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Diffstat (limited to 'runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealQZ.h')
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diff --git a/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealQZ.h b/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealQZ.h deleted file mode 100644 index b3a910dd9..000000000 --- a/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/RealQZ.h +++ /dev/null @@ -1,654 +0,0 @@ -// This file is part of Eigen, a lightweight C++ template library -// for linear algebra. -// -// Copyright (C) 2012 Alexey Korepanov <kaikaikai@yandex.ru> -// -// 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_QZ_H -#define EIGEN_REAL_QZ_H - -namespace Eigen { - - /** \eigenvalues_module \ingroup Eigenvalues_Module - * - * - * \class RealQZ - * - * \brief Performs a real QZ decomposition of a pair of square matrices - * - * \tparam _MatrixType the type of the matrix of which we are computing the - * real QZ decomposition; this is expected to be an instantiation of the - * Matrix class template. - * - * Given a real square matrices A and B, this class computes the real QZ - * decomposition: \f$ A = Q S Z \f$, \f$ B = Q T Z \f$ where Q and Z are - * real orthogonal matrixes, T is upper-triangular matrix, and S is upper - * quasi-triangular matrix. An orthogonal matrix is a matrix whose - * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular - * matrix is a block-triangular matrix whose diagonal consists of 1-by-1 - * blocks and 2-by-2 blocks where further reduction is impossible due to - * complex eigenvalues. - * - * The eigenvalues of the pencil \f$ A - z B \f$ can be obtained from - * 1x1 and 2x2 blocks on the diagonals of S and T. - * - * Call the function compute() to compute the real QZ decomposition of a - * given pair of matrices. Alternatively, you can use the - * RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ) - * constructor which computes the real QZ decomposition at construction - * time. Once the decomposition is computed, you can use the matrixS(), - * matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices - * S, T, Q and Z in the decomposition. If computeQZ==false, some time - * is saved by not computing matrices Q and Z. - * - * Example: \include RealQZ_compute.cpp - * Output: \include RealQZ_compute.out - * - * \note The implementation is based on the algorithm in "Matrix Computations" - * by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for - * generalized eigenvalue problems" by C.B.Moler and G.W.Stewart. - * - * \sa class RealSchur, class ComplexSchur, class EigenSolver, class ComplexEigenSolver - */ - - template<typename _MatrixType> class RealQZ - { - 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 QZ 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 RealQZ(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) : - m_S(size, size), - m_T(size, size), - m_Q(size, size), - m_Z(size, size), - m_workspace(size*2), - m_maxIters(400), - m_isInitialized(false) - { } - - /** \brief Constructor; computes real QZ decomposition of given matrices - * - * \param[in] A Matrix A. - * \param[in] B Matrix B. - * \param[in] computeQZ If false, A and Z are not computed. - * - * This constructor calls compute() to compute the QZ decomposition. - */ - RealQZ(const MatrixType& A, const MatrixType& B, bool computeQZ = true) : - m_S(A.rows(),A.cols()), - m_T(A.rows(),A.cols()), - m_Q(A.rows(),A.cols()), - m_Z(A.rows(),A.cols()), - m_workspace(A.rows()*2), - m_maxIters(400), - m_isInitialized(false) { - compute(A, B, computeQZ); - } - - /** \brief Returns matrix Q in the QZ decomposition. - * - * \returns A const reference to the matrix Q. - */ - const MatrixType& matrixQ() const { - eigen_assert(m_isInitialized && "RealQZ is not initialized."); - eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition."); - return m_Q; - } - - /** \brief Returns matrix Z in the QZ decomposition. - * - * \returns A const reference to the matrix Z. - */ - const MatrixType& matrixZ() const { - eigen_assert(m_isInitialized && "RealQZ is not initialized."); - eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition."); - return m_Z; - } - - /** \brief Returns matrix S in the QZ decomposition. - * - * \returns A const reference to the matrix S. - */ - const MatrixType& matrixS() const { - eigen_assert(m_isInitialized && "RealQZ is not initialized."); - return m_S; - } - - /** \brief Returns matrix S in the QZ decomposition. - * - * \returns A const reference to the matrix S. - */ - const MatrixType& matrixT() const { - eigen_assert(m_isInitialized && "RealQZ is not initialized."); - return m_T; - } - - /** \brief Computes QZ decomposition of given matrix. - * - * \param[in] A Matrix A. - * \param[in] B Matrix B. - * \param[in] computeQZ If false, A and Z are not computed. - * \returns Reference to \c *this - */ - RealQZ& compute(const MatrixType& A, const MatrixType& B, bool computeQZ = true); - - /** \brief Reports whether previous computation was successful. - * - * \returns \c Success if computation was succesful, \c NoConvergence otherwise. - */ - ComputationInfo info() const - { - eigen_assert(m_isInitialized && "RealQZ is not initialized."); - return m_info; - } - - /** \brief Returns number of performed QR-like iterations. - */ - Index iterations() const - { - eigen_assert(m_isInitialized && "RealQZ is not initialized."); - return m_global_iter; - } - - /** Sets the maximal number of iterations allowed to converge to one eigenvalue - * or decouple the problem. - */ - RealQZ& setMaxIterations(Index maxIters) - { - m_maxIters = maxIters; - return *this; - } - - private: - - MatrixType m_S, m_T, m_Q, m_Z; - Matrix<Scalar,Dynamic,1> m_workspace; - ComputationInfo m_info; - Index m_maxIters; - bool m_isInitialized; - bool m_computeQZ; - Scalar m_normOfT, m_normOfS; - Index m_global_iter; - - typedef Matrix<Scalar,3,1> Vector3s; - typedef Matrix<Scalar,2,1> Vector2s; - typedef Matrix<Scalar,2,2> Matrix2s; - typedef JacobiRotation<Scalar> JRs; - - void hessenbergTriangular(); - void computeNorms(); - Index findSmallSubdiagEntry(Index iu); - Index findSmallDiagEntry(Index f, Index l); - void splitOffTwoRows(Index i); - void pushDownZero(Index z, Index f, Index l); - void step(Index f, Index l, Index iter); - - }; // RealQZ - - /** \internal Reduces S and T to upper Hessenberg - triangular form */ - template<typename MatrixType> - void RealQZ<MatrixType>::hessenbergTriangular() - { - - const Index dim = m_S.cols(); - - // perform QR decomposition of T, overwrite T with R, save Q - HouseholderQR<MatrixType> qrT(m_T); - m_T = qrT.matrixQR(); - m_T.template triangularView<StrictlyLower>().setZero(); - m_Q = qrT.householderQ(); - // overwrite S with Q* S - m_S.applyOnTheLeft(m_Q.adjoint()); - // init Z as Identity - if (m_computeQZ) - m_Z = MatrixType::Identity(dim,dim); - // reduce S to upper Hessenberg with Givens rotations - for (Index j=0; j<=dim-3; j++) { - for (Index i=dim-1; i>=j+2; i--) { - JRs G; - // kill S(i,j) - if(m_S.coeff(i,j) != 0) - { - G.makeGivens(m_S.coeff(i-1,j), m_S.coeff(i,j), &m_S.coeffRef(i-1, j)); - m_S.coeffRef(i,j) = Scalar(0.0); - m_S.rightCols(dim-j-1).applyOnTheLeft(i-1,i,G.adjoint()); - m_T.rightCols(dim-i+1).applyOnTheLeft(i-1,i,G.adjoint()); - // update Q - if (m_computeQZ) - m_Q.applyOnTheRight(i-1,i,G); - } - // kill T(i,i-1) - if(m_T.coeff(i,i-1)!=Scalar(0)) - { - G.makeGivens(m_T.coeff(i,i), m_T.coeff(i,i-1), &m_T.coeffRef(i,i)); - m_T.coeffRef(i,i-1) = Scalar(0.0); - m_S.applyOnTheRight(i,i-1,G); - m_T.topRows(i).applyOnTheRight(i,i-1,G); - // update Z - if (m_computeQZ) - m_Z.applyOnTheLeft(i,i-1,G.adjoint()); - } - } - } - } - - /** \internal Computes vector L1 norms of S and T when in Hessenberg-Triangular form already */ - template<typename MatrixType> - inline void RealQZ<MatrixType>::computeNorms() - { - const Index size = m_S.cols(); - m_normOfS = Scalar(0.0); - m_normOfT = Scalar(0.0); - for (Index j = 0; j < size; ++j) - { - m_normOfS += m_S.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum(); - m_normOfT += m_T.row(j).segment(j, size - j).cwiseAbs().sum(); - } - } - - - /** \internal Look for single small sub-diagonal element S(res, res-1) and return res (or 0) */ - template<typename MatrixType> - inline Index RealQZ<MatrixType>::findSmallSubdiagEntry(Index iu) - { - using std::abs; - Index res = iu; - while (res > 0) - { - Scalar s = abs(m_S.coeff(res-1,res-1)) + abs(m_S.coeff(res,res)); - if (s == Scalar(0.0)) - s = m_normOfS; - if (abs(m_S.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s) - break; - res--; - } - return res; - } - - /** \internal Look for single small diagonal element T(res, res) for res between f and l, and return res (or f-1) */ - template<typename MatrixType> - inline Index RealQZ<MatrixType>::findSmallDiagEntry(Index f, Index l) - { - using std::abs; - Index res = l; - while (res >= f) { - if (abs(m_T.coeff(res,res)) <= NumTraits<Scalar>::epsilon() * m_normOfT) - break; - res--; - } - return res; - } - - /** \internal decouple 2x2 diagonal block in rows i, i+1 if eigenvalues are real */ - template<typename MatrixType> - inline void RealQZ<MatrixType>::splitOffTwoRows(Index i) - { - using std::abs; - using std::sqrt; - const Index dim=m_S.cols(); - if (abs(m_S.coeff(i+1,i))==Scalar(0)) - return; - Index j = findSmallDiagEntry(i,i+1); - if (j==i-1) - { - // block of (S T^{-1}) - Matrix2s STi = m_T.template block<2,2>(i,i).template triangularView<Upper>(). - template solve<OnTheRight>(m_S.template block<2,2>(i,i)); - Scalar p = Scalar(0.5)*(STi(0,0)-STi(1,1)); - Scalar q = p*p + STi(1,0)*STi(0,1); - if (q>=0) { - Scalar z = sqrt(q); - // one QR-like iteration for ABi - lambda I - // is enough - when we know exact eigenvalue in advance, - // convergence is immediate - JRs G; - if (p>=0) - G.makeGivens(p + z, STi(1,0)); - else - G.makeGivens(p - z, STi(1,0)); - m_S.