// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012-2016 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2016 Tobias Wood <tobias@spinicist.org.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_GENERALIZEDEIGENSOLVER_H
#define EIGEN_GENERALIZEDEIGENSOLVER_H
#include "./RealQZ.h"
namespace Eigen {
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class GeneralizedEigenSolver
*
* \brief Computes the generalized eigenvalues and eigenvectors of a pair of general matrices
*
* \tparam _MatrixType the type of the matrices of which we are computing the
* eigen-decomposition; this is expected to be an instantiation of the Matrix
* class template. Currently, only real matrices are supported.
*
* The generalized eigenvalues and eigenvectors of a matrix pair \f$ A \f$ and \f$ B \f$ are scalars
* \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda Bv \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 =
* B V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
* have \f$ A = B V D V^{-1} \f$. This is called the generalized eigen-decomposition.
*
* The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the
* matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is
* singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex \f$ \alpha \f$
* and real \f$ \beta \f$ such that: \f$ \lambda_i = \alpha_i / \beta_i \f$. If \f$ \beta_i \f$ is (nearly) zero,
* then one can consider the well defined left eigenvalue \f$ \mu = \beta_i / \alpha_i\f$ such that:
* \f$ \mu_i A v_i = B v_i \f$, or even \f$ \mu_i u_i^T A = u_i^T B \f$ where \f$ u_i \f$ is
* called the left eigenvector.
*
* Call the function compute() to compute the generalized eigenvalues and eigenvectors of
* a given matrix pair. Alternatively, you can use the
* GeneralizedEigenSolver(const MatrixType&, 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.
*
* Here is an usage example of this class:
* Example: \include GeneralizedEigenSolver.cpp
* Output: \verbinclude GeneralizedEigenSolver.out
*
* \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
*/
template<typename _MatrixType> class GeneralizedEigenSolver
{
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 Eigen::Index Index; ///< \deprecated since Eigen 3.3
/** \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 real scalar values eigenvalues as returned by betas().
*
* This is a column vector with entries of type #Scalar.
* The length of the vector is the size of #MatrixType.
*/
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> VectorType;
/** \brief Type for vector of complex scalar values eigenvalues as returned by alphas().
*
* 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> ComplexVectorType;
/** \brief Expression type for the eigenvalues as returned by eigenvalues().
*/
typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> 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.
*/
GeneralizedEigenSolver()
: m_eivec(),
m_alphas(),
m_betas(),
m_valuesOkay(false),
m_vectorsOkay(false),
m_realQZ()
{}
/** \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 GeneralizedEigenSolver()
*/
explicit GeneralizedEigenSolver(Index size)
: m_eivec(size, size),
m_alphas(size),
m_betas(size),
m_valuesOkay(false),
m_vectorsOkay(false),
m_realQZ(size),
m_tmp(size)
{}
/** \brief Constructor; computes the generalized eigendecomposition of given matrix pair.
*
* \param[in] A Square matrix whose eigendecomposition is to be computed.
* \param[in] B 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 generalized eigenvalues
* and eigenvectors.
*
* \sa compute()
*/
GeneralizedEigenSolver(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true)
: m_eivec(A.rows(), A.cols()),
m_alphas(A.cols()),
m_betas(A.cols()),
m_valuesOkay(false),
m_vectorsOkay(false),
m_realQZ(A.cols()),
m_tmp(A.cols())
{
compute(A, B, computeEigenvectors);
}
/* \brief Returns the computed generalized eigenvectors.
*
* \returns %Matrix whose columns are the (possibly complex) right eigenvectors.
* i.e. the eigenvectors that solve (A - l*B)x = 0. The ordering matches the eigenvalues.
*
* \pre Either the constructor
* GeneralizedEigenSolver(const MatrixType&,const MatrixType&, bool) or the member function
* compute(const MatrixType&, const MatrixType& bool) has been called before, and
* \p computeEigenvectors was set to true (the default).
