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EigenSolver.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// 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 EIGEN_EIGENSOLVER_H
#define EIGEN_EIGENSOLVER_H
#include "./RealSchur.h"
namespace Eigen {
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class EigenSolver
*
* \brief Computes eigenvalues and eigenvectors of general matrices
*
* \tparam _MatrixType the type of the matrix of which we are computing the
* eigendecomposition; this is expected to be an instantiation of the Matrix
* class template. Currently, only real matrices are supported.
*
* The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars
* \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$. If
* \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and
* \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V =
* V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
* have \f$ A = V D V^{-1} \f$. This is called the eigendecomposition.
*
* The eigenvalues and eigenvectors of a matrix may be complex, even when the
* matrix is real. However, we can choose real matrices \f$ V \f$ and \f$ D
* \f$ satisfying \f$ A V = V D \f$, just like the eigendecomposition, if the
* matrix \f$ D \f$ is not required to be diagonal, but if it is allowed to
* have blocks of the form
* \f[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f]
* (where \f$ u \f$ and \f$ v \f$ are real numbers) on the diagonal. These
* blocks correspond to complex eigenvalue pairs \f$ u \pm iv \f$. We call
* this variant of the eigendecomposition the pseudo-eigendecomposition.
*
* Call the function compute() to compute the eigenvalues and eigenvectors of
* a given matrix. Alternatively, you can use the
* EigenSolver(const MatrixType&, bool) constructor which computes the
* eigenvalues and eigenvectors at construction time. Once the eigenvalue and
* eigenvectors are computed, they can be retrieved with the eigenvalues() and
* eigenvectors() functions. The pseudoEigenvalueMatrix() and
* pseudoEigenvectors() methods allow the construction of the
* pseudo-eigendecomposition.
*
* The documentation for EigenSolver(const MatrixType&, bool) contains an
* example of the typical use of this class.
*
* \note The implementation is adapted from
* <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
* Their code is based on EISPACK.
*
* \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
*/
template<typename _MatrixType> class EigenSolver
{
public:
/** \brief Synonym for the template parameter \p _MatrixType. */
typedef _MatrixType MatrixType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
/** \brief Scalar type for matrices of type #MatrixType. */
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef typename MatrixType::Index Index;
/** \brief Complex scalar type for #MatrixType.
*
* This is \c std::complex<Scalar> if #Scalar is real (e.g.,
* \c float or \c double) and just \c Scalar if #Scalar is
* complex.
*/
typedef std::complex<RealScalar> ComplexScalar;
/** \brief Type for vector of eigenvalues as returned by eigenvalues().
*
* This is a column vector with entries of type #ComplexScalar.
* The length of the vector is the size of #MatrixType.
*/
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
/** \brief Type for matrix of eigenvectors as returned by eigenvectors().
*
* This is a square matrix with entries of type #ComplexScalar.
* The size is the same as the size of #MatrixType.
*/
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType;
/** \brief Default constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via EigenSolver::compute(const MatrixType&, bool).
*
* \sa compute() for an example.
*/
EigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false), m_realSchur(), m_matT(), m_tmp() {}
/** \brief Default constructor with memory preallocation
*
* Like the default constructor but with preallocation of the internal data
* according to the specified problem \a size.
* \sa EigenSolver()
*/
EigenSolver(Index size)
: m_eivec(size, size),
m_eivalues(size),
m_isInitialized(false),
m_eigenvectorsOk(false),
m_realSchur(size),
m_matT(size, size),
m_tmp(size)
{}
/** \brief Constructor; computes eigendecomposition of given matrix.
*
* \param[in] matrix Square matrix whose eigendecomposition is to be computed.
* \param[in] computeEigenvectors If true, both the eigenvectors and the
* eigenvalues are computed; if false, only the eigenvalues are
* computed.
*
* This constructor calls compute() to compute the eigenvalues
* and eigenvectors.
*
* Example: \include EigenSolver_EigenSolver_MatrixType.cpp
* Output: \verbinclude EigenSolver_EigenSolver_MatrixType.out
*
* \sa compute()
*/
EigenSolver(const MatrixType& matrix, bool computeEigenvectors = true)
: m_eivec(matrix.rows(), matrix.cols()),
m_eivalues(matrix.cols()),
m_isInitialized(false),
m_eigenvectorsOk(false),
m_realSchur(matrix.cols()),
m_matT(matrix.rows(), matrix.cols()),
m_tmp(matrix.cols())
{
compute(matrix, computeEigenvectors);
}
/** \brief Returns the eigenvectors of given matrix.
