// 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_SCHUR_H
#define EIGEN_COMPLEX_SCHUR_H
#include "./HessenbergDecomposition.h"
namespace Eigen {
namespace internal {
template<typename MatrixType, bool IsComplex> struct complex_schur_reduce_to_hessenberg;
}
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class ComplexSchur
*
* \brief Performs a complex Schur decomposition of a real or complex square matrix
*
* \tparam _MatrixType the type of the matrix of which we are
* computing the Schur decomposition; this is expected to be an
* instantiation of the Matrix class template.
*
* Given a real or complex square matrix A, this class computes the
* Schur decomposition: \f$ A = U T U^*\f$ where U is a unitary
* complex matrix, and T is a complex upper triangular matrix. The
* diagonal of the matrix T corresponds to the eigenvalues of the
* matrix A.
*
* Call the function compute() to compute the Schur decomposition of
* a given matrix. Alternatively, you can use the
* ComplexSchur(const MatrixType&, bool) constructor which computes
* the Schur decomposition at construction time. Once the
* decomposition is computed, you can use the matrixU() and matrixT()
* functions to retrieve the matrices U and V in the decomposition.
*
* \note This code is inspired from Jampack
*
* \sa class RealSchur, class EigenSolver, class ComplexEigenSolver
*/
template<typename _MatrixType> class ComplexSchur
{
public:
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 \p _MatrixType. */
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef typename MatrixType::Index Index;
/** \brief Complex scalar type for \p _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 the matrices in the Schur decomposition.
*
* This is a square matrix with entries of type #ComplexScalar.
* The size is the same as the size of \p _MatrixType.
*/
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType;
/** \brief Default constructor.
*
* \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
*
* The default constructor is useful in cases in which the user
* intends to perform decompositions via compute(). The \p size
* parameter is only used as a hint. It is not an error to give a
* wrong \p size, but it may impair performance.
*
* \sa compute() for an example.
*/
ComplexSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
: m_matT(size,size),
m_matU(size,size),
m_hess(size),
m_isInitialized(false),
m_matUisUptodate(false)
{}
/** \brief Constructor; computes Schur decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
*
* This constructor calls compute() to compute the Schur decomposition.
*
* \sa matrixT() and matrixU() for examples.
*/
ComplexSchur(const MatrixType& matrix, bool computeU = true)
: m_matT(matrix.rows(),matrix.cols()),
m_matU(matrix.rows(),matrix.cols()),
m_hess(matrix.rows()),
m_isInitialized(false),
m_matUisUptodate(false)
{
compute(matrix, computeU);
}
/** \brief Returns the unitary matrix in the Schur decomposition.
*
* \returns A const reference to the matrix U.
*
* It is assumed that either the constructor
* ComplexSchur(const MatrixType& matrix, bool computeU) or the
* member function compute(const MatrixType& matrix, bool computeU)
* has been called before to compute the Schur decomposition of a
* matrix, and that \p computeU was set to true (the default
* value).
*
* Example: \include ComplexSchur_matrixU.cpp
* Output: \verbinclude ComplexSchur_matrixU.out
*/
const ComplexMatrixType& matrixU() const
{
eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the ComplexSchur decomposition.");
return m_matU;
}
/** \brief Returns the triangular matrix in the Schur decomposition.
*
* \returns A const reference to the matrix T.
*
* It is assumed that either the constructor
* ComplexSchur(const MatrixType& matrix, bool computeU) or the
* member function compute(const MatrixType& matrix, bool computeU)
* has been called before to compute the Schur decomposition of a
* matrix.
