-i) += f * A.col(i).tail(m-i);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
A.col(i).tail(m-i) *= g;
|
|
|
|
|
}
|
|
|
|
|
else
|
|
|
|
|
A.col(i).tail(m-i).setZero();
|
|
|
|
|
++A(i,i);
|
|
|
|
|
}
|
|
|
|
|
// Diagonalization of the bidiagonal form: Loop over
|
|
|
|
|
// singular values, and over allowed iterations.
|
|
|
|
|
for (k=n-1; k>=0; k--)
|
|
|
|
|
{
|
|
|
|
|
for (its=0; its<max_iters; its++)
|
|
|
|
|
{
|
|
|
|
|
flag = true;
|
|
|
|
|
for (l=k; l>=0; l--)
|
|
|
|
|
{
|
|
|
|
|
// Test for splitting.
|
|
|
|
|
nm = l-1;
|
|
|
|
|
// Note that rv1[1] is always zero.
|
|
|
|
|
//if ((double)(ei_abs(rv1[l])+anorm) == anorm)
|
|
|
|
|
if (l==0 || ei_abs(m_rv1[l]) <= eps*anorm)
|
|
|
|
|
{
|
|
|
|
|
flag = false;
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
//if ((double)(ei_abs(W[nm])+anorm) == anorm)
|
|
|
|
|
if (ei_abs(W[nm]) <= eps*anorm)
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
if (flag)
|
|
|
|
|
{
|
|
|
|
|
c = 0.0; //Cancellation of rv1[l], if l > 0.
|
|
|
|
|
s = 1.0;
|
|
|
|
|
for (i=l ;i<k+1; i++)
|
|
|
|
|
{
|
|
|
|
|
f = s*m_rv1[i];
|
|
|
|
|
m_rv1[i] = c*m_rv1[i];
|
|
|
|
|
//if ((double)(ei_abs(f)+anorm) == anorm)
|
|
|
|
|
if (ei_abs(f) <= eps*anorm)
|
|
|
|
|
break;
|
|
|
|
|
g = W[i];
|
|
|
|
|
h = pythag(f,g);
|
|
|
|
|
W[i] = h;
|
|
|
|
|
h = Scalar(1.0)/h;
|
|
|
|
|
c = g*h;
|
|
|
|
|
s = -f*h;
|
|
|
|
|
V.applyOnTheRight(i,nm,PlanarRotation<Scalar>(c,s));
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
z = W[k];
|
|
|
|
|
if (l == k) //Convergence.
|
|
|
|
|
{
|
|
|
|
|
if (z < 0.0) { // Singular value is made nonnegative.
|
|
|
|
|
W[k] = -z;
|
|
|
|
|
V.col(k) = -V.col(k);
|
|
|
|
|
}
|
|
|
|
|
break;
|
|
|
|
|
}
|
|
|
|
|
if (its+1 == max_iters)
|
|
|
|
|
{
|
|
|
|
|
convergence = false;
|
|
|
|
|
}
|
|
|
|
|
x = W[l]; // Shift from bottom 2-by-2 minor.
|
|
|
|
|
nm = k-1;
|
|
|
|
|
y = W[nm];
|
|
|
|
|
g = m_rv1[nm];
|
|
|
|
|
h = m_rv1[k];
|
|
|
|
|
f = ((y-z)*(y+z) + (g-h)*(g+h))/(Scalar(2.0)*h*y);
|
|
|
|
|
g = pythag(f,1.0);
|
|
|
|
|
f = ((x-z)*(x+z) + h*((y/(f+sign(g,f)))-h))/x;
|
|
|
|
|
c = s = 1.0;
|
|
|
|
|
//Next QR transformation:
|
|
|
|
|
for (j=l; j<=nm; j++)
|
|
|
|
|
{
|
|
|
|
|
i = j+1;
|
|
|
|
|
g = m_rv1[i];
|
|
|
|
|
y = W[i];
|
|
|
|
|
h = s*g;
|
|
|
|
|
g = c*g;
|
|
|
|
|
|
|
|
|
|
z = pythag(f,h);
|
|
|
|
|
m_rv1[j] = z;
|
|
|
|
|
c = f/z;
|
|
|
|
|
s = h/z;
|
|
|
|
|
f = x*c + g*s;
|
|
|
|
|
g = g*c - x*s;
|
|
|
|
|
h = y*s;
|
|
|
|
|
y *= c;
|
|
|
|
|
V.applyOnTheRight(i,j,PlanarRotation<Scalar>(c,s));
|
|
|
|
|
|
|
|
|
|
z = pythag(f,h);
|
|
|
|
|
W[j] = z;
|
|
|
|
|
// Rotation can be arbitrary if z = 0.