*
* \returns %Matrix whose columns are the (possibly complex) eigenvectors.
*
* \pre Either the constructor
* EigenSolver(const MatrixType&,bool) or the member function
* compute(const MatrixType&, bool) has been called before, and
* \p computeEigenvectors was set to true (the default).
*
* Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
* to eigenvalue number \f$ k \f$ as returned by eigenvalues(). The
* eigenvectors are normalized to have (Euclidean) norm equal to one. The
* matrix returned by this function is the matrix \f$ V \f$ in the
* eigendecomposition \f$ A = V D V^{-1} \f$, if it exists.
*
* Example: \include EigenSolver_eigenvectors.cpp
* Output: \verbinclude EigenSolver_eigenvectors.out
*
* \sa eigenvalues(), pseudoEigenvectors()
*/
EigenvectorsType eigenvectors() const;
/** \brief Returns the pseudo-eigenvectors of given matrix.
*
* \returns Const reference to matrix whose columns are the pseudo-eigenvectors.
*
* \pre Either the constructor
* EigenSolver(const MatrixType&,bool) or the member function
* compute(const MatrixType&, bool) has been called before, and
* \p computeEigenvectors was set to true (the default).
*
* The real matrix \f$ V \f$ returned by this function and the
* block-diagonal matrix \f$ D \f$ returned by pseudoEigenvalueMatrix()
* satisfy \f$ AV = VD \f$.
*
* Example: \include EigenSolver_pseudoEigenvectors.cpp
* Output: \verbinclude EigenSolver_pseudoEigenvectors.out
*
* \sa pseudoEigenvalueMatrix(), eigenvectors()
*/
const MatrixType& pseudoEigenvectors() const
{
eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
return m_eivec;
}
/** \brief Returns the block-diagonal matrix in the pseudo-eigendecomposition.
*
* \returns A block-diagonal matrix.
*
* \pre Either the constructor
* EigenSolver(const MatrixType&,bool) or the member function
* compute(const MatrixType&, bool) has been called before.
*
* The matrix \f$ D \f$ returned by this function is real and
* block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2
* blocks of the form
* \f$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f$.
* These blocks are not sorted in any particular order.
* The matrix \f$ D \f$ and the matrix \f$ V \f$ returned by
* pseudoEigenvectors() satisfy \f$ AV = VD \f$.
*
* \sa pseudoEigenvectors() for an example, eigenvalues()
*/
MatrixType pseudoEigenvalueMatrix() const;
/** \brief Returns the eigenvalues of given matrix.
*
* \returns A const reference to the column vector containing the eigenvalues.
*
* \pre Either the constructor
* EigenSolver(const MatrixType&,bool) or the member function
* compute(const MatrixType&, bool) has been called before.
*
* The eigenvalues are repeated according to their algebraic multiplicity,
* so there are as many eigenvalues as rows in the matrix. The eigenvalues
* are not sorted in any particular order.
*
* Example: \include EigenSolver_eigenvalues.cpp
* Output: \verbinclude EigenSolver_eigenvalues.out
*
* \sa eigenvectors(), pseudoEigenvalueMatrix(),
* MatrixBase::eigenvalues()
*/
const EigenvalueType& eigenvalues() const
{
eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
return m_eivalues;
}
/** \brief Computes eigendecomposition of given matrix.
*
* \param[in] matrix Square matrix whose eigendecomposition is to be computed.
* \param[in] computeEigenvectors If true, both the eigenvectors and the
* eigenvalues are computed; if false, only the eigenvalues are
* computed.
* \returns Reference to \c *this
*
* This function computes the eigenvalues of the real matrix \p matrix.
* The eigenvalues() function can be used to retrieve them. If
* \p computeEigenvectors is true, then the eigenvectors are also computed
* and can be retrieved by calling eigenvectors().
*
* The matrix is first reduced to real Schur form using the RealSchur
* class. The Schur decomposition is then used to compute the eigenvalues
* and eigenvectors.
*
* The cost of the computation is dominated by the cost of the
* Schur decomposition, which is very approximately \f$ 25n^3 \f$
* (where \f$ n \f$ is the size of the matrix) if \p computeEigenvectors
* is true, and \f$ 10n^3 \f$ if \p computeEigenvectors is false.
*
* This method reuses of the allocated data in the EigenSolver object.
