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638 lines
18 KiB
638 lines
18 KiB
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra. Eigen itself is part of the KDE project.
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//
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// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN2_SVD_H
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#define EIGEN2_SVD_H
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namespace Eigen {
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/** \ingroup SVD_Module
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* \nonstableyet
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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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* with \c M \>= \c N.
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*
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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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private:
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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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enum {
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PacketSize = internal::packet_traits<Scalar>::size,
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AlignmentMask = int(PacketSize)-1,
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MinSize = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)
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};
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typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVector;
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typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> RowVector;
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typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MinSize> MatrixUType;
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typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixVType;
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typedef Matrix<Scalar, MinSize, 1> SingularValuesType;
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public:
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SVD() {} // a user who relied on compiler-generated default compiler reported problems with MSVC in 2.0.7
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SVD(const MatrixType& matrix)
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: m_matU(matrix.rows(), (std::min)(matrix.rows(), matrix.cols())),
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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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{
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compute(matrix);
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}
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template<typename OtherDerived, typename ResultType>
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bool solve(const MatrixBase<OtherDerived> &b, ResultType* result) const;
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const MatrixUType& matrixU() const { return m_matU; }
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const SingularValuesType& singularValues() const { return m_sigma; }
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const MatrixVType& matrixV() const { return m_matV; }
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void compute(const MatrixType& matrix);
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SVD& sort();
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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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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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};
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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 JAMA (public domain)
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*/
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template<typename MatrixType>
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void SVD<MatrixType>::compute(const MatrixType& matrix)
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{
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const int m = matrix.rows();
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const int n = matrix.cols();
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const int nu = (std::min)(m,n);
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ei_assert(m>=n && "In Eigen 2.0, SVD only works for MxN matrices with M>=N. Sorry!");
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ei_assert(m>1 && "In Eigen 2.0, SVD doesn't work on 1x1 matrices");
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m_matU.resize(m, nu);
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m_matU.setZero();
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m_sigma.resize((std::min)(m,n));
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m_matV.resize(n,n);
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RowVector e(n);
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ColVector work(m);
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MatrixType matA(matrix);
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const bool wantu = true;
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const bool wantv = true;
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int i=0, j=0, k=0;
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// Reduce A to bidiagonal form, storing the diagonal elements
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// in s and the super-diagonal elements in e.
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int nct = (std::min)(m-1,n);
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int nrt = (std::max)(0,(std::min)(n-2,m));
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for (k = 0; k < (std::max)(nct,nrt); ++k)
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{
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if (k < nct)
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{
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// Compute the transformation for the k-th column and
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// place the k-th diagonal in m_sigma[k].
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m_sigma[k] = matA.col(k).end(m-k).norm();
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if (m_sigma[k] != 0.0) // FIXME
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{
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if (matA(k,k) < 0.0)
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m_sigma[k] = -m_sigma[k];
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matA.col(k).end(m-k) /= m_sigma[k];
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matA(k,k) += 1.0;
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}
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m_sigma[k] = -m_sigma[k];
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}
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for (j = k+1; j < n; ++j)
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{
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if ((k < nct) && (m_sigma[k] != 0.0))
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{
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// Apply the transformation.
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Scalar t = matA.col(k).end(m-k).eigen2_dot(matA.col(j).end(m-k)); // FIXME dot product or cwise prod + .sum() ??
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t = -t/matA(k,k);
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matA.col(j).end(m-k) += t * matA.col(k).end(m-k);
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}
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// Place the k-th row of A into e for the
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// subsequent calculation of the row transformation.
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e[j] = matA(k,j);
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}
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// Place the transformation in U for subsequent back multiplication.
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if (wantu & (k < nct))
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m_matU.col(k).end(m-k) = matA.col(k).end(m-k);
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if (k < nrt)
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{
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// Compute the k-th row transformation and place the
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// k-th super-diagonal in e[k].
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e[k] = e.end(n-k-1).norm();
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if (e[k] != 0.0)
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{
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if (e[k+1] < 0.0)
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e[k] = -e[k];
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e.end(n-k-1) /= e[k];
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e[k+1] += 1.0;
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}
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e[k] = -e[k];
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if ((k+1 < m) & (e[k] != 0.0))
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{
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// Apply the transformation.
