1c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This file is part of Eigen, a lightweight C++ template library 2c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// for linear algebra. 3c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 4c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr> 57faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk> 6c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 7c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This Source Code Form is subject to the terms of the Mozilla 8c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Public License v. 2.0. If a copy of the MPL was not distributed 9c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 11c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#ifndef EIGEN_REAL_SCHUR_H 12c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#define EIGEN_REAL_SCHUR_H 13c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 14c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#include "./HessenbergDecomposition.h" 15c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 16c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathnamespace Eigen { 17c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 18c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \eigenvalues_module \ingroup Eigenvalues_Module 19c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 20c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 21c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \class RealSchur 22c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 23c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \brief Performs a real Schur decomposition of a square matrix 24c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 25c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \tparam _MatrixType the type of the matrix of which we are computing the 26c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * real Schur decomposition; this is expected to be an instantiation of the 27c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Matrix class template. 28c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 29c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Given a real square matrix A, this class computes the real Schur 30c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and 31c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose 32c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular 33c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * matrix is a block-triangular matrix whose diagonal consists of 1-by-1 34c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the 35c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * blocks on the diagonal of T are the same as the eigenvalues of the matrix 36c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * A, and thus the real Schur decomposition is used in EigenSolver to compute 37c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * the eigendecomposition of a matrix. 38c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 39c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Call the function compute() to compute the real Schur decomposition of a 40c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool) 41c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * constructor which computes the real Schur decomposition at construction 42c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * time. Once the decomposition is computed, you can use the matrixU() and 43c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * matrixT() functions to retrieve the matrices U and T in the decomposition. 44c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 45c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * The documentation of RealSchur(const MatrixType&, bool) contains an example 46c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * of the typical use of this class. 47c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 48c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note The implementation is adapted from 49c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain). 50c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Their code is based on EISPACK. 51c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 52c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver 53c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 54c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename _MatrixType> class RealSchur 55c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 56c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath public: 57c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef _MatrixType MatrixType; 58c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath enum { 59c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RowsAtCompileTime = MatrixType::RowsAtCompileTime, 60c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ColsAtCompileTime = MatrixType::ColsAtCompileTime, 61c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Options = MatrixType::Options, 62c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 63c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime 64c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath }; 65c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename MatrixType::Scalar Scalar; 66c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; 67c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename MatrixType::Index Index; 68c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 69c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType; 70c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType; 71c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 72c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Default constructor. 73c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 74c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed. 75c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 76c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * The default constructor is useful in cases in which the user intends to 77c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * perform decompositions via compute(). The \p size parameter is only 78c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * used as a hint. It is not an error to give a wrong \p size, but it may 79c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * impair performance. 80c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 81c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa compute() for an example. 82c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 83c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) 84c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : m_matT(size, size), 85c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matU(size, size), 86c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_workspaceVector(size), 87c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_hess(size), 88c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_isInitialized(false), 897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_matUisUptodate(false), 907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_maxIters(-1) 91c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { } 92c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 93c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Constructor; computes real Schur decomposition of given matrix. 94c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 95c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] matrix Square matrix whose Schur decomposition is to be computed. 96c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] computeU If true, both T and U are computed; if false, only T is computed. 97c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 98c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * This constructor calls compute() to compute the Schur decomposition. 99c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 100c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Example: \include RealSchur_RealSchur_MatrixType.cpp 101c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Output: \verbinclude RealSchur_RealSchur_MatrixType.out 102c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 103c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealSchur(const MatrixType& matrix, bool computeU = true) 104c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : m_matT(matrix.rows(),matrix.cols()), 105c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matU(matrix.rows(),matrix.cols()), 106c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_workspaceVector(matrix.rows()), 107c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_hess(matrix.rows()), 108c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_isInitialized(false), 1097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_matUisUptodate(false), 1107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_maxIters(-1) 111c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 112c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath compute(matrix, computeU); 113c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 114c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 115c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Returns the orthogonal matrix in the Schur decomposition. 116c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 117c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns A const reference to the matrix U. 118c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 119c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \pre Either the constructor RealSchur(const MatrixType&, bool) or the 120c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * member function compute(const MatrixType&, bool) has been called before 121c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * to compute the Schur decomposition of a matrix, and \p computeU was set 122c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * to true (the default value). 