*
* Example: \include EigenSolver_compute.cpp
* Output: \verbinclude EigenSolver_compute.out
*/
EigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
return m_realSchur.info();
}
/** \brief Sets the maximum number of iterations allowed. */
EigenSolver& setMaxIterations(Index maxIters)
{
m_realSchur.setMaxIterations(maxIters);
return *this;
}
/** \brief Returns the maximum number of iterations. */
Index getMaxIterations()
{
return m_realSchur.getMaxIterations();
}
private:
void doComputeEigenvectors();
protected:
static void check_template_parameters()
{
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
}
MatrixType m_eivec;
EigenvalueType m_eivalues;
bool m_isInitialized;
bool m_eigenvectorsOk;
RealSchur<MatrixType> m_realSchur;
MatrixType m_matT;
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
ColumnVectorType m_tmp;
};
template<typename MatrixType>
MatrixType EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const
{
eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
Index n = m_eivalues.rows();
MatrixType matD = MatrixType::Zero(n,n);
for (Index i=0; i<n; ++i)
{
if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i))))
matD.coeffRef(i,i) = numext::real(m_eivalues.coeff(i));
else
{
matD.template block<2,2>(i,i) << numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)),
-numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i));
++i;
}
}
return matD;
}
template<typename MatrixType>
typename EigenSolver<MatrixType>::EigenvectorsType EigenSolver<MatrixType>::eigenvectors() const
{
eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
Index n = m_eivec.cols();
EigenvectorsType matV(n,n);
for (Index j=0; j<n; ++j)
{
if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j))) || j+1==n)
{
// we have a real eigen value
matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
matV.col(j).normalize();
}
else
{
// we have a pair of complex eigen values
for (Index i=0; i<n; ++i)
{
matV.coeffRef(i,j) = ComplexScalar(m_eivec.coeff(i,j), m_eivec.coeff(i,j+1));
matV.coeffRef(i,j+1) = ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
}
matV.col(j).normalize();
matV.col(j+1).normalize();
++j;
}
}
return matV;
}
template<typename MatrixType>
EigenSolver<MatrixType>&
EigenSolver<MatrixType>::compute(const MatrixType& matrix, bool computeEigenvectors)
{
check_template_parameters();
using std::sqrt;
using std::abs;
eigen_assert(matrix.cols() == matrix.rows());
// Reduce to real Schur form.
m_realSchur.compute(matrix, computeEigenvectors);
if (m_realSchur.info() == Success)
{
m_matT = m_realSchur.matrixT();
if (computeEigenvectors)
m_eivec = m_realSchur.matrixU();
// Compute eigenvalues from matT
m_eivalues.resize(matrix.cols());
Index i = 0;
while (i < matrix.cols())
{
if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) == Scalar(0))
{
m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
++i;
}
else
{
Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1));
Scalar z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
m_eivalues.coeffRef(i) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, z);
m_eivalues.coeffRef(i+1) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z);
i += 2;
}
}
// Compute eigenvectors.
if (computeEigenvectors)
doComputeEigenvectors();
}
m_isInitialized = true;
m_eigenvectorsOk = computeEigenvectors;
return *this;
}
// Complex scalar division.
template<typename Scalar>
std::complex<Scalar> cdiv(const Scalar& xr, const Scalar& xi, const Scalar& yr, const Scalar& yi)
{
using std::abs;
Scalar r,d;
if (abs(yr) > abs(yi))
{
r = yi/yr;
d = yr + r*yi;
return std::complex<Scalar>((xr + r*xi)/d, (xi - r*xr)/d);
}
else
{
r = yr/yi;
d = yi + r*yr;
return std::complex<Scalar>((r*xr + xi)/d, (r*xi - xr)/d);
}
}
template<typename MatrixType>
void EigenSolver<MatrixType>::doComputeEigenvectors()
{
using std::abs;
const Index size = m_eivec.cols();
const Scalar eps = NumTraits<Scalar>::epsilon();
