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-// This file is part of Eigen, a lightweight C++ template library
-// for linear algebra.
-//
-// Copyright (C) 2008 Gael Guennebaud <g.gael@free.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 EIGEN2_SVD_H
-#define EIGEN2_SVD_H
-
-namespace Eigen {
-
-/** \ingroup SVD_Module
- * \nonstableyet
- *
- * \class SVD
- *
- * \brief Standard SVD decomposition of a matrix and associated features
- *
- * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
- *
- * This class performs a standard SVD decomposition of a real matrix A of size \c M x \c N
- * with \c M \>= \c N.
- *
- *
- * \sa MatrixBase::SVD()
- */
-template<typename MatrixType> class SVD
-{
- private:
- typedef typename MatrixType::Scalar Scalar;
- typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
-
- enum {
- PacketSize = internal::packet_traits<Scalar>::size,
- AlignmentMask = int(PacketSize)-1,
- MinSize = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)
- };
-
- typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVector;
- typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> RowVector;
-
- typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MinSize> MatrixUType;
- typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixVType;
- typedef Matrix<Scalar, MinSize, 1> SingularValuesType;
-
- public:
-
- SVD() {} // a user who relied on compiler-generated default compiler reported problems with MSVC in 2.0.7
-
- SVD(const MatrixType& matrix)
- : m_matU(matrix.rows(), (std::min)(matrix.rows(), matrix.cols())),
- m_matV(matrix.cols(),matrix.cols()),
- m_sigma((std::min)(matrix.rows(),matrix.cols()))
- {
- compute(matrix);
- }
-
- template<typename OtherDerived, typename ResultType>
- bool solve(const MatrixBase<OtherDerived> &b, ResultType* result) const;
-
- const MatrixUType& matrixU() const { return m_matU; }
- const SingularValuesType& singularValues() const { return m_sigma; }
- const MatrixVType& matrixV() const { return m_matV; }
-
- void compute(const MatrixType& matrix);
- SVD& sort();
-
- template<typename UnitaryType, typename PositiveType>
- void computeUnitaryPositive(UnitaryType *unitary, PositiveType *positive) const;
- template<typename PositiveType, typename UnitaryType>
- void computePositiveUnitary(PositiveType *positive, UnitaryType *unitary) const;
- template<typename RotationType, typename ScalingType>
- void computeRotationScaling(RotationType *unitary, ScalingType *positive) const;
- template<typename ScalingType, typename RotationType>
- void computeScalingRotation(ScalingType *positive, RotationType *unitary) const;
-
- protected:
- /** \internal */
- MatrixUType m_matU;
- /** \internal */
- MatrixVType m_matV;
- /** \internal */
- SingularValuesType m_sigma;
-};
-
-/** Computes / recomputes the SVD decomposition A = U S V^* of \a matrix
- *
- * \note this code has been adapted from JAMA (public domain)
- */
-template<typename MatrixType>
-void SVD<MatrixType>::compute(const MatrixType& matrix)
-{
- const int m = matrix.rows();
- const int n = matrix.cols();
- const int nu = (std::min)(m,n);
- ei_assert(m>=n && "In Eigen 2.0, SVD only works for MxN matrices with M>=N. Sorry!");
- ei_assert(m>1 && "In Eigen 2.0, SVD doesn't work on 1x1 matrices");
-
- m_matU.resize(m, nu);
- m_matU.setZero();
- m_sigma.resize((std::min)(m,n));
- m_matV.resize(n,n);
-
- RowVector e(n);
- ColVector work(m);
- MatrixType matA(matrix);
- const bool wantu = true;
- const bool wantv = true;
- int i=0, j=0, k=0;
-
- // Reduce A to bidiagonal form, storing the diagonal elements
- // in s and the super-diagonal elements in e.
- int nct = (std::min)(m-1,n);
- int nrt = (std::max)(0,(std::min)(n-2,m));
- for (k = 0; k < (std::max)(nct,nrt); ++k)
- {
- if (k < nct)
- {
- // Compute the transformation for the k-th column and
- // place the k-th diagonal in m_sigma[k].
- m_sigma[k] = matA.col(k).end(m-k).norm();
- if (m_sigma[k] != 0.0) // FIXME
- {
- if (matA(k,k) < 0.0)
- m_sigma[k] = -m_sigma[k];
- matA.col(k).end(m-k) /= m_sigma[k];
- matA(k,k) += 1.0;
- }
- m_sigma[k] = -m_sigma[k];
- }
-
- for (j = k+1; j < n; ++j)
- {
- if ((k < nct) && (m_sigma[k] != 0.0))
- {
- // Apply the transformation.
- Scalar t = matA.col(k).end(m-k).eigen2_dot(matA.col(j).end(m-k)); // FIXME dot product or cwise prod + .sum() ??
- t = -t/matA(k,k);
- matA.col(j).end(m-k) += t * matA.col(k).end(m-k);
- }
-
- // Place the k-th row of A into e for the
- // subsequent calculation of the row transformation.
- e[j] = matA(k,j);
- }
-
- // Place the transformation in U for subsequent back multiplication.