123c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 124c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa RealSchur(const MatrixType&, bool) for an example 125c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 126c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const MatrixType& matrixU() const 127c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 128c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "RealSchur is not initialized."); 129c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition."); 130c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_matU; 131c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 132c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 133c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Returns the quasi-triangular matrix in the Schur decomposition. 134c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 135c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns A const reference to the matrix T. 136c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 137c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \pre Either the constructor RealSchur(const MatrixType&, bool) or the 138c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * member function compute(const MatrixType&, bool) has been called before 139c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * to compute the Schur decomposition of a matrix. 140c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 141c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa RealSchur(const MatrixType&, bool) for an example 142c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 143c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const MatrixType& matrixT() const 144c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 145c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "RealSchur is not initialized."); 146c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_matT; 147c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 148c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 149c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Computes Schur decomposition of given matrix. 150c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 151c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] matrix Square matrix whose Schur decomposition is to be computed. 152c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] computeU If true, both T and U are computed; if false, only T is computed. 153c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns Reference to \c *this 154c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 155c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * The Schur decomposition is computed by first reducing the matrix to 156c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Hessenberg form using the class HessenbergDecomposition. The Hessenberg 157c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * matrix is then reduced to triangular form by performing Francis QR 158c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * iterations with implicit double shift. The cost of computing the Schur 159c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * decomposition depends on the number of iterations; as a rough guide, it 160c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * may be taken to be \f$25n^3\f$ flops if \a computeU is true and 161c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \f$10n^3\f$ flops if \a computeU is false. 162c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 163c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Example: \include RealSchur_compute.cpp 164c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Output: \verbinclude RealSchur_compute.out 1657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 1667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \sa compute(const MatrixType&, bool, Index) 167c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 168c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealSchur& compute(const MatrixType& matrix, bool computeU = true); 169c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 1707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T 1717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] matrixH Matrix in Hessenberg form H 1727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T 1737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param computeU Computes the matriX U of the Schur vectors 1747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \return Reference to \c *this 1757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 1767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This routine assumes that the matrix is already reduced in Hessenberg form matrixH 1777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * using either the class HessenbergDecomposition or another mean. 1787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * It computes the upper quasi-triangular matrix T of the Schur decomposition of H 1797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * When computeU is true, this routine computes the matrix U such that 1807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix 1817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 1827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix 1837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * is not available, the user should give an identity matrix (Q.setIdentity()) 1847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 1857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \sa compute(const MatrixType&, bool) 1867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 1877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename HessMatrixType, typename OrthMatrixType> 1887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU); 189c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Reports whether previous computation was successful. 190c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 191c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns \c Success if computation was succesful, \c NoConvergence otherwise. 192c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 193c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ComputationInfo info() const 194c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 195c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "RealSchur is not initialized."); 196c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_info; 197c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 198c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 1997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** \brief Sets the maximum number of iterations allowed. 2007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size 2027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * of the matrix. 2037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 2047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealSchur& setMaxIterations(Index maxIters) 2057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { 2067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_maxIters = maxIters; 2077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return *this; 2087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 2097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** \brief Returns the maximum number of iterations. */ 2117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index getMaxIterations() 2127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { 2137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return m_maxIters; 2147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 2157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** \brief Maximum number of iterations per row. 217c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 2187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * If not otherwise specified, the maximum number of iterations is this number times the size of the 2197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * matrix. It is currently set to 40. 220c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 2217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static const int m_maxIterationsPerRow = 40; 222c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 223c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath private: 224c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 225c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType m_matT; 226c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType m_matU; 227c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ColumnVectorType m_workspaceVector; 228c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath HessenbergDecomposition<MatrixType> m_hess; 229c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ComputationInfo m_info; 230c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath bool m_isInitialized; 231c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath bool m_matUisUptodate; 2327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index m_maxIters; 233c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 234c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef Matrix<Scalar,3,1> Vector3s; 235c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 236c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar computeNormOfT(); 2377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index findSmallSubdiagEntry(Index iu, const Scalar& norm); 2387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift); 239c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo); 240c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector); 241c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace); 242c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 243c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 244c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 245c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 246c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathRealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU) 247c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 