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diff --git a/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealSchur.h b/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealSchur.h new file mode 100644 index 000000000..f5c86041d --- /dev/null +++ b/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealSchur.h @@ -0,0 +1,546 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr> +// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk> +// +// This Source Code Form is subject to the terms of the Mozilla +// Public License v. 2.0. If a copy of the MPL was not distributed +// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. + +#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 + * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain). + * Their code is based on EISPACK. + * + * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver + */ +template<typename _MatrixType> 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<typename NumTraits<Scalar>::Real> ComplexScalar; + typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 + + typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType; + typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> 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. + */ + explicit 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 + */ + template<typename InputType> + explicit RealSchur(const EigenBase<InputType>& 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.derived(), 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) + */ + template<typename InputType> + RealSchur& compute(const EigenBase<InputType>& 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<typename HessMatrixType, typename OrthMatrixType> + 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<MatrixType> m_hess; + ComputationInfo m_info; + bool m_isInitialized; + bool m_matUisUptodate; + Index m_maxIters; + + typedef Matrix<Scalar,3,1> 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<typename MatrixType> +template<typename InputType> +RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU) +{ + const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)(); + + eigen_assert(matrix.cols() == matrix.rows()); + Index maxIters = m_maxIters; + if (maxIters == -1) + maxIters = m_maxIterationsPerRow * matrix.rows(); + + Scalar scale = matrix.derived().cwiseAbs().maxCoeff(); + if(scale<considerAsZero) + { + m_matT.setZero(matrix.rows(),matrix.cols()); + if(computeU) + m_matU.setIdentity(matrix.rows(),matrix.cols()); + m_info = Success; + m_isInitialized = true; + m_matUisUptodate = computeU; + return *this; + } + + // Step 1. Reduce to Hessenberg form + m_hess.compute(matrix.derived()/scale); + + // Step 2. Reduce to real Schur form + computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU); + + m_matT *= scale; + + return *this; +} +template<typename MatrixType> +template<typename HessMatrixType, typename OrthMatrixType> +RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU) +{ + using std::abs; + + 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<typename MatrixType> +inline typename MatrixType::Scalar RealSchur<MatrixType>::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<typename MatrixType> +inline Index RealSchur<MatrixType>::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<Scalar>::epsilon() * s) + break; + res--; + } + return res; +} + +/** \internal Update T given that rows iu-1 and iu decouple from the rest. */ +template<typename MatrixType> +inline void RealSchur<MatrixType>::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<Scalar> 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<typename MatrixType> +inline void RealSchur<MatrixType>::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<typename MatrixType> +inline void RealSchur<MatrixType>::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<Scalar>::epsilon() * rhs) + break; + } +} + +/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */ +template<typename MatrixType> +inline void RealSchur<MatrixType>::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<Scalar, 2, 1> 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<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2); + Scalar tau, beta; + Matrix<Scalar, 1, 1> 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 |