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// This file is part of Eigen, a lightweight C++ template library
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// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
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// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_REAL_SCHUR_H
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#define EIGEN_REAL_SCHUR_H
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#include "./EigenvaluesCommon.h"
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#include "./HessenbergDecomposition.h"
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/** \eigenvalues_module \ingroup Eigenvalues_Module
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* \brief Performs a real Schur decomposition of a square matrix
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* \tparam _MatrixType the type of the matrix of which we are computing the
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* real Schur decomposition; this is expected to be an instantiation of the
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* Matrix class template.
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* Given a real square matrix A, this class computes the real Schur
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* decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
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* T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
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* inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
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* matrix is a block-triangular matrix whose diagonal consists of 1-by-1
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* blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
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* blocks on the diagonal of T are the same as the eigenvalues of the matrix
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* A, and thus the real Schur decomposition is used in EigenSolver to compute
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* the eigendecomposition of a matrix.
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* Call the function compute() to compute the real Schur decomposition of a
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* given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
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* constructor which computes the real Schur decomposition at construction
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* time. Once the decomposition is computed, you can use the matrixU() and
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* matrixT() functions to retrieve the matrices U and T in the decomposition.
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* The documentation of RealSchur(const MatrixType&, bool) contains an example
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* of the typical use of this class.
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* \note The implementation is adapted from
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* <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
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* Their code is based on EISPACK.
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* \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
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template<typename _MatrixType> class RealSchur
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typedef _MatrixType MatrixType;
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RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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Options = MatrixType::Options,
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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typedef typename MatrixType::Scalar Scalar;
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typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
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typedef typename MatrixType::Index Index;
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typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
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typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
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/** \brief Default constructor.
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* \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via compute(). The \p size parameter is only
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* used as a hint. It is not an error to give a wrong \p size, but it may
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* \sa compute() for an example.
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RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
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m_workspaceVector(size),
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m_isInitialized(false),
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m_matUisUptodate(false)
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/** \brief Constructor; computes real Schur decomposition of given matrix.
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* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
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* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
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* This constructor calls compute() to compute the Schur decomposition.
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* Example: \include RealSchur_RealSchur_MatrixType.cpp
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* Output: \verbinclude RealSchur_RealSchur_MatrixType.out
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RealSchur(const MatrixType& matrix, bool computeU = true)
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: m_matT(matrix.rows(),matrix.cols()),
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m_matU(matrix.rows(),matrix.cols()),
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m_workspaceVector(matrix.rows()),
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m_hess(matrix.rows()),
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m_isInitialized(false),
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m_matUisUptodate(false)
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compute(matrix, computeU);
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/** \brief Returns the orthogonal matrix in the Schur decomposition.
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* \returns A const reference to the matrix U.
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* \pre Either the constructor RealSchur(const MatrixType&, bool) or the
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* member function compute(const MatrixType&, bool) has been called before
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* to compute the Schur decomposition of a matrix, and \p computeU was set
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* to true (the default value).
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* \sa RealSchur(const MatrixType&, bool) for an example
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const MatrixType& matrixU() const
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eigen_assert(m_isInitialized && "RealSchur is not initialized.");
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eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
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/** \brief Returns the quasi-triangular matrix in the Schur decomposition.
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* \returns A const reference to the matrix T.
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* \pre Either the constructor RealSchur(const MatrixType&, bool) or the
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* member function compute(const MatrixType&, bool) has been called before
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* to compute the Schur decomposition of a matrix.
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* \sa RealSchur(const MatrixType&, bool) for an example
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const MatrixType& matrixT() const
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eigen_assert(m_isInitialized && "RealSchur is not initialized.");
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/** \brief Computes Schur decomposition of given matrix.
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* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
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* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
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* \returns Reference to \c *this
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* The Schur decomposition is computed by first reducing the matrix to
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* Hessenberg form using the class HessenbergDecomposition. The Hessenberg
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* matrix is then reduced to triangular form by performing Francis QR
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* iterations with implicit double shift. The cost of computing the Schur
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* decomposition depends on the number of iterations; as a rough guide, it
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* may be taken to be \f$25n^3\f$ flops if \a computeU is true and
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* \f$10n^3\f$ flops if \a computeU is false.
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* Example: \include RealSchur_compute.cpp
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* Output: \verbinclude RealSchur_compute.out
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RealSchur& compute(const MatrixType& matrix, bool computeU = true);
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/** \brief Reports whether previous computation was successful.
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* \returns \c Success if computation was succesful, \c NoConvergence otherwise.
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ComputationInfo info() const
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eigen_assert(m_isInitialized && "RealSchur is not initialized.");
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/** \brief Maximum number of iterations.
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* Maximum number of iterations allowed for an eigenvalue to converge.
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static const int m_maxIterations = 40;
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ColumnVectorType m_workspaceVector;
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HessenbergDecomposition<MatrixType> m_hess;
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ComputationInfo m_info;
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bool m_isInitialized;
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bool m_matUisUptodate;
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typedef Matrix<Scalar,3,1> Vector3s;
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Scalar computeNormOfT();
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Index findSmallSubdiagEntry(Index iu, Scalar norm);
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void splitOffTwoRows(Index iu, bool computeU, Scalar exshift);
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void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
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void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
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void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
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template<typename MatrixType>
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RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
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assert(matrix.cols() == matrix.rows());
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// Step 1. Reduce to Hessenberg form
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m_hess.compute(matrix);
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m_matT = m_hess.matrixH();
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m_matU = m_hess.matrixQ();
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// Step 2. Reduce to real Schur form
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m_workspaceVector.resize(m_matT.cols());
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Scalar* workspace = &m_workspaceVector.coeffRef(0);
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// The matrix m_matT is divided in three parts.