- if (wantu & (k < nct))
- m_matU.col(k).end(m-k) = matA.col(k).end(m-k);
-
- if (k < nrt)
- {
- // Compute the k-th row transformation and place the
- // k-th super-diagonal in e[k].
- e[k] = e.end(n-k-1).norm();
- if (e[k] != 0.0)
- {
- if (e[k+1] < 0.0)
- e[k] = -e[k];
- e.end(n-k-1) /= e[k];
- e[k+1] += 1.0;
- }
- e[k] = -e[k];
- if ((k+1 < m) & (e[k] != 0.0))
- {
- // Apply the transformation.
- work.end(m-k-1) = matA.corner(BottomRight,m-k-1,n-k-1) * e.end(n-k-1);
- for (j = k+1; j < n; ++j)
- matA.col(j).end(m-k-1) += (-e[j]/e[k+1]) * work.end(m-k-1);
- }
-
- // Place the transformation in V for subsequent back multiplication.
- if (wantv)
- m_matV.col(k).end(n-k-1) = e.end(n-k-1);
- }
- }
-
-
- // Set up the final bidiagonal matrix or order p.
- int p = (std::min)(n,m+1);
- if (nct < n)
- m_sigma[nct] = matA(nct,nct);
- if (m < p)
- m_sigma[p-1] = 0.0;
- if (nrt+1 < p)
- e[nrt] = matA(nrt,p-1);
- e[p-1] = 0.0;
-
- // If required, generate U.
- if (wantu)
- {
- for (j = nct; j < nu; ++j)
- {
- m_matU.col(j).setZero();
- m_matU(j,j) = 1.0;
- }
- for (k = nct-1; k >= 0; k--)
- {
- if (m_sigma[k] != 0.0)
- {
- for (j = k+1; j < nu; ++j)
- {
- 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 ?
- t = -t/m_matU(k,k);
- m_matU.col(j).end(m-k) += t * m_matU.col(k).end(m-k);
- }
- m_matU.col(k).end(m-k) = - m_matU.col(k).end(m-k);
- m_matU(k,k) = Scalar(1) + m_matU(k,k);
- if (k-1>0)
- m_matU.col(k).start(k-1).setZero();
- }
- else
- {
- m_matU.col(k).setZero();
- m_matU(k,k) = 1.0;
- }
- }
- }
-
- // If required, generate V.
- if (wantv)
- {
- for (k = n-1; k >= 0; k--)
- {
- if ((k < nrt) & (e[k] != 0.0))
- {
- for (j = k+1; j < nu; ++j)
- {
- 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 ?
- t = -t/m_matV(k+1,k);
- m_matV.col(j).end(n-k-1) += t * m_matV.col(k).end(n-k-1);
- }
- }
- m_matV.col(k).setZero();
- m_matV(k,k) = 1.0;
- }
- }
-
- // Main iteration loop for the singular values.
- int pp = p-1;
- int iter = 0;
- Scalar eps = ei_pow(Scalar(2),ei_is_same_type<Scalar,float>::ret ? Scalar(-23) : Scalar(-52));
- while (p > 0)
- {
- int k=0;
- int kase=0;
-
- // Here is where a test for too many iterations would go.
-
- // This section of the program inspects for
- // negligible elements in the s and e arrays. On
- // completion the variables kase and k are set as follows.
-
- // kase = 1 if s(p) and e[k-1] are negligible and k<p
- // kase = 2 if s(k) is negligible and k<p
- // kase = 3 if e[k-1] is negligible, k<p, and
- // s(k), ..., s(p) are not negligible (qr step).
- // kase = 4 if e(p-1) is negligible (convergence).
-
- for (k = p-2; k >= -1; --k)
- {
- if (k == -1)
- break;
- if (ei_abs(e[k]) <= eps*(ei_abs(m_sigma[k]) + ei_abs(m_sigma[k+1])))
- {
- e[k] = 0.0;
- break;
- }
- }
- if (k == p-2)
- {
- kase = 4;
- }
- else
- {
- int ks;
- for (ks = p-1; ks >= k; --ks)
- {
- if (ks == k)
- break;
- Scalar t = (ks != p ? ei_abs(e[ks]) : Scalar(0)) + (ks != k+1 ? ei_abs(e[ks-1]) : Scalar(0));
- if (ei_abs(m_sigma[ks]) <= eps*t)
- {
- m_sigma[ks] = 0.0;
- break;
- }
- }
- if (ks == k)
- {
- kase = 3;
- }
- else if (ks == p-1)
- {
- kase = 1;
- }
- else
- {
- kase = 2;
- k = ks;
- }
- }
- ++k;
-
- // Perform the task indicated by kase.
- switch (kase)
- {
-
- // Deflate negligible s(p).