*
* \sa eigenvalues()
*/
EigenvectorsType eigenvectors() const {
eigen_assert(m_vectorsOkay && "Eigenvectors for GeneralizedEigenSolver were not calculated.");
return m_eivec;
}
/** \brief Returns an expression of the computed generalized eigenvalues.
*
* \returns An expression of the column vector containing the eigenvalues.
*
* It is a shortcut for \code this->alphas().cwiseQuotient(this->betas()); \endcode
* Not that betas might contain zeros. It is therefore not recommended to use this function,
* but rather directly deal with the alphas and betas vectors.
*
* \pre Either the constructor
* GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function
* compute(const MatrixType&,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.
*
* \sa alphas(), betas(), eigenvectors()
*/
EigenvalueType eigenvalues() const
{
eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
return EigenvalueType(m_alphas,m_betas);
}
/** \returns A const reference to the vectors containing the alpha values
*
* This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
*
* \sa betas(), eigenvalues() */
ComplexVectorType alphas() const
{
eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
return m_alphas;
}
/** \returns A const reference to the vectors containing the beta values
*
* This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
*
* \sa alphas(), eigenvalues() */
VectorType betas() const
{
eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
return m_betas;
}
/** \brief Computes generalized eigendecomposition of given matrix.
*
* \param[in] A Square matrix whose eigendecomposition is to be computed.
* \param[in] B 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 generalized Schur form using the RealQZ
* class. The generalized Schur decomposition is then used to compute the eigenvalues
* and eigenvectors.
*
* The cost of the computation is dominated by the cost of the
* generalized Schur decomposition.
*
* This method reuses of the allocated data in the GeneralizedEigenSolver object.
*/
GeneralizedEigenSolver& compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true);
ComputationInfo info() const
{
eigen_assert(m_valuesOkay && "EigenSolver is not initialized.");
return m_realQZ.info();
}
/** Sets the maximal number of iterations allowed.
*/
GeneralizedEigenSolver& setMaxIterations(Index maxIters)
{
m_realQZ.setMaxIterations(maxIters);
return *this;
}
protected:
static void check_template_parameters()
{
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
}
EigenvectorsType m_eivec;
ComplexVectorType m_alphas;
VectorType m_betas;
bool m_valuesOkay, m_vectorsOkay;
RealQZ<MatrixType> m_realQZ;
ComplexVectorType m_tmp;
};
template<typename MatrixType>
GeneralizedEigenSolver<MatrixType>&
GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors)
{
check_template_parameters();
using std::sqrt;
using std::abs;
eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows());
Index size = A.cols();
m_valuesOkay = false;
m_vectorsOkay = false;
// Reduce to generalized real Schur form:
// A = Q S Z and B = Q T Z
m_realQZ.compute(A, B, computeEigenvectors);
if (m_realQZ.info() == Success)
{
// Resize storage
m_alphas.resize(size);
m_betas.resize(size);
if (computeEigenvectors)
{
m_eivec.resize(size,size);
m_tmp.resize(size);
}
// Aliases:
Map<VectorType> v(reinterpret_cast<Scalar*>(m_tmp.data()), size);
ComplexVectorType &cv = m_tmp;
const MatrixType &mZ = m_realQZ.matrixZ();
const MatrixType &mS = m_realQZ.matrixS();
