// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2008 Gael Guennebaud // Copyright (C) 2010,2012 Jitse Niesen // // 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_REAL_SCHUR_H #define EIGEN_REAL_SCHUR_H #include "./HessenbergDecomposition.h" namespace Eigen { /** \eigenvalues_module \ingroup Eigenvalues_Module * * * \class RealSchur * * \brief Performs a real Schur decomposition of a square matrix * * \tparam _MatrixType the type of the matrix of which we are computing the * real Schur decomposition; this is expected to be an instantiation of the * Matrix class template. * * Given a real square matrix A, this class computes the real Schur * decomposition: \f\$ A = U T U^T \f\$ where U is a real orthogonal matrix and * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose * inverse is equal to its transpose, \f\$ U^{-1} = U^T \f\$. A quasi-triangular * matrix is a block-triangular matrix whose diagonal consists of 1-by-1 * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the * blocks on the diagonal of T are the same as the eigenvalues of the matrix * A, and thus the real Schur decomposition is used in EigenSolver to compute * the eigendecomposition of a matrix. * * Call the function compute() to compute the real Schur decomposition of a * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool) * constructor which computes the real Schur decomposition at construction * time. Once the decomposition is computed, you can use the matrixU() and * matrixT() functions to retrieve the matrices U and T in the decomposition. * * The documentation of RealSchur(const MatrixType&, bool) contains an example * of the typical use of this class. * * \note The implementation is adapted from * JAMA (public domain). * Their code is based on EISPACK. * * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver */ template class RealSchur { public: typedef _MatrixType MatrixType; enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }; typedef typename MatrixType::Scalar Scalar; typedef std::complex::Real> ComplexScalar; typedef typename MatrixType::Index Index; typedef Matrix EigenvalueType; typedef Matrix ColumnVectorType; /** \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. */ RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) : m_matT(size, size), m_matU(size, size), m_workspaceVector(size), m_hess(size), m_isInitialized(false), m_matUisUptodate(false), m_maxIters(-1) { } /** \brief Constructor; computes real 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. * * Example: \include RealSchur_RealSchur_MatrixType.cpp * Output: \verbinclude RealSchur_RealSchur_MatrixType.out */ RealSchur(const MatrixType& matrix, bool computeU = true) : m_matT(matrix.rows(),matrix.cols()), m_matU(matrix.rows(),matrix.cols()), m_workspaceVector(matrix.rows()), m_hess(matrix.rows()), m_isInitialized(false), m_matUisUptodate(false), m_maxIters(-1) { compute(matrix, computeU); } /** \brief Returns the orthogonal matrix in the Schur decomposition. * * \returns A const reference to the matrix U. * * \pre Either the constructor RealSchur(const MatrixType&, bool) or the * member function compute(const MatrixType&, bool) has been called before * to compute the Schur decomposition of a matrix, and \p computeU was set * to true (the default value). * * \sa RealSchur(const MatrixType&, bool) for an example */ const MatrixType& matrixU() const { eigen_assert(m_isInitialized && "RealSchur is not initialized."); eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition."); return m_matU; } /** \brief Returns the quasi-triangular matrix in the Schur decomposition. * * \returns A const reference to the matrix T. * * \pre Either the constructor RealSchur(const MatrixType&, bool) or the * member function compute(const MatrixType&, bool) has been called before * to compute the Schur decomposition of a matrix. * * \sa RealSchur(const MatrixType&, bool) for an example */ const MatrixType& matrixT() const { eigen_assert(m_isInitialized && "RealSchur 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 Francis QR * iterations with implicit double shift. The cost of computing the Schur * decomposition depends on the number of iterations; as a rough guide, it * may be taken to be \f\$25n^3\f\$ flops if \a computeU is true and * \f\$10n^3\f\$ flops if \a computeU is false. * * Example: \include RealSchur_compute.cpp * Output: \verbinclude RealSchur_compute.out * * \sa compute(const MatrixType&, bool, Index) */ RealSchur& compute(const MatrixType& matrix, bool computeU = true); /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T * \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 RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU); /** \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 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. */ RealSchur& 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 40. */ static const int m_maxIterationsPerRow = 40; private: MatrixType m_matT; MatrixType m_matU; ColumnVectorType m_workspaceVector; HessenbergDecomposition m_hess; ComputationInfo m_info; bool m_isInitialized; bool m_matUisUptodate; Index m_maxIters; typedef Matrix Vector3s; Scalar computeNormOfT(); Index findSmallSubdiagEntry(Index iu); void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift); void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo); void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector); void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace); }; template RealSchur& RealSchur::compute(const MatrixType& matrix, bool computeU) { eigen_assert(matrix.cols() == matrix.rows()); Index maxIters = m_maxIters; if (maxIters == -1) maxIters = m_maxIterationsPerRow * matrix.rows(); // Step 1. Reduce to Hessenberg form m_hess.compute(matrix); // Step 2. Reduce to real Schur form computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU); return *this; } template template RealSchur& RealSchur::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU) { m_matT = matrixH; if(computeU) m_matU = matrixQ; Index maxIters = m_maxIters; if (maxIters == -1) maxIters = m_maxIterationsPerRow * matrixH.rows(); m_workspaceVector.resize(m_matT.cols()); Scalar* workspace = &m_workspaceVector.coeffRef(0); // 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 window). // Rows iu+1,...,end are already brought in triangular form. Index iu = m_matT.cols() - 1; Index iter = 0; // iteration count for current eigenvalue Index totalIter = 0; // iteration count for whole matrix Scalar exshift(0); // sum of exceptional shifts Scalar norm = computeNormOfT(); if(norm!