2487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(matrix.cols() == matrix.rows()); 2497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index maxIters = m_maxIters; 2507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (maxIters == -1) 2517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez maxIters = m_maxIterationsPerRow * matrix.rows(); 252c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 253c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Step 1. Reduce to Hessenberg form 254c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_hess.compute(matrix); 255c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 256c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Step 2. Reduce to real Schur form 2577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU); 2587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return *this; 2607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename HessMatrixType, typename OrthMatrixType> 2637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezRealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU) 2647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_matT = matrixH; 2667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if(computeU) 2677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_matU = matrixQ; 2687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index maxIters = m_maxIters; 2707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (maxIters == -1) 2717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez maxIters = m_maxIterationsPerRow * matrixH.rows(); 272c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_workspaceVector.resize(m_matT.cols()); 273c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar* workspace = &m_workspaceVector.coeffRef(0); 274c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 275c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // The matrix m_matT is divided in three parts. 276c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. 277c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Rows il,...,iu is the part we are working on (the active window). 278c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Rows iu+1,...,end are already brought in triangular form. 279c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index iu = m_matT.cols() - 1; 2807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index iter = 0; // iteration count for current eigenvalue 2817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index totalIter = 0; // iteration count for whole matrix 2827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Scalar exshift(0); // sum of exceptional shifts 283c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar norm = computeNormOfT(); 284c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 285c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(norm!=0) 286c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 287c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath while (iu >= 0) 288c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 289c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index il = findSmallSubdiagEntry(iu, norm); 290c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 291c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Check for convergence 292c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (il == iu) // One root found 293c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 294c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift; 295c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (iu > 0) 296c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu, iu-1) = Scalar(0); 297c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath iu--; 298c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath iter = 0; 299c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 300c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else if (il == iu-1) // Two roots found 301c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 302c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath splitOffTwoRows(iu, computeU, exshift); 303c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath iu -= 2; 304c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath iter = 0; 305c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 306c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else // No convergence yet 307c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 308c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG ) 309c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Vector3s firstHouseholderVector(0,0,0), shiftInfo; 310c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath computeShift(iu, iter, exshift, shiftInfo); 311c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath iter = iter + 1; 3127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez totalIter = totalIter + 1; 3137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (totalIter > maxIters) break; 314c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index im; 315c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector); 316c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace); 317c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 318c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 319c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 3207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if(totalIter <= maxIters) 321c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_info = Success; 322c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else 323c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_info = NoConvergence; 324c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 325c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_isInitialized = true; 326c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matUisUptodate = computeU; 327c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return *this; 328c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 329c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 330c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \internal Computes and returns vector L1 norm of T */ 331c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 332c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT() 333c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 334c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index size = m_matT.cols(); 335c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // FIXME to be efficient the following would requires a triangular reduxion code 336c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Scalar norm = m_matT.upper().cwiseAbs().sum() 337c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum(); 338c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar norm(0); 339c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (Index j = 0; j < size; ++j) 3407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum(); 341c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return norm; 342c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 343c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 344c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \internal Look for single small sub-diagonal element and returns its index */ 345c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 3467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& norm) 347c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 3487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 349c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index res = iu; 350c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath while (res > 0) 351c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 3527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res)); 353c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (s == 0.0) 354c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath s = norm; 3557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s) 356c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath break; 357c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath res--; 358c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 359c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return res; 360c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 361c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 362c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \internal Update T given that rows iu-1 and iu decouple from the rest. */ 363c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 3647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift) 365c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 3667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::sqrt; 3677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 368c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index size = m_matT.cols(); 369c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 370c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // The eigenvalues of the 2x2 matrix [a b; c d] are 371c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc 372c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu)); 373c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4 374c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu,iu) += exshift; 375c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu-1,iu-1) += exshift; 376c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 377c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (q >= Scalar(0)) // Two real eigenvalues 378c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 3797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Scalar z = sqrt(abs(q)); 380c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath JacobiRotation<Scalar> rot; 381c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (p >= Scalar(0)) 382c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath rot.makeGivens(p + z, m_matT.coeff(iu, iu-1)); 383c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else 384c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath rot.makeGivens(p - z, m_matT.coeff(iu, iu-1)); 385c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 