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint()); - m_T.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint()); - // update Q - if (m_computeQZ) - m_Q.applyOnTheRight(i,i+1,G); - - G.makeGivens(m_T.coeff(i+1,i+1), m_T.coeff(i+1,i)); - m_S.topRows(i+2).applyOnTheRight(i+1,i,G); - m_T.topRows(i+2).applyOnTheRight(i+1,i,G); - // update Z - if (m_computeQZ) - m_Z.applyOnTheLeft(i+1,i,G.adjoint()); - - m_S.coeffRef(i+1,i) = Scalar(0.0); - m_T.coeffRef(i+1,i) = Scalar(0.0); - } - } - else - { - pushDownZero(j,i,i+1); - } - } - - /** \internal use zero in T(z,z) to zero S(l,l-1), working in block f..l */ - template<typename MatrixType> - inline void RealQZ<MatrixType>::pushDownZero(Index z, Index f, Index l) - { - JRs G; - const Index dim = m_S.cols(); - for (Index zz=z; zz<l; zz++) - { - // push 0 down - Index firstColS = zz>f ? (zz-1) : zz; - G.makeGivens(m_T.coeff(zz, zz+1), m_T.coeff(zz+1, zz+1)); - m_S.rightCols(dim-firstColS).applyOnTheLeft(zz,zz+1,G.adjoint()); - m_T.rightCols(dim-zz).applyOnTheLeft(zz,zz+1,G.adjoint()); - m_T.coeffRef(zz+1,zz+1) = Scalar(0.0); - // update Q - if (m_computeQZ) - m_Q.applyOnTheRight(zz,zz+1,G); - // kill S(zz+1, zz-1) - if (zz>f) - { - G.makeGivens(m_S.coeff(zz+1, zz), m_S.coeff(zz+1,zz-1)); - m_S.topRows(zz+2).applyOnTheRight(zz, zz-1,G); - m_T.topRows(zz+1).applyOnTheRight(zz, zz-1,G); - m_S.coeffRef(zz+1,zz-1) = Scalar(0.0); - // update Z - if (m_computeQZ) - m_Z.applyOnTheLeft(zz,zz-1,G.adjoint()); - } - } - // finally kill S(l,l-1) - G.makeGivens(m_S.coeff(l,l), m_S.coeff(l,l-1)); - m_S.applyOnTheRight(l,l-1,G); - m_T.applyOnTheRight(l,l-1,G); - m_S.coeffRef(l,l-1)=Scalar(0.0); - // update Z - if (m_computeQZ) - m_Z.applyOnTheLeft(l,l-1,G.adjoint()); - } - - /** \internal QR-like iterative step for block f..l */ - template<typename MatrixType> - inline void RealQZ<MatrixType>::step(Index f, Index l, Index iter) - { - using std::abs; - const Index dim = m_S.cols(); - - // x, y, z - Scalar x, y, z; - if (iter==10) - { - // Wilkinson ad hoc shift - const Scalar - a11=m_S.coeff(f+0,f+0), a12=m_S.coeff(f+0,f+1), - a21=m_S.coeff(f+1,f+0), a22=m_S.coeff(f+1,f+1), a32=m_S.coeff(f+2,f+1), - b12=m_T.coeff(f+0,f+1), - b11i=Scalar(1.0)/m_T.coeff(f+0,f+0), - b22i=Scalar(1.0)/m_T.coeff(f+1,f+1), - a87=m_S.coeff(l-1,l-2), - a98=m_S.coeff(l-0,l-1), - b77i=Scalar(1.0)/m_T.coeff(l-2,l-2), - b88i=Scalar(1.0)/m_T.coeff(l-1,l-1); - Scalar ss = abs(a87*b77i) + abs(a98*b88i), - lpl = Scalar(1.5)*ss, - ll = ss*ss; - x = ll + a11*a11*b11i*b11i - lpl*a11*b11i + a12*a21*b11i*b22i - - a11*a21*b12*b11i*b11i*b22i; - y = a11*a21*b11i*b11i - lpl*a21*b11i + a21*a22*b11i*b22i - - a21*a21*b12*b11i*b11i*b22i; - z = a21*a32*b11i*b22i; - } - else if (iter==16) - { - // another exceptional shift - x = m_S.coeff(f,f)/m_T.coeff(f,f)-m_S.coeff(l,l)/m_T.coeff(l,l) + m_S.coeff(l,l-1)*m_T.coeff(l-1,l) / - (m_T.coeff(l-1,l-1)*m_T.coeff(l,l)); - y = m_S.coeff(f+1,f)/m_T.coeff(f,f); - z = 0; - } - else if (iter>23 && !