*
* Note that this function returns a plain square matrix. If you want to reference
* only the upper triangular part, use:
* \code schur.matrixT().triangularView<Upper>() \endcode
*
* Example: \include ComplexSchur_matrixT.cpp
* Output: \verbinclude ComplexSchur_matrixT.out
*/
const ComplexMatrixType& matrixT() const
{
eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
return m_matT;
}
/** \brief Computes Schur decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
* \returns Reference to \c *this
*
* The Schur decomposition is computed by first reducing the
* matrix to Hessenberg form using the class
* HessenbergDecomposition. The Hessenberg matrix is then reduced
* to triangular form by performing QR iterations with a single
* shift. The cost of computing the Schur decomposition depends
* on the number of iterations; as a rough guide, it may be taken
* on the number of iterations; as a rough guide, it may be taken
* to be \f$25n^3\f$ complex flops, or \f$10n^3\f$ complex flops
* if \a computeU is false.
*
* Example: \include ComplexSchur_compute.cpp
* Output: \verbinclude ComplexSchur_compute.out
*/
ComplexSchur& compute(const MatrixType& matrix, bool computeU = 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 && "RealSchur is not initialized.");
return m_info;
}
/** \brief Maximum number of iterations.
*
* Maximum number of iterations allowed for an eigenvalue to converge.
*/
static const int m_maxIterations = 30;
protected:
ComplexMatrixType m_matT, m_matU;
HessenbergDecomposition<MatrixType> m_hess;
ComputationInfo m_info;
bool m_isInitialized;
bool m_matUisUptodate;
private:
bool subdiagonalEntryIsNeglegible(Index i);
ComplexScalar computeShift(Index iu, Index iter);
void reduceToTriangularForm(bool computeU);
friend struct internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>;
};
/** If m_matT(i+1,i) is neglegible in floating point arithmetic
* compared to m_matT(i,i) and m_matT(j,j), then set it to zero and
* return true, else return false. */
template<typename MatrixType>
inline bool ComplexSchur<MatrixType>::subdiagonalEntryIsNeglegible(Index i)
{
RealScalar d = internal::norm1(m_matT.coeff(i,i)) + internal::norm1(m_matT.coeff(i+1,i+1));
RealScalar sd = internal::norm1(m_matT.coeff(i+1,i));
if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
{
m_matT.coeffRef(i+1,i) = ComplexScalar(0);
return true;
}
return false;
}
/** Compute the shift in the current QR iteration. */
template<typename MatrixType>
typename ComplexSchur<MatrixType>::ComplexScalar ComplexSchur<MatrixType>::computeShift(Index iu, Index iter)
{
if (iter == 10 || iter == 20)
{
// exceptional shift, taken from http://www.netlib.org/eispack/comqr.f
return internal::abs(internal::real(m_matT.coeff(iu,iu-1))) + internal::abs(internal::real(m_matT.coeff(iu-1,iu-2)));
}
// compute the shift as one of the eigenvalues of t, the 2x2
// diagonal block on the bottom of the active submatrix
Matrix<ComplexScalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
RealScalar normt = t.cwiseAbs().sum();
t /= normt; // the normalization by sf is to avoid under/overflow
ComplexScalar b = t.coeff(0,1) * t.coeff(1,0);
ComplexScalar c = t.coeff(0,0) - t.coeff(1,1);
ComplexScalar disc = sqrt(c*c + RealScalar(4)*b);
ComplexScalar det = t.coeff(0,0) * t.coeff(1,1) - b;
ComplexScalar trace = t.coeff(0,0) + t.coeff(1,1);
ComplexScalar eival1 = (trace + disc) / RealScalar(2);
ComplexScalar eival2 = (trace - disc) / RealScalar(2);
if(internal::norm1(eival1) > internal::norm1(eival2))
eival2 = det / eival1;
else
eival1 = det / eival2;
// choose the eigenvalue closest to the bottom entry of the diagonal
if(internal::norm1(eival1-t.coeff(1,1)) < internal::norm1(eival2-t.coeff(1,1)))
return normt * eival1;
else