=0) { while (iu >= 0) { Index il = findSmallSubdiagEntry(iu); // Check for convergence if (il == iu) // One root found { m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift; if (iu > 0) m_matT.coeffRef(iu, iu-1) = Scalar(0); iu--; iter = 0; } else if (il == iu-1) // Two roots found { splitOffTwoRows(iu, computeU, exshift); iu -= 2; iter = 0; } else // No convergence yet { // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG ) Vector3s firstHouseholderVector(0,0,0), shiftInfo; computeShift(iu, iter, exshift, shiftInfo); iter = iter + 1; totalIter = totalIter + 1; if (totalIter > maxIters) break; Index im; initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector); performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace); } } } if(totalIter <= maxIters) m_info = Success; else m_info = NoConvergence; m_isInitialized = true; m_matUisUptodate = computeU; return *this; } /** \internal Computes and returns vector L1 norm of T */ template inline typename MatrixType::Scalar RealSchur::computeNormOfT() { const Index size = m_matT.cols(); // FIXME to be efficient the following would requires a triangular reduxion code // Scalar norm = m_matT.upper().cwiseAbs().sum() // + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum(); Scalar norm(0); for (Index j = 0; j < size; ++j) norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum(); return norm; } /** \internal Look for single small sub-diagonal element and returns its index */ template inline typename MatrixType::Index RealSchur::findSmallSubdiagEntry(Index iu) { using std::abs; Index res = iu; while (res > 0) { Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res)); if (abs(m_matT.coeff(res,res-1)) <= NumTraits::epsilon() * s) break; res--; } return res; } /** \internal Update T given that rows iu-1 and iu decouple from the rest. */ template inline void RealSchur::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift) { using std::sqrt; using std::abs; const Index size = m_matT.cols(); // The eigenvalues of the 2x2 matrix [a b; c d] are // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu)); Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4 m_matT.coeffRef(iu,iu) += exshift; m_matT.coeffRef(iu-1,iu-1) += exshift; if (q >= Scalar(0)) // Two real eigenvalues { Scalar z = sqrt(abs(q)); JacobiRotation rot; if (p >= Scalar(0)) rot.makeGivens(p + z, m_matT.coeff(iu, iu-1)); else rot.makeGivens(p - z, m_matT.coeff(iu, iu-1)); m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint()); m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot); m_matT.coeffRef(iu, iu-1) = Scalar(0); if (computeU) m_matU.applyOnTheRight(iu-1, iu, rot); } if (iu > 1) m_matT.coeffRef(iu-1, iu-2) = Scalar(0); } /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */ template inline void RealSchur::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo) { using std::sqrt; using std::abs; shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu); shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1); shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // Wilkinson's original ad hoc shift if (iter == 10) { exshift += shiftInfo.coeff(0); for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i,i) -= shiftInfo.coeff(0); Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2)); shiftInfo.coeffRef(0) = Scalar(0.75) * s; shiftInfo.coeffRef(1) = Scalar(0.75) * s; shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s; } // MATLAB's new ad hoc shift if (iter == 30) { Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0); s = s * s + shiftInfo.coeff(2); if (s > Scalar(0)) { s = sqrt(s); if (shiftInfo.coeff(1) < shiftInfo.coeff(0)) s = -s; s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0); s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s; exshift += s; for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i,i) -= s; shiftInfo.setConstant(Scalar(0.964)); } } } /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */ template inline void RealSchur::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector) { using std::abs; Vector3s& v = firstHouseholderVector; // alias to save typing for (im = iu-2; im >= il; --im) { const Scalar Tmm = m_matT.coeff(im,im); const Scalar r = shiftInfo.coeff(0) - Tmm; const Scalar s = shiftInfo.coeff(1) - Tmm; v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1); v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s; v.coeffRef(2) = m_matT.coeff(im+2,im+1); if (im == il) { break; } const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2))); const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1))); if (abs(lhs) < NumTraits::epsilon() * rhs) break; } } /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */ template inline void RealSchur::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace) { eigen_assert(im >= il); eigen_assert(im <= iu-2); const Index size = m_matT.cols(); for (Index k = im; k <= iu-2; ++k) { bool firstIteration = (k == im); Vector3s v; if (firstIteration) v = firstHouseholderVector; else v = m_matT.template block<3,1>(k,k-1); Scalar tau, beta; Matrix ess; v.makeHouseholder(ess, tau, beta); if (beta != Scalar(0)) // if v is not zero { if (firstIteration && k > il) m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1); else if (!firstIteration) m_matT.coeffRef(k,k-1) = beta; // These Householder transformations form the O(n^3) part of the algorithm m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace); m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace); if (computeU) m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace); } } Matrix v = m_matT.template block<2,1>(iu-1, iu-2); Scalar tau, beta; Matrix ess; v.makeHouseholder(ess, tau, beta); if (beta != Scalar(0)) // if v is not zero { m_matT.coeffRef(iu-1, iu-2) = beta; m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace); m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace); if (computeU) m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace); } // clean up pollution due to round-off errors for (Index i = im+2; i <= iu; ++i) { m_matT.coeffRef(i,i-2) = Scalar(0); if (i > im+2) m_matT.coeffRef(i,i-3) = Scalar(0); } } } // end namespace Eigen #endif // EIGEN_REAL_SCHUR_H