// 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) 2010,2012 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/.
/** \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. */ typedeftypename MatrixType::Scalar Scalar; typedeftypename NumTraits<Scalar>::Real RealScalar; typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
/** \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.
*/ explicit ComplexSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
: m_matT(size,size),
m_matU(size,size),
m_hess(size),
m_isInitialized(false),
m_matUisUptodate(false),
m_maxIters(-1)
{}
/** \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.
*/ template<typename InputType> explicit ComplexSchur(const EigenBase<InputType>& 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),
m_maxIters(-1)
{
compute(matrix.derived(), 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 * * \sa compute(const MatrixType&, bool, Index)
*/ template<typename InputType>
ComplexSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);
/** \brief Compute Schur decomposition from a given Hessenberg matrix * \param[in] matrixH Matrix in Hessenberg form H * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T * \param computeU Computes the matriX U of the Schur vectors * \return Reference to \c *this * * This routine assumes that the matrix is already reduced in Hessenberg form matrixH * using either the class HessenbergDecomposition or another mean. * It computes the upper quasi-triangular matrix T of the Schur decomposition of H * When computeU is true, this routine computes the matrix U such that * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix * * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix * is not available, the user should give an identity matrix (Q.setIdentity()) * * \sa compute(const MatrixType&, bool)
*/ template<typename HessMatrixType, typename OrthMatrixType>
ComplexSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU=true);
/** \brief Reports whether previous computation was successful. * * \returns \c Success if computation was successful, \c NoConvergence otherwise.
*/
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "ComplexSchur is not initialized."); return m_info;
}
/** \brief Sets the maximum number of iterations allowed. * * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size * of the matrix.
*/
ComplexSchur& setMaxIterations(Index maxIters)
{
m_maxIters = maxIters; return *this;
}
/** \brief Returns the maximum number of iterations. */
Index getMaxIterations()
{ return m_maxIters;
}
/** \brief Maximum number of iterations per row. * * If not otherwise specified, the maximum number of iterations is this number times the size of the * matrix. It is currently set to 30.
*/ staticconstint m_maxIterationsPerRow = 30;
/** 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> inlinebool ComplexSchur<MatrixType>::subdiagonalEntryIsNeglegible(Index i)
{
RealScalar d = numext::norm1(m_matT.coeff(i,i)) + numext::norm1(m_matT.coeff(i+1,i+1));
RealScalar sd = numext::norm1(m_matT.coeff(i+1,i)); if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
{
m_matT.coeffRef(i+1,i) = ComplexScalar(0); returntrue;
} returnfalse;
}
/** Compute the shift in the current QR iteration. */ template<typename MatrixType> typename ComplexSchur<MatrixType>::ComplexScalar ComplexSchur<MatrixType>::computeShift(Index iu, Index iter)
{ using std::abs; if (iter == 10 || iter == 20)
{ // exceptional shift, taken from http://www.netlib.org/eispack/comqr.f return abs(numext::real(m_matT.coeff(iu,iu-1))) + abs(numext::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);
RealScalar eival1_norm = numext::norm1(eival1);
RealScalar eival2_norm = numext::norm1(eival2); // A division by zero can only occur if eival1==eival2==0. // In this case, det==0, and all we have to do is checking that eival2_norm!=0 if(eival1_norm > eival2_norm)
eival2 = det / eival1; elseif(eival2_norm!=RealScalar(0))
eival1 = det / eival2;
// choose the eigenvalue closest to the bottom entry of the diagonal if(numext::norm1(eival1-t.coeff(1,1)) < numext::norm1(eival2-t.coeff(1,1))) return normt * eival1; else return normt * eival2;
}
/* 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 staticvoid 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();
}
};
// 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)
{
Index maxIters = m_maxIters; if (maxIters == -1)
maxIters = m_maxIterationsPerRow * m_matT.rows();
// 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
Index totalIter = 0; // number of iterations for whole matrix
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, we give up
iter++;
totalIter++; if(totalIter > maxIters) 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. */
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