386c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint()); 387c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot); 388c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu, iu-1) = Scalar(0); 389c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (computeU) 390c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matU.applyOnTheRight(iu-1, iu, rot); 391c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 392c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 393c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (iu > 1) 394c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu-1, iu-2) = Scalar(0); 395c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 396c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 397c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */ 398c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 399c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo) 400c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 4017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::sqrt; 4027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 403c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu); 404c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1); 405c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); 406c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 407c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Wilkinson's original ad hoc shift 408c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (iter == 10) 409c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 410c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath exshift += shiftInfo.coeff(0); 411c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (Index i = 0; i <= iu; ++i) 412c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(i,i) -= shiftInfo.coeff(0); 4137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2)); 414c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.coeffRef(0) = Scalar(0.75) * s; 415c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.coeffRef(1) = Scalar(0.75) * s; 416c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s; 417c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 418c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 419c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // MATLAB's new ad hoc shift 420c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (iter == 30) 421c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 422c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0); 423c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath s = s * s + shiftInfo.coeff(2); 424c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (s > Scalar(0)) 425c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 4267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez s = sqrt(s); 427c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (shiftInfo.coeff(1) < shiftInfo.coeff(0)) 428c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath s = -s; 429c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0); 430c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s; 431c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath exshift += s; 432c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (Index i = 0; i <= iu; ++i) 433c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(i,i) -= s; 434c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath shiftInfo.setConstant(Scalar(0.964)); 435c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 436c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 437c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 438c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 439c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */ 440c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 441c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector) 442c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 4437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 444c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Vector3s& v = firstHouseholderVector; // alias to save typing 445c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 446c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (im = iu-2; im >= il; --im) 447c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 448c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Scalar Tmm = m_matT.coeff(im,im); 449c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Scalar r = shiftInfo.coeff(0) - Tmm; 450c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Scalar s = shiftInfo.coeff(1) - Tmm; 451c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1); 452c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s; 453c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v.coeffRef(2) = m_matT.coeff(im+2,im+1); 454c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (im == il) { 455c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath break; 456c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 4577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2))); 4587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1))); 4597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs) 460c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 461c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath break; 462c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 463c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 464c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 465c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 466c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */ 467c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 468c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace) 469c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 4707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(im >= il); 4717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(im <= iu-2); 472c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 473c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index size = m_matT.cols(); 474c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 475c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (Index k = im; k <= iu-2; ++k) 476c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 477c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath bool firstIteration = (k == im); 478c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 479c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Vector3s v; 480c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (firstIteration) 481c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v = firstHouseholderVector; 482c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else 483c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v = m_matT.template block<3,1>(k,k-1); 484c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 485c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar tau, beta; 486c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Matrix<Scalar, 2, 1> ess; 487c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v.makeHouseholder(ess, tau, beta); 488c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 489c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (beta != Scalar(0)) // if v is not zero 490c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 491c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (firstIteration && k > il) 492c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1); 493c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else if (!firstIteration) 494c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(k,k-1) = beta; 495c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 496c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // These Householder transformations form the O(n^3) part of the algorithm 497c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace); 498c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace); 499c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (computeU) 500c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace); 501c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 502c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 503c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 504c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2); 505c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar tau, beta; 506c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Matrix<Scalar, 1, 1> ess; 507c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath v.makeHouseholder(ess, tau, beta); 508c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 509c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (beta != Scalar(0)) // if v is not zero 510c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 511c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(iu-1, iu-2) = beta; 512c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace); 513c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace); 514c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (computeU) 515c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace); 516c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 517c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 518c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // clean up pollution due to round-off errors 519c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (Index i = im+2; i <= iu; ++i) 520c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 521c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(i,i-2) = Scalar(0); 522c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (i > im+2) 523c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_matT.coeffRef(i,i-3) = Scalar(0); 524c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 525c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 526c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 527c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} // end namespace Eigen 528c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 529c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#endif // EIGEN_REAL_SCHUR_H 530