|
|
|
|
|
if (z!=Scalar(0))
|
|
|
|
|
{
|
|
|
|
|
z = Scalar(1.0)/z;
|
|
|
|
|
c = f*z;
|
|
|
|
|
s = h*z;
|
|
|
|
|
}
|
|
|
|
|
f = c*g + s*y;
|
|
|
|
|
x = c*y - s*g;
|
|
|
|
|
A.applyOnTheRight(i,j,PlanarRotation<Scalar>(c,s));
|
|
|
|
|
}
|
|
|
|
|
m_rv1[l] = 0.0;
|
|
|
|
|
m_rv1[k] = f;
|
|
|
|
|
W[k] = x;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// sort the singular values:
|
|
|
|
|
{
|
|
|
|
|
for (Index i=0; i<n; i++)
|
|
|
|
|
{
|
|
|
|
|
Index k;
|
|
|
|
|
W.tail(n-i).maxCoeff(&k);
|
|
|
|
|
if (k != 0)
|
|
|
|
|
{
|
|
|
|
|
k += i;
|
|
|
|
|
std::swap(W[k],W[i]);
|
|
|
|
|
A.col(i).swap(A.col(k));
|
|
|
|
|
V.col(i).swap(V.col(k));
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
m_matU.leftCols(n) = A;
|
|
|
|
|
m_matU.rightCols(m-n).setZero();
|
|
|
|
|
|
|
|
|
|
// Gram Schmidt orthogonalization to fill up U
|
|
|
|
|
for (int col = A.cols(); col < A.rows(); ++col)
|
|
|
|
|
{
|
|
|
|
|
typename MatrixUType::ColXpr colVec = m_matU.col(col);
|
|
|
|
|
colVec(col) = 1;
|
|
|
|
|
for (int prevCol = 0; prevCol < col; ++prevCol)
|
|
|
|
|
{
|
|
|
|
|
typename MatrixUType::ColXpr prevColVec = m_matU.col(prevCol);
|
|
|
|
|
colVec -= colVec.dot(prevColVec)*prevColVec;
|
|
|
|
|
}
|
|
|
|
|
// Here we can run into troubles when colVec.norm() = 0.
|
|
|
|
|
m_matU.col(col) = colVec.normalized();
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
m_isInitialized = true;
|
|
|
|
|
return *this;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
template<typename _MatrixType, typename Rhs>
|
|
|
|
|
struct ei_solve_retval<SVD<_MatrixType>, Rhs>
|
|
|
|
|
: ei_solve_retval_base<SVD<_MatrixType>, Rhs>
|
|
|
|
|
{
|
|
|
|
|
EIGEN_MAKE_SOLVE_HELPERS(SVD<_MatrixType>,Rhs)
|
|
|
|
|
|
|
|
|
|
template<typename Dest> void evalTo(Dest& dst) const
|
|
|
|
|
{
|
|
|
|
|
ei_assert(rhs().rows() == dec().rows());
|
|
|
|
|
|
|
|
|
|
for (Index j=0; j<cols(); ++j)
|
|
|
|
|
{
|
|
|
|
|
Matrix<Scalar,MatrixType::RowsAtCompileTime,1> aux = dec().matrixU().adjoint() * rhs().col(j);
|
|
|
|
|
|
|
|
|
|
for (Index i = 0; i < dec().singularValues().size(); ++i)
|
|
|
|
|
{
|
|
|
|
|
Scalar si = dec().singularValues().coeff(i);
|
|
|
|
|
if(si == RealScalar(0))
|
|
|
|
|
aux.coeffRef(i) = Scalar(0);
|
|
|
|
|
else
|
|
|
|
|
aux.coeffRef(i) /= si;
|
|
|
|
|
}
|
|
|
|
|
aux.tail(aux.size() - dec().singularValues().size()).setZero();
|
|
|
|
|
|
|
|
|
|
const Index minsize = std::min(dec().rows(),dec().cols());
|
|
|
|
|
dst.col(j).head(minsize) = aux.head(minsize);
|
|
|
|
|
if(dec().cols()>dec().rows()) dst.col(j).tail(cols()-minsize).setZero();
|
|
|
|
|
dst.col(j) = dec().matrixV() * dst.col(j);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
/** Computes the polar decomposition of the matrix, as a product unitary x positive.
|
|
|
|
|
*
|
|
|
|
|
* If either pointer is zero, the corresponding computation is skipped.
|
|
|
|
|
*
|
|
|
|
|
* Only for square matrices.
|
|
|
|
|
*
|
|
|
|
|
* \sa computePositiveUnitary(), computeRotationScaling()
|
|
|
|
|
*/
|
|
|
|
|
template<typename MatrixType>
|
|
|
|
|
template<typename UnitaryType, typename PositiveType>
|
|
|
|
|
void SVD<MatrixType>::computeUnitaryPositive(UnitaryType *unitary,
|
|
|
|
|
PositiveType *positive) const
|
|
|
|
|
{
|
|
|
|
|
ei_assert(m_isInitialized && "SVD is not initialized.");
|
|
|
|
|
ei_assert(m_matU.cols() == m_matV.cols() && "Polar decomposition is only for square matrices");
|
|
|
|
|
if(unitary) *unitary = m_matU * m_matV.adjoint();
|
|
|
|
|
if(positive) *positive = m_matV * m_sigma.asDiagonal() * m_matV.adjoint();
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/** Computes the polar decomposition of the matrix, as a product positive x unitary.