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// Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
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// Rows il,...,iu is the part we are working on (the active window).
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// Rows iu+1,...,end are already brought in triangular form.
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Index iu = m_matT.cols() - 1;
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Index iter = 0; // iteration count
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Scalar exshift = 0.0; // sum of exceptional shifts
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Scalar norm = computeNormOfT();
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Index il = findSmallSubdiagEntry(iu, norm);
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// Check for convergence
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if (il == iu) // One root found
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m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
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m_matT.coeffRef(iu, iu-1) = Scalar(0);
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else if (il == iu-1) // Two roots found
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splitOffTwoRows(iu, computeU, exshift);
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else // No convergence yet
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// The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
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Vector3s firstHouseholderVector(0,0,0), shiftInfo;
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computeShift(iu, iter, exshift, shiftInfo);
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if (iter > m_maxIterations) break;
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initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
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performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
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if(iter <= m_maxIterations)
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m_info = NoConvergence;
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m_isInitialized = true;
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m_matUisUptodate = computeU;
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/** \internal Computes and returns vector L1 norm of T */
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template<typename MatrixType>
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inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
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const Index size = m_matT.cols();
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// FIXME to be efficient the following would requires a triangular reduxion code
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// Scalar norm = m_matT.upper().cwiseAbs().sum()
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// + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
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for (Index j = 0; j < size; ++j)
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norm += m_matT.row(j).segment(std::max(j-1,Index(0)), size-std::max(j-1,Index(0))).cwiseAbs().sum();
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/** \internal Look for single small sub-diagonal element and returns its index */
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template<typename MatrixType>
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inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, Scalar norm)
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Scalar s = internal::abs(m_matT.coeff(res-1,res-1)) + internal::abs(m_matT.coeff(res,res));
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if (internal::abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
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/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, Scalar exshift)
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const Index size = m_matT.cols();
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// The eigenvalues of the 2x2 matrix [a b; c d] are
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// trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
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Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
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Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4
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m_matT.coeffRef(iu,iu) += exshift;
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m_matT.coeffRef(iu-1,iu-1) += exshift;
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if (q >= Scalar(0)) // Two real eigenvalues
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Scalar z = internal::sqrt(internal::abs(q));
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JacobiRotation<Scalar> rot;
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rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
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rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
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m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
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m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
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m_matT.coeffRef(iu, iu-1) = Scalar(0);
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m_matU.applyOnTheRight(iu-1, iu, rot);
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m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
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/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
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shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
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shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
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shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
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// Wilkinson's original ad hoc shift
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exshift += shiftInfo.coeff(0);
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for (Index i = 0; i <= iu; ++i)
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m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
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Scalar s = internal::abs(m_matT.coeff(iu,iu-1)) + internal::abs(m_matT.coeff(iu-1,iu-2));
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shiftInfo.coeffRef(0) = Scalar(0.75) * s;
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shiftInfo.coeffRef(1) = Scalar(0.75) * s;
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shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
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// MATLAB's new ad hoc shift
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Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
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s = s * s + shiftInfo.coeff(2);
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s = internal::sqrt(s);
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if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
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s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
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s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
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for (Index i = 0; i <= iu; ++i)
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m_matT.coeffRef(i,i) -= s;
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shiftInfo.setConstant(Scalar(0.964));
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/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
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Vector3s& v = firstHouseholderVector; // alias to save typing
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for (im = iu-2; im >= il; --im)
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const Scalar Tmm = m_matT.coeff(im,im);
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const Scalar r = shiftInfo.coeff(0) - Tmm;
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const Scalar s = shiftInfo.coeff(1) - Tmm;
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v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
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v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
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v.coeffRef(2) = m_matT.coeff(im+2,im+1);
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const Scalar lhs = m_matT.coeff(im,im-1) * (internal::abs(v.coeff(1)) + internal::abs(v.coeff(2)));
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const Scalar rhs = v.coeff(0) * (internal::abs(m_matT.coeff(im-1,im-1)) + internal::abs(Tmm) + internal::abs(m_matT.coeff(im+1,im+1)));
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if (internal::abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
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/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
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const Index size = m_matT.cols();
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for (Index k = im; k <= iu-2; ++k)
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bool firstIteration = (k == im);
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v = firstHouseholderVector;
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v = m_matT.template block<3,1>(k,k-1);
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Matrix<Scalar, 2, 1> ess;
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v.makeHouseholder(ess, tau, beta);
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if (beta != Scalar(0)) // if v is not zero
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if (firstIteration && k > il)
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m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
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else if (!firstIteration)
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m_matT.coeffRef(k,k-1) = beta;
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// These Householder transformations form the O(n^3) part of the algorithm
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m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
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m_matT.block(0, k, std::min(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
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m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
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Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
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Matrix<Scalar, 1, 1> ess;
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v.makeHouseholder(ess, tau, beta);
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if (beta != Scalar(0)) // if v is not zero
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m_matT.coeffRef(iu-1, iu-2) = beta;
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m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
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m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
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m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
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// clean up pollution due to round-off errors
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for (Index i = im+2; i <= iu; ++i)
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m_matT.coeffRef(i,i-2) = Scalar(0);
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m_matT.coeffRef(i,i-3) = Scalar(0);
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#endif // EIGEN_REAL_SCHUR_H
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thirdparty/eigen3/Eigen/src/Eigenvalues/RealSchur.h