const MatrixType &mT = m_realQZ.matrixT();
Index i = 0;
while (i < size)
{
if (i == size - 1 || mS.coeff(i+1, i) == Scalar(0))
{
// Real eigenvalue
m_alphas.coeffRef(i) = mS.diagonal().coeff(i);
m_betas.coeffRef(i) = mT.diagonal().coeff(i);
if (computeEigenvectors)
{
v.setConstant(Scalar(0.0));
v.coeffRef(i) = Scalar(1.0);
// For singular eigenvalues do nothing more
if(abs(m_betas.coeffRef(i)) >= (std::numeric_limits<RealScalar>::min)())
{
// Non-singular eigenvalue
const Scalar alpha = real(m_alphas.coeffRef(i));
const Scalar beta = m_betas.coeffRef(i);
for (Index j = i-1; j >= 0; j--)
{
const Index st = j+1;
const Index sz = i-j;
if (j > 0 && mS.coeff(j, j-1) != Scalar(0))
{
// 2x2 block
Matrix<Scalar, 2, 1> rhs = (alpha*mT.template block<2,Dynamic>(j-1,st,2,sz) - beta*mS.template block<2,Dynamic>(j-1,st,2,sz)) .lazyProduct( v.segment(st,sz) );
Matrix<Scalar, 2, 2> lhs = beta * mS.template block<2,2>(j-1,j-1) - alpha * mT.template block<2,2>(j-1,j-1);
v.template segment<2>(j-1) = lhs.partialPivLu().solve(rhs);
j--;
}
else
{
v.coeffRef(j) = -v.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum() / (beta*mS.coeffRef(j,j) - alpha*mT.coeffRef(j,j));
}
}
}
m_eivec.col(i).real().noalias() = mZ.transpose() * v;
m_eivec.col(i).real().normalize();
m_eivec.col(i).imag().setConstant(0);
}
++i;
}
else
{
// We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a positive diagonal 2x2 block T
// Then taking beta=T_00*T_11, we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00):
// T = [a 0]
// [0 b]
RealScalar a = mT.diagonal().coeff(i),
b = mT.diagonal().coeff(i+1);
const RealScalar beta = m_betas.coeffRef(i) = m_betas.coeffRef(i+1) = a*b;
// ^^ NOTE: using diagonal()(i) instead of coeff(i,i) workarounds a MSVC bug.
Matrix<RealScalar,2,2> S2 = mS.template block<2,2>(i,i) * Matrix<Scalar,2,1>(b,a).asDiagonal();
Scalar p = Scalar(0.5) * (S2.coeff(0,0) - S2.coeff(1,1));
Scalar z = sqrt(abs(p * p + S2.coeff(1,0) * S2.coeff(0,1)));
const ComplexScalar alpha = ComplexScalar(S2.coeff(1,1) + p, (beta > 0) ? z : -z);
m_alphas.coeffRef(i) = conj(alpha);
m_alphas.coeffRef(i+1) = alpha;
if (computeEigenvectors) {
// Compute eigenvector in position (i+1) and then position (i) is just the conjugate
cv.setZero();
cv.coeffRef(i+1) = Scalar(1.0);
// here, the "static_cast" workaound expression template issues.
cv.coeffRef(i) = -(static_cast<Scalar>(beta*mS.coeffRef(i,i+1)) - alpha*mT.coeffRef(i,i+1))
/ (static_cast<Scalar>(beta*mS.coeffRef(i,i)) - alpha*mT.coeffRef(i,i));
for (Index j = i-1; j >= 0; j--)
{
const Index st = j+1;
const Index sz = i+1-j;
if (j > 0 && mS.coeff(j, j-1) != Scalar(0))
{
// 2x2 block
Matrix<ComplexScalar, 2, 1> rhs = (alpha*mT.template block<2,Dynamic>(j-1,st,2,sz) - beta*mS.template block<2,Dynamic>(j-1,st,2,sz)) .lazyProduct( cv.segment(st,sz) );
Matrix<ComplexScalar, 2, 2> lhs = beta * mS.template block<2,2>(j-1,j-1) - alpha * mT.template block<2,2>(j-1,j-1);
cv.template segment<2>(j-1) = lhs.partialPivLu().solve(rhs);
j--;
} else {
cv.coeffRef(j) = cv.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum()
/ (alpha*mT.coeffRef(j,j) - static_cast<Scalar>(beta*mS.coeffRef(j,j)));
}
}
m_eivec.col(i+1).noalias() = (mZ.transpose() * cv);
m_eivec.col(i+1).normalize();
m_eivec.col(i) = m_eivec.col(i+1).conjugate();
}
i += 2;
}
}
m_valuesOkay = true;
m_vectorsOkay = computeEigenvectors;
}
return *this;
}
} // end namespace Eigen
#endif // EIGEN_GENERALIZEDEIGENSOLVER_H