|
|
|
|
|
*
|
|
|
|
|
* If either pointer is zero, the corresponding computation is skipped.
|
|
|
|
|
*
|
|
|
|
|
* Only for square matrices.
|
|
|
|
|
*
|
|
|
|
|
* \sa computeUnitaryPositive(), computeRotationScaling()
|
|
|
|
|
*/
|
|
|
|
|
template<typename MatrixType>
|
|
|
|
|
template<typename UnitaryType, typename PositiveType>
|
|
|
|
|
void SVD<MatrixType>::computePositiveUnitary(UnitaryType *positive,
|
|
|
|
|
PositiveType *unitary) const
|
|
|
|
|
{
|
|
|
|
|
ei_assert(m_isInitialized && "SVD is not initialized.");
|
|
|
|
|
ei_assert(m_matU.rows() == m_matV.rows() && "Polar decomposition is only for square matrices");
|
|
|
|
|
if(unitary) *unitary = m_matU * m_matV.adjoint();
|
|
|
|
|
if(positive) *positive = m_matU * m_sigma.asDiagonal() * m_matU.adjoint();
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/** decomposes the matrix as a product rotation x scaling, the scaling being
|
|
|
|
|
* not necessarily positive.
|
|
|
|
|
*
|
|
|
|
|
* If either pointer is zero, the corresponding computation is skipped.
|
|
|
|
|
*
|
|
|
|
|
* This method requires the Geometry module.
|
|
|
|
|
*
|
|
|
|
|
* \sa computeScalingRotation(), computeUnitaryPositive()
|
|
|
|
|
*/
|
|
|
|
|
template<typename MatrixType>
|
|
|
|
|
template<typename RotationType, typename ScalingType>
|
|
|
|
|
void SVD<MatrixType>::computeRotationScaling(RotationType *rotation, ScalingType *scaling) const
|
|
|
|
|
{
|
|
|
|
|
ei_assert(m_isInitialized && "SVD is not initialized.");
|
|
|
|
|
ei_assert(m_matU.rows() == m_matV.rows() && "Polar decomposition is only for square matrices");
|
|
|
|
|
Scalar x = (m_matU * m_matV.adjoint()).determinant(); // so x has absolute value 1
|
|
|
|
|
Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> sv(m_sigma);
|
|
|
|
|
sv.coeffRef(0) *= x;
|
|
|
|
|
if(scaling) scaling->lazyAssign(m_matV * sv.asDiagonal() * m_matV.adjoint());
|
|
|
|
|
if(rotation)
|
|
|
|
|
{
|
|
|
|
|
MatrixType m(m_matU);
|
|
|
|
|
m.col(0) /= x;
|
|
|
|
|
rotation->lazyAssign(m * m_matV.adjoint());
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/** decomposes the matrix as a product scaling x rotation, the scaling being
|
|
|
|
|
* not necessarily positive.
|
|
|
|
|
*
|
|
|
|
|
* If either pointer is zero, the corresponding computation is skipped.
|
|
|
|
|
*
|
|
|
|
|
* This method requires the Geometry module.
|
|
|
|
|
*
|
|
|
|
|
* \sa computeRotationScaling(), computeUnitaryPositive()
|
|
|
|
|
*/
|
|
|
|
|
template<typename MatrixType>
|
|
|
|
|
template<typename ScalingType, typename RotationType>
|
|
|
|
|
void SVD<MatrixType>::computeScalingRotation(ScalingType *scaling, RotationType *rotation) const
|
|
|
|
|
{
|
|
|
|
|
ei_assert(m_isInitialized && "SVD is not initialized.");
|
|
|
|
|
ei_assert(m_matU.rows() == m_matV.rows() && "Polar decomposition is only for square matrices");
|
|
|
|
|
Scalar x = (m_matU * m_matV.adjoint()).determinant(); // so x has absolute value 1
|
|
|
|
|
Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> sv(m_sigma);
|
|
|
|
|
sv.coeffRef(0) *= x;
|
|
|
|
|
if(scaling) scaling->lazyAssign(m_matU * sv.asDiagonal() * m_matU.adjoint());
|
|
|
|
|
if(rotation)
|
|
|
|
|
{
|
|
|
|
|
MatrixType m(m_matU);
|
|
|
|
|
m.col(0) /= x;
|
|
|
|
|
rotation->lazyAssign(m * m_matV.adjoint());
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
/** \svd_module
|
|
|
|
|
* \returns the SVD decomposition of \c *this
|
|
|
|
|
*/
|
|
|
|
|
template<typename Derived>
|
|
|
|
|
inline SVD<typename MatrixBase<Derived>::PlainObject>
|
|
|
|
|
MatrixBase<Derived>::svd() const
|
|
|
|
|
{
|
|
|
|
|
return SVD<PlainObject>(derived());
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
#endif // EIGEN_SVD_H
|
Benoit Jacob