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work.end(m-k-1) = matA.corner(BottomRight,m-k-1,n-k-1) * e.end(n-k-1);
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for (j = k+1; j < n; ++j)
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matA.col(j).end(m-k-1) += (-e[j]/e[k+1]) * work.end(m-k-1);
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}
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// Place the transformation in V for subsequent back multiplication.
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if (wantv)
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m_matV.col(k).end(n-k-1) = e.end(n-k-1);
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}
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}
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// Set up the final bidiagonal matrix or order p.
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int p = (std::min)(n,m+1);
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if (nct < n)
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m_sigma[nct] = matA(nct,nct);
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if (m < p)
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m_sigma[p-1] = 0.0;
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if (nrt+1 < p)
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e[nrt] = matA(nrt,p-1);
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e[p-1] = 0.0;
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// If required, generate U.
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if (wantu)
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{
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for (j = nct; j < nu; ++j)
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{
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m_matU.col(j).setZero();
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m_matU(j,j) = 1.0;
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}
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for (k = nct-1; k >= 0; k--)
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{
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if (m_sigma[k] != 0.0)
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{
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for (j = k+1; j < nu; ++j)
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{
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Scalar t = m_matU.col(k).end(m-k).eigen2_dot(m_matU.col(j).end(m-k)); // FIXME is it really a dot product we want ?
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t = -t/m_matU(k,k);
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m_matU.col(j).end(m-k) += t * m_matU.col(k).end(m-k);
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}
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m_matU.col(k).end(m-k) = - m_matU.col(k).end(m-k);
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m_matU(k,k) = Scalar(1) + m_matU(k,k);
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if (k-1>0)
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m_matU.col(k).start(k-1).setZero();
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}
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else
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{
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m_matU.col(k).setZero();
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m_matU(k,k) = 1.0;
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}
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}
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}
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// If required, generate V.
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if (wantv)
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{
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for (k = n-1; k >= 0; k--)
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{
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if ((k < nrt) & (e[k] != 0.0))
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{
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for (j = k+1; j < nu; ++j)
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{
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Scalar t = m_matV.col(k).end(n-k-1).eigen2_dot(m_matV.col(j).end(n-k-1)); // FIXME is it really a dot product we want ?
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t = -t/m_matV(k+1,k);
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m_matV.col(j).end(n-k-1) += t * m_matV.col(k).end(n-k-1);
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}
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}
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m_matV.col(k).setZero();
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m_matV(k,k) = 1.0;
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}
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}
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// Main iteration loop for the singular values.
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int pp = p-1;
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int iter = 0;
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Scalar eps = ei_pow(Scalar(2),ei_is_same_type<Scalar,float>::ret ? Scalar(-23) : Scalar(-52));
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while (p > 0)
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{
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int k=0;
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int kase=0;
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// Here is where a test for too many iterations would go.
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// This section of the program inspects for
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// negligible elements in the s and e arrays. On
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// completion the variables kase and k are set as follows.
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// kase = 1 if s(p) and e[k-1] are negligible and k<p
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// kase = 2 if s(k) is negligible and k<p
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// kase = 3 if e[k-1] is negligible, k<p, and
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// s(k), ..., s(p) are not negligible (qr step).
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// kase = 4 if e(p-1) is negligible (convergence).
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for (k = p-2; k >= -1; --k)
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{
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if (k == -1)
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break;
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if (ei_abs(e[k]) <= eps*(ei_abs(m_sigma[k]) + ei_abs(m_sigma[k+1])))
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{
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e[k] = 0.0;
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break;
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}
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}
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if (k == p-2)
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{
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kase = 4;
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}
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else
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{
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int ks;
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for (ks = p-1; ks >= k; --ks)
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{
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if (ks == k)
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break;
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Scalar t = (ks != p ? ei_abs(e[ks]) : Scalar(0)) + (ks != k+1 ? ei_abs(e[ks-1]) : Scalar(0));
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if (ei_abs(m_sigma[ks]) <= eps*t)
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{
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m_sigma[ks] = 0.0;
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break;
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}
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}
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if (ks == k)
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{
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kase = 3;
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}
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else if (ks == p-1)
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{
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kase = 1;
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}
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else
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{
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kase = 2;
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k = ks;
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}
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}
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++k;
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// Perform the task indicated by kase.