// inefficient! this is already computed in RealSchur
Scalar norm(0);
for (Index j = 0; j < size; ++j)
{
norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
}
// Backsubstitute to find vectors of upper triangular form
if (norm == 0.0)
{
return;
}
for (Index n = size-1; n >= 0; n--)
{
Scalar p = m_eivalues.coeff(n).real();
Scalar q = m_eivalues.coeff(n).imag();
// Scalar vector
if (q == Scalar(0))
{
Scalar lastr(0), lastw(0);
Index l = n;
m_matT.coeffRef(n,n) = 1.0;
for (Index i = n-1; i >= 0; i--)
{
Scalar w = m_matT.coeff(i,i) - p;
Scalar r = m_matT.row(i).segment(l,n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
if (m_eivalues.coeff(i).imag() < 0.0)
{
lastw = w;
lastr = r;
}
else
{
l = i;
if (m_eivalues.coeff(i).imag() == 0.0)
{
if (w != 0.0)
m_matT.coeffRef(i,n) = -r / w;
else
m_matT.coeffRef(i,n) = -r / (eps * norm);
}
else // Solve real equations
{
Scalar x = m_matT.coeff(i,i+1);
Scalar y = m_matT.coeff(i+1,i);
Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
Scalar t = (x * lastr - lastw * r) / denom;
m_matT.coeffRef(i,n) = t;
if (abs(x) > abs(lastw))
m_matT.coeffRef(i+1,n) = (-r - w * t) / x;
else
m_matT.coeffRef(i+1,n) = (-lastr - y * t) / lastw;
}
// Overflow control
Scalar t = abs(m_matT.coeff(i,n));
if ((eps * t) * t > Scalar(1))
m_matT.col(n).tail(size-i) /= t;
}
}
}
else if (q < Scalar(0) && n > 0) // Complex vector
{
Scalar lastra(0), lastsa(0), lastw(0);
Index l = n-1;
// Last vector component imaginary so matrix is triangular
if (abs(m_matT.coeff(n,n-1)) > abs(m_matT.coeff(n-1,n)))
{
m_matT.coeffRef(n-1,n-1) = q / m_matT.coeff(n,n-1);
m_matT.coeffRef(n-1,n) = -(m_matT.coeff(n,n) - p) / m_matT.coeff(n,n-1);
}
else
{
std::complex<Scalar> cc = cdiv<Scalar>(0.0,-m_matT.coeff(n-1,n),m_matT.coeff(n-1,n-1)-p,q);
m_matT.coeffRef(n-1,n-1) = numext::real(cc);
m_matT.coeffRef(n-1,n) = numext::imag(cc);
}
m_matT.coeffRef(n,n-1) = 0.0;
m_matT.coeffRef(n,n) = 1.0;
for (Index i = n-2; i >= 0; i--)
{
Scalar ra = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n-1).segment(l, n-l+1));
Scalar sa = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
Scalar w = m_matT.coeff(i,i) - p;
if (m_eivalues.coeff(i).imag() < 0.0)
{
lastw = w;
lastra = ra;
lastsa = sa;
}
else
{
l = i;
if (m_eivalues.coeff(i).imag() == RealScalar(0))
{
std::complex<Scalar> cc = cdiv(-ra,-sa,w,q);
m_matT.coeffRef(i,n-1) = numext::real(cc);
m_matT.coeffRef(i,n) = numext::imag(cc);
}
else
{
// Solve complex equations
Scalar x = m_matT.coeff(i,i+1);
Scalar y = m_matT.coeff(i+1,i);
Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
if ((vr == 0.0) && (vi == 0.0))
vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw));
std::complex<Scalar> cc = cdiv(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra,vr,vi);
m_matT.coeffRef(i,n-1) = numext::real(cc);
m_matT.coeffRef(i,n) = numext::imag(cc);
if (abs(x) > (abs(lastw) + abs(q)))
{
m_matT.coeffRef(i+1,n-1) = (-ra - w * m_matT.coeff(i,n-1) + q * m_matT.coeff(i,n)) / x;
m_matT.coeffRef(i+1,n) = (-sa - w * m_matT.coeff(i,n) - q * m_matT.coeff(i,n-1)) / x;
}
else
{
cc = cdiv(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n),lastw,q);
m_matT.coeffRef(i+1,n-1) = numext::real(cc);
m_matT.coeffRef(i+1,n) = numext::imag(cc);
}
}
// Overflow control
using std::max;
Scalar t = (max)(abs(m_matT.coeff(i,n-1)),abs(m_matT.coeff(i,n)));
if ((eps * t) * t > Scalar(1))
m_matT.block(i, n-1, size-i, 2) /= t;
}
}
// We handled a pair of complex conjugate eigenvalues, so need to skip them both
n--;
}
else
{
eigen_assert(0 && "Internal bug in EigenSolver"); // this should not happen
}
}
// Back transformation to get eigenvectors of original matrix
for (Index j = size-1; j >= 0; j--)
{
m_tmp.noalias() = m_eivec.leftCols(j+1) * m_matT.col(j).segment(0, j+1);
m_eivec.col(j) = m_tmp;
}
}
} // end namespace Eigen
#endif // EIGEN_EIGENSOLVER_H
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