// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Claire Maurice
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 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_COMPLEX_EIGEN_SOLVER_H
#define EIGEN_COMPLEX_EIGEN_SOLVER_H
#include "./ComplexSchur.h"
namespace Eigen {
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class ComplexEigenSolver
*
* \brief Computes eigenvalues and eigenvectors of general complex 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.
*
* 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 main function in this class is compute(), which computes the
* eigenvalues and eigenvectors of a given function. The
* documentation for that function contains an example showing the
* main features of the class.
*
* \sa class EigenSolver, class SelfAdjointEigenSolver
*/
template<typename _MatrixType> class ComplexEigenSolver
{
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> EigenvectorType;
/** \brief Default constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via compute().
*/
ComplexEigenSolver()
: m_eivec(),
m_eivalues(),
m_schur(),
m_isInitialized(false),
m_eigenvectorsOk(false),
m_matX()
{}
/** \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 ComplexEigenSolver()
*/
ComplexEigenSolver(Index size)
: m_eivec(size, size),
m_eivalues(size),
m_schur(size),
m_isInitialized(false),
m_eigenvectorsOk(false),
m_matX(size, 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 eigendecomposition.
*/
ComplexEigenSolver(const MatrixType& matrix, bool computeEigenvectors = true)
: m_eivec(matrix.rows(),matrix.cols()),
m_eivalues(matrix.cols()),
m_schur(matrix.rows()),
m_isInitialized(false),
m_eigenvectorsOk(false),
m_matX(matrix.rows(),matrix.cols())
{
compute(matrix, computeEigenvectors);
}
/** \brief Returns the eigenvectors of given matrix.
*
* \returns A const reference to the matrix whose columns are the eigenvectors.
*
* \pre Either the constructor
* ComplexEigenSolver(const MatrixType& matrix, bool) or the member
* function compute(const MatrixType& matrix, bool) has been called before
* to compute the eigendecomposition of a matrix, and
* \p computeEigenvectors was set to true (the default).
*
* This function returns a matrix whose columns are the eigenvectors. Column
* \f$ k \f$ 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 ComplexEigenSolver_eigenvectors.cpp
* Output: \verbinclude ComplexEigenSolver_eigenvectors.out
*/
const EigenvectorType& eigenvectors() const
{
eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
return m_eivec;
}
/** \brief Returns the eigenvalues of given matrix.
*
* \returns A const reference to the column vector containing the eigenvalues.
*
* \pre Either the constructor
* ComplexEigenSolver(const MatrixType& matrix, bool) or the member
* function compute(const MatrixType& matrix, bool) has been called before
* to compute the eigendecomposition of a matrix.
*
* This function returns a column vector containing the
* eigenvalues. 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 ComplexEigenSolver_eigenvalues.cpp
* Output: \verbinclude ComplexEigenSolver_eigenvalues.out
*/
const EigenvalueType& eigenvalues() const
{
eigen_assert(m_isInitialized && "ComplexEigenSolver 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 complex 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 Schur form using the
* ComplexSchur 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 \f$ O(n^3) \f$ where \f$ n \f$
* is the size of the matrix.
*
* Example: \include ComplexEigenSolver_compute.cpp
* Output: \verbinclude ComplexEigenSolver_compute.out
*/
ComplexEigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);
/** \brief Reports whether previous computation was successful.
*
* \returns \c Success if computation was succesful, \c NoConvergence otherwise.
*/
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
return m_schur.info();
}
protected:
EigenvectorType m_eivec;
EigenvalueType m_eivalues;
ComplexSchur<MatrixType> m_schur;
bool m_isInitialized;
bool m_eigenvectorsOk;
EigenvectorType m_matX;
private:
void doComputeEigenvectors(RealScalar matrixnorm);
void sortEigenvalues(bool computeEigenvectors);
};
template<typename MatrixType>
ComplexEigenSolver<MatrixType>& ComplexEigenSolver<MatrixType>::compute(const MatrixType& matrix, bool computeEigenvectors)
{
// this code is inspired from Jampack
assert(matrix.cols() == matrix.rows());
// Do a complex Schur decomposition, A = U T U^*
// The eigenvalues are on the diagonal of T.
m_schur.compute(matrix, computeEigenvectors);
if(m_schur.info() == Success)
{
m_eivalues = m_schur.matrixT().diagonal();
if(computeEigenvectors)
doComputeEigenvectors(matrix.norm());
sortEigenvalues(computeEigenvectors);
}
m_isInitialized = true;
m_eigenvectorsOk = computeEigenvectors;
return *this;
}
template<typename MatrixType>
void ComplexEigenSolver<MatrixType>::doComputeEigenvectors(RealScalar matrixnorm)
{
const Index n = m_eivalues.size();
// Compute X such that T = X D X^(-1), where D is the diagonal of T.
// The matrix X is unit triangular.
m_matX = EigenvectorType::Zero(n, n);
for(Index k=n-1 ; k>=0 ; k--)
{
m_matX.coeffRef(k,k) = ComplexScalar(1.0,0.0);
// Compute X(i,k) using the (i,k) entry of the equation X T = D X
for(Index i=k-1 ; i>=0 ; i--)
{
m_matX.coeffRef(i,k) = -m_schur.matrixT().coeff(i,k);
if(k-i-1>0)
m_matX.coeffRef(i,k) -= (m_schur.matrixT().row(i).segment(i+1,k-i-1) * m_matX.col(k).segment(i+1,k-i-1)).value();
ComplexScalar z = m_schur.matrixT().coeff(i,i) - m_schur.matrixT().coeff(k,k);
if(z==ComplexScalar(0))
{
// If the i-th and k-th eigenvalue are equal, then z equals 0.
// Use a small value instead, to prevent division by zero.
internal::real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
}
m_matX.coeffRef(i,k) = m_matX.coeff(i,k) / z;
}
}
// Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
m_eivec.noalias() = m_schur.matrixU() * m_matX;
// .. and normalize the eigenvectors
for(Index k=0 ; k<n ; k++)
{
m_eivec.col(k).normalize();
}
}
template<typename MatrixType>
void ComplexEigenSolver<MatrixType>::sortEigenvalues(bool computeEigenvectors)
{
const Index n = m_eivalues.size();
for (Index i=0; i<n; i++)
{
Index k;
m_eivalues.cwiseAbs().tail(n-i).minCoeff(&k);
if (k != 0)
{
k += i;
std::swap(m_eivalues[k],m_eivalues[i]);
if(computeEigenvectors)
m_eivec.col(i).swap(m_eivec.col(k));
}
}
}
} // end namespace Eigen
#endif // EIGEN_COMPLEX_EIGEN_SOLVER_H