// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// 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(internal::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(internal::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 = internal::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 = internal::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
{
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