449 lines
16 KiB
C++
449 lines
16 KiB
C++
// 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>
|
|
//
|
|
// Eigen is free software; you can redistribute it and/or
|
|
// modify it under the terms of the GNU Lesser General Public
|
|
// License as published by the Free Software Foundation; either
|
|
// version 3 of the License, or (at your option) any later version.
|
|
//
|
|
// Alternatively, you can redistribute it and/or
|
|
// modify it under the terms of the GNU General Public License as
|
|
// published by the Free Software Foundation; either version 2 of
|
|
// the License, or (at your option) any later version.
|
|
//
|
|
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
|
|
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
|
|
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
|
|
// GNU General Public License for more details.
|
|
//
|
|
// You should have received a copy of the GNU Lesser General Public
|
|
// License and a copy of the GNU General Public License along with
|
|
// Eigen. If not, see <http://www.gnu.org/licenses/>.
|
|
|
|
#ifndef EIGEN_COMPLEX_SCHUR_H
|
|
#define EIGEN_COMPLEX_SCHUR_H
|
|
|
|
#include "./EigenvaluesCommon.h"
|
|
#include "./HessenbergDecomposition.h"
|
|
|
|
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>;
|
|
};
|
|
|
|
namespace internal {
|
|
|
|
/** Computes the principal value of the square root of the complex \a z. */
|
|
template<typename RealScalar>
|
|
std::complex<RealScalar> sqrt(const std::complex<RealScalar> &z)
|
|
{
|
|
RealScalar t, tre, tim;
|
|
|
|
t = abs(z);
|
|
|
|
if (abs(real(z)) <= abs(imag(z)))
|
|
{
|
|
// No cancellation in these formulas
|
|
tre = sqrt(RealScalar(0.5)*(t + real(z)));
|
|
tim = sqrt(RealScalar(0.5)*(t - real(z)));
|
|
}
|
|
else
|
|
{
|
|
// Stable computation of the above formulas
|
|
if (z.real() > RealScalar(0))
|
|
{
|
|
tre = t + z.real();
|
|
tim = abs(imag(z))*sqrt(RealScalar(0.5)/tre);
|
|
tre = sqrt(RealScalar(0.5)*tre);
|
|
}
|
|
else
|
|
{
|
|
tim = t - z.real();
|
|
tre = abs(imag(z))*sqrt(RealScalar(0.5)/tim);
|
|
tim = sqrt(RealScalar(0.5)*tim);
|
|
}
|
|
}
|
|
if(z.imag() < RealScalar(0))
|
|
tim = -tim;
|
|
|
|
return (std::complex<RealScalar>(tre,tim));
|
|
}
|
|
} // end namespace internal
|
|
|
|
|
|
/** 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 = internal::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) 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_info = Success;
|
|
else
|
|
m_info = NoConvergence;
|
|
|
|
m_isInitialized = true;
|
|
m_matUisUptodate = computeU;
|
|
}
|
|
|
|
#endif // EIGEN_COMPLEX_SCHUR_H
|