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switch (kase)
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{
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// Deflate negligible s(p).
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case 1:
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{
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Scalar f(e[p-2]);
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e[p-2] = 0.0;
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for (j = p-2; j >= k; --j)
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{
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Scalar t(internal::hypot(m_sigma[j],f));
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Scalar cs(m_sigma[j]/t);
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Scalar sn(f/t);
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m_sigma[j] = t;
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if (j != k)
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{
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f = -sn*e[j-1];
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e[j-1] = cs*e[j-1];
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}
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if (wantv)
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{
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for (i = 0; i < n; ++i)
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{
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t = cs*m_matV(i,j) + sn*m_matV(i,p-1);
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m_matV(i,p-1) = -sn*m_matV(i,j) + cs*m_matV(i,p-1);
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m_matV(i,j) = t;
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}
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}
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}
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}
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break;
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// Split at negligible s(k).
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case 2:
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{
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Scalar f(e[k-1]);
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e[k-1] = 0.0;
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for (j = k; j < p; ++j)
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{
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Scalar t(internal::hypot(m_sigma[j],f));
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Scalar cs( m_sigma[j]/t);
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Scalar sn(f/t);
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m_sigma[j] = t;
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f = -sn*e[j];
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e[j] = cs*e[j];
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if (wantu)
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{
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for (i = 0; i < m; ++i)
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{
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t = cs*m_matU(i,j) + sn*m_matU(i,k-1);
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m_matU(i,k-1) = -sn*m_matU(i,j) + cs*m_matU(i,k-1);
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m_matU(i,j) = t;
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}
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}
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}
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}
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break;
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// Perform one qr step.
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case 3:
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{
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// Calculate the shift.
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Scalar scale = (std::max)((std::max)((std::max)((std::max)(
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ei_abs(m_sigma[p-1]),ei_abs(m_sigma[p-2])),ei_abs(e[p-2])),
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ei_abs(m_sigma[k])),ei_abs(e[k]));
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Scalar sp = m_sigma[p-1]/scale;
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Scalar spm1 = m_sigma[p-2]/scale;
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Scalar epm1 = e[p-2]/scale;
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Scalar sk = m_sigma[k]/scale;
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Scalar ek = e[k]/scale;
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Scalar b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/Scalar(2);
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Scalar c = (sp*epm1)*(sp*epm1);
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Scalar shift(0);
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if ((b != 0.0) || (c != 0.0))
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{
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shift = ei_sqrt(b*b + c);
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if (b < 0.0)
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shift = -shift;
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shift = c/(b + shift);
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}
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Scalar f = (sk + sp)*(sk - sp) + shift;
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Scalar g = sk*ek;
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// Chase zeros.
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for (j = k; j < p-1; ++j)
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{
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Scalar t = internal::hypot(f,g);
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Scalar cs = f/t;
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Scalar sn = g/t;
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if (j != k)
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e[j-1] = t;
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f = cs*m_sigma[j] + sn*e[j];
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e[j] = cs*e[j] - sn*m_sigma[j];
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g = sn*m_sigma[j+1];
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m_sigma[j+1] = cs*m_sigma[j+1];
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if (wantv)
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{
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for (i = 0; i < n; ++i)
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{
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t = cs*m_matV(i,j) + sn*m_matV(i,j+1);
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m_matV(i,j+1) = -sn*m_matV(i,j) + cs*m_matV(i,j+1);
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m_matV(i,j) = t;
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}
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}
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t = internal::hypot(f,g);
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cs = f/t;
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sn = g/t;
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m_sigma[j] = t;
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f = cs*e[j] + sn*m_sigma[j+1];
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m_sigma[j+1] = -sn*e[j] + cs*m_sigma[j+1];
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g = sn*e[j+1];
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e[j+1] = cs*e[j+1];
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if (wantu && (j < m-1))
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{
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for (i = 0; i < m; ++i)
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{
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t = cs*m_matU(i,j) + sn*m_matU(i,j+1);
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m_matU(i,j+1) = -sn*m_matU(i,j) + cs*m_matU(i,j+1);
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m_matU(i,j) = t;
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}
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}
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}
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e[p-2] = f;
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iter = iter + 1;
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}
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break;
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// Convergence.