- case 1:
- {
- Scalar f(e[p-2]);
- e[p-2] = 0.0;
- for (j = p-2; j >= k; --j)
- {
- Scalar t(numext::hypot(m_sigma[j],f));
- Scalar cs(m_sigma[j]/t);
- Scalar sn(f/t);
- m_sigma[j] = t;
- if (j != k)
- {
- f = -sn*e[j-1];
- e[j-1] = cs*e[j-1];
- }
- if (wantv)
- {
- for (i = 0; i < n; ++i)
- {
- t = cs*m_matV(i,j) + sn*m_matV(i,p-1);
- m_matV(i,p-1) = -sn*m_matV(i,j) + cs*m_matV(i,p-1);
- m_matV(i,j) = t;
- }
- }
- }
- }
- break;
-
- // Split at negligible s(k).
- case 2:
- {
- Scalar f(e[k-1]);
- e[k-1] = 0.0;
- for (j = k; j < p; ++j)
- {
- Scalar t(numext::hypot(m_sigma[j],f));
- Scalar cs( m_sigma[j]/t);
- Scalar sn(f/t);
- m_sigma[j] = t;
- f = -sn*e[j];
- e[j] = cs*e[j];
- if (wantu)
- {
- for (i = 0; i < m; ++i)
- {
- t = cs*m_matU(i,j) + sn*m_matU(i,k-1);
- m_matU(i,k-1) = -sn*m_matU(i,j) + cs*m_matU(i,k-1);
- m_matU(i,j) = t;
- }
- }
- }
- }
- break;
-
- // Perform one qr step.
- case 3:
- {
- // Calculate the shift.
- Scalar scale = (std::max)((std::max)((std::max)((std::max)(
- ei_abs(m_sigma[p-1]),ei_abs(m_sigma[p-2])),ei_abs(e[p-2])),
- ei_abs(m_sigma[k])),ei_abs(e[k]));
- Scalar sp = m_sigma[p-1]/scale;
- Scalar spm1 = m_sigma[p-2]/scale;
- Scalar epm1 = e[p-2]/scale;
- Scalar sk = m_sigma[k]/scale;
- Scalar ek = e[k]/scale;
- Scalar b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/Scalar(2);
- Scalar c = (sp*epm1)*(sp*epm1);
- Scalar shift(0);
- if ((b != 0.0) || (c != 0.0))
- {
- shift = ei_sqrt(b*b + c);
- if (b < 0.0)
- shift = -shift;
- shift = c/(b + shift);
- }
- Scalar f = (sk + sp)*(sk - sp) + shift;
- Scalar g = sk*ek;
-
- // Chase zeros.
-
- for (j = k; j < p-1; ++j)
- {
- Scalar t = numext::hypot(f,g);
- Scalar cs = f/t;
- Scalar sn = g/t;
- if (j != k)
- e[j-1] = t;
- f = cs*m_sigma[j] + sn*e[j];
- e[j] = cs*e[j] - sn*m_sigma[j];
- g = sn*m_sigma[j+1];
- m_sigma[j+1] = cs*m_sigma[j+1];
- if (wantv)
- {
- for (i = 0; i < n; ++i)
- {
- t = cs*m_matV(i,j) + sn*m_matV(i,j+1);
- m_matV(i,j+1) = -sn*m_matV(i,j) + cs*m_matV(i,j+1);
- m_matV(i,j) = t;
- }
- }
- t = numext::hypot(f,g);
- cs = f/t;
- sn = g/t;
- m_sigma[j] = t;
- f = cs*e[j] + sn*m_sigma[j+1];
- m_sigma[j+1] = -sn*e[j] + cs*m_sigma[j+1];
- g = sn*e[j+1];
- e[j+1] = cs*e[j+1];
- if (wantu && (j < m-1))
- {
- for (i = 0; i < m; ++i)
- {
- t = cs*m_matU(i,j) + sn*m_matU(i,j+1);
- m_matU(i,j+1) = -sn*m_matU(i,j) + cs*m_matU(i,j+1);
- m_matU(i,j) = t;
- }
- }
- }
- e[p-2] = f;
- iter = iter + 1;
- }
- break;
-
- // Convergence.
- case 4:
- {
- // Make the singular values positive.
- if (m_sigma[k] <= 0.0)
- {
- m_sigma[k] = m_sigma[k] < Scalar(0) ? -m_sigma[k] : Scalar(0);
- if (wantv)
- m_matV.col(k).start(pp+1) = -m_matV.col(k).start(pp+1);
- }
-
- // Order the singular values.
- while (k < pp)
- {
- if (m_sigma[k] >= m_sigma[k+1])
- break;
- Scalar t = m_sigma[k];
- m_sigma[k] = m_sigma[k+1];
- m_sigma[k+1] = t;
- if (wantv && (k < n-1))
- m_matV.col(k).swap(m_matV.col(k+1));
- if (wantu && (k < m-1))
- m_matU.col(k).swap(m_matU.col(k+1));
- ++k;
- }
- iter = 0;
- p--;
- }
- break;
- } // end big switch
- } // end iterations
-}
-
-template<typename MatrixType>
-SVD<MatrixType>& SVD<MatrixType>::sort()
-{
- int mu = m_matU.rows();
- int mv = m_matV.rows();
- int n = m_matU.cols();
-
- for (int i=0; i<n; ++i)
- {
- int k = i;
- Scalar p = m_sigma.coeff(i);
-
- 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
-{
- ei_assert(b.rows() == m_matU.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
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