return normt * eival2;
}
template<typename MatrixType>
ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
{
m_matUisUptodate = false;
eigen_assert(matrix.cols() == matrix.rows());
if(matrix.cols() == 1)
{
m_matT = matrix.template cast<ComplexScalar>();
if(computeU) m_matU = ComplexMatrixType::Identity(1,1);
m_info = Success;
m_isInitialized = true;
m_matUisUptodate = computeU;
return *this;
}
internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>::run(*this, matrix, computeU);
reduceToTriangularForm(computeU);
return *this;
}
namespace internal {
/* Reduce given matrix to Hessenberg form */
template<typename MatrixType, bool IsComplex>
struct complex_schur_reduce_to_hessenberg
{
// this is the implementation for the case IsComplex = true
static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
{
_this.m_hess.compute(matrix);
_this.m_matT = _this.m_hess.matrixH();
if(computeU) _this.m_matU = _this.m_hess.matrixQ();
}
};
template<typename MatrixType>
struct complex_schur_reduce_to_hessenberg<MatrixType, false>
{
static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
{
typedef typename ComplexSchur<MatrixType>::ComplexScalar ComplexScalar;
typedef typename ComplexSchur<MatrixType>::ComplexMatrixType ComplexMatrixType;
// Note: m_hess is over RealScalar; m_matT and m_matU is over ComplexScalar
_this.m_hess.compute(matrix);
_this.m_matT = _this.m_hess.matrixH().template cast<ComplexScalar>();
if(computeU)
{
// This may cause an allocation which seems to be avoidable
MatrixType Q = _this.m_hess.matrixQ();
_this.m_matU = Q.template cast<ComplexScalar>();
}
}
};
} // end namespace internal
// Reduce the Hessenberg matrix m_matT to triangular form by QR iteration.
template<typename MatrixType>
void ComplexSchur<MatrixType>::reduceToTriangularForm(bool computeU)
{
// The matrix m_matT is divided in three parts.
// Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
// Rows il,...,iu is the part we are working on (the active submatrix).
// Rows iu+1,...,end are already brought in triangular form.
Index iu = m_matT.cols() - 1;
Index il;
Index iter = 0; // number of iterations we are working on the (iu,iu) element
while(true)
{
// find iu, the bottom row of the active submatrix
while(iu > 0)
{
if(!subdiagonalEntryIsNeglegible(iu-1)) break;
iter = 0;
--iu;
}
// if iu is zero then we are done; the whole matrix is triangularized
if(iu==0) break;
// if we spent too many iterations on the current element, we give up
iter++;
if(iter > m_maxIterations * m_matT.cols()) break;
// find il, the top row of the active submatrix
il = iu-1;
while(il > 0 && !subdiagonalEntryIsNeglegible(il-1))
{
--il;
}
/* perform the QR step using Givens rotations. The first rotation
creates a bulge; the (il+2,il) element becomes nonzero. This
bulge is chased down to the bottom of the active submatrix. */
ComplexScalar shift = computeShift(iu, iter);
JacobiRotation<ComplexScalar> rot;
rot.makeGivens(m_matT.coeff(il,il) - shift, m_matT.coeff(il+1,il));
m_matT.rightCols(m_matT.cols()-il).applyOnTheLeft(il, il+1, rot.adjoint());
m_matT.topRows((std::min)(il+2,iu)+1).applyOnTheRight(il, il+1, rot);
if(computeU) m_matU.applyOnTheRight(il, il+1, rot);
for(Index i=il+1 ; i<iu ; i++)
{
rot.makeGivens(m_matT.coeffRef(i,i-1), m_matT.coeffRef(i+1,i-1), &m_matT.coeffRef(i,i-1));
m_matT.coeffRef(i+1,i-1) = ComplexScalar(0);
m_matT.rightCols(m_matT.cols()-i).applyOnTheLeft(i, i+1, rot.adjoint());
m_matT.topRows((std::min)(i+2,iu)+1).applyOnTheRight(i, i+1, rot);
if(computeU) m_matU.applyOnTheRight(i, i+1, rot);
}
}
if(iter <= m_maxIterations * m_matT.cols())
m_info = Success;
else
m_info = NoConvergence;
m_isInitialized = true;
m_matUisUptodate = computeU;
}
} // end namespace Eigen
#endif // EIGEN_COMPLEX_SCHUR_H