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case 4:
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{
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// Make the singular values positive.
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if (m_sigma[k] <= 0.0)
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{
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m_sigma[k] = m_sigma[k] < Scalar(0) ? -m_sigma[k] : Scalar(0);
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if (wantv)
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m_matV.col(k).start(pp+1) = -m_matV.col(k).start(pp+1);
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}
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// Order the singular values.
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while (k < pp)
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{
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if (m_sigma[k] >= m_sigma[k+1])
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break;
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Scalar t = m_sigma[k];
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m_sigma[k] = m_sigma[k+1];
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m_sigma[k+1] = t;
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if (wantv && (k < n-1))
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m_matV.col(k).swap(m_matV.col(k+1));
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if (wantu && (k < m-1))
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m_matU.col(k).swap(m_matU.col(k+1));
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++k;
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}
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iter = 0;
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p--;
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}
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break;
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} // end big switch
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} // end iterations
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}
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template<typename MatrixType>
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SVD<MatrixType>& SVD<MatrixType>::sort()
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{
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int mu = m_matU.rows();
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int mv = m_matV.rows();
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int n = m_matU.cols();
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for (int i=0; i<n; ++i)
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{
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int k = i;
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Scalar p = m_sigma.coeff(i);
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for (int j=i+1; j<n; ++j)
|
|
{
|
|
if (m_sigma.coeff(j) > p)
|
|
{
|
|
k = j;
|
|
p = m_sigma.coeff(j);
|
|
}
|
|
}
|
|
if (k != i)
|
|
{
|
|
m_sigma.coeffRef(k) = m_sigma.coeff(i); // i.e.
|
|
m_sigma.coeffRef(i) = p; // swaps the i-th and the k-th elements
|
|
|
|
int j = mu;
|
|
for(int s=0; j!=0; ++s, --j)
|
|
std::swap(m_matU.coeffRef(s,i), m_matU.coeffRef(s,k));
|
|
|
|
j = mv;
|
|
for (int s=0; j!=0; ++s, --j)
|
|
std::swap(m_matV.coeffRef(s,i), m_matV.coeffRef(s,k));
|
|
}
|
|
}
|
|
return *this;
|
|
}
|
|
|
|
/** \returns the solution of \f$ A x = b \f$ using the current SVD decomposition of A.
|
|
* The parts of the solution corresponding to zero singular values are ignored.
|
|
*
|
|
* \sa MatrixBase::svd(), LU::solve(), LLT::solve()
|
|
*/
|
|
template<typename MatrixType>
|
|
template<typename OtherDerived, typename ResultType>
|
|
bool SVD<MatrixType>::solve(const MatrixBase<OtherDerived> &b, ResultType* result) const
|
|
{
|
|
const int rows = m_matU.rows();
|
|
ei_assert(b.rows() == rows);
|
|
|
|
Scalar maxVal = m_sigma.cwise().abs().maxCoeff();
|
|
for (int j=0; j<b.cols(); ++j)
|
|
{
|
|
Matrix<Scalar,MatrixUType::RowsAtCompileTime,1> aux = m_matU.transpose() * b.col(j);
|
|
|
|
for (int i = 0; i <m_matU.cols(); ++i)
|
|
{
|
|
Scalar si = m_sigma.coeff(i);
|
|
if (ei_isMuchSmallerThan(ei_abs(si),maxVal))
|
|
aux.coeffRef(i) = 0;
|
|
else
|
|
aux.coeffRef(i) /= si;
|
|
}
|
|
|
|
result->col(j) = m_matV * aux;
|
|
}
|
|
return true;
|
|
}
|
|
|
|
/** 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_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_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_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_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());
|
|
}
|
|
|
|
} // end namespace Eigen
|
|
|
|
#endif // EIGEN2_SVD_H
|