(iter%8)) - { - // extremely exceptional shift - x = internal::random<Scalar>(-1.0,1.0); - y = internal::random<Scalar>(-1.0,1.0); - z = internal::random<Scalar>(-1.0,1.0); - } - else - { - // Compute the shifts: (x,y,z,0...) = (AB^-1 - l1 I) (AB^-1 - l2 I) e1 - // where l1 and l2 are the eigenvalues of the 2x2 matrix C = U V^-1 where - // U and V are 2x2 bottom right sub matrices of A and B. Thus: - // = AB^-1AB^-1 + l1 l2 I - (l1+l2)(AB^-1) - // = AB^-1AB^-1 + det(M) - tr(M)(AB^-1) - // Since we are only interested in having x, y, z with a correct ratio, we have: - const Scalar - a11 = m_S.coeff(f,f), a12 = m_S.coeff(f,f+1), - a21 = m_S.coeff(f+1,f), a22 = m_S.coeff(f+1,f+1), - a32 = m_S.coeff(f+2,f+1), - - a88 = m_S.coeff(l-1,l-1), a89 = m_S.coeff(l-1,l), - a98 = m_S.coeff(l,l-1), a99 = m_S.coeff(l,l), - - b11 = m_T.coeff(f,f), b12 = m_T.coeff(f,f+1), - b22 = m_T.coeff(f+1,f+1), - - b88 = m_T.coeff(l-1,l-1), b89 = m_T.coeff(l-1,l), - b99 = m_T.coeff(l,l); - - x = ( (a88/b88 - a11/b11)*(a99/b99 - a11/b11) - (a89/b99)*(a98/b88) + (a98/b88)*(b89/b99)*(a11/b11) ) * (b11/a21) - + a12/b22 - (a11/b11)*(b12/b22); - y = (a22/b22-a11/b11) - (a21/b11)*(b12/b22) - (a88/b88-a11/b11) - (a99/b99-a11/b11) + (a98/b88)*(b89/b99); - z = a32/b22; - } - - JRs G; - - for (Index k=f; k<=l-2; k++) - { - // variables for Householder reflections - Vector2s essential2; - Scalar tau, beta; - - Vector3s hr(x,y,z); - - // Q_k to annihilate S(k+1,k-1) and S(k+2,k-1) - hr.makeHouseholderInPlace(tau, beta); - essential2 = hr.template bottomRows<2>(); - Index fc=(std::max)(k-1,Index(0)); // first col to update - m_S.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data()); - m_T.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data()); - if (m_computeQZ) - m_Q.template middleCols<3>(k).applyHouseholderOnTheRight(essential2, tau, m_workspace.data()); - if (k>f) - m_S.coeffRef(k+2,k-1) = m_S.coeffRef(k+1,k-1) = Scalar(0.0); - - // Z_{k1} to annihilate T(k+2,k+1) and T(k+2,k) - hr << m_T.coeff(k+2,k+2),m_T.coeff(k+2,k),m_T.coeff(k+2,k+1); - hr.makeHouseholderInPlace(tau, beta); - essential2 = hr.template bottomRows<2>(); - { - Index lr = (std::min)(k+4,dim); // last row to update - Map<Matrix<Scalar,Dynamic,1> > tmp(m_workspace.data(),lr); - // S - tmp = m_S.template middleCols<2>(k).topRows(lr) * essential2; - tmp += m_S.col(k+2).head(lr); - m_S.col(k+2).head(lr) -= tau*tmp; - m_S.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint(); - // T - tmp = m_T.template middleCols<2>(k).topRows(lr) * essential2; - tmp += m_T.col(k+2).head(lr); - m_T.col(k+2).head(lr) -= tau*tmp; - m_T.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint(); - } - if (m_computeQZ) - { - // Z - Map<Matrix<Scalar,1,Dynamic> > tmp(m_workspace.data(),dim); - tmp = essential2.adjoint()*(m_Z.template middleRows<2>(k)); - tmp += m_Z.row(k+2); - m_Z.row(k+2) -= tau*tmp; - m_Z.template middleRows<2>(k) -= essential2 * (tau*tmp); - } - m_T.coeffRef(k+2,k) = m_T.coeffRef(k+2,k+1) = Scalar(0.0); - - // Z_{k2} to annihilate T(k+1,k) - G.makeGivens(m_T.coeff(k+1,k+1), m_T.coeff(k+1,k)); - m_S.applyOnTheRight(k+1,k,G); - m_T.applyOnTheRight(k+1,k,G); - // update Z - if (m_computeQZ) - m_Z.applyOnTheLeft(k+1,k,G.adjoint()); - m_T.coeffRef(k+1,k) = Scalar(0.0); - - // update x,y,z - x = m_S.coeff(k+1,k); - y = m_S.coeff(k+2,k); - if (k < l-2) - z = m_S.coeff(k+3,k); - } // loop over k - - // Q_{n-1} to annihilate y = S(l,l-2) - G.makeGivens(x,y); - m_S.applyOnTheLeft(l-1,l,G.adjoint()); - m_T.applyOnTheLeft(l-1,l,G.adjoint()); - if (m_computeQZ) - m_Q.applyOnTheRight(l-1,l,G); - m_S.coeffRef(l,l-2) = Scalar(0.0); - - // Z_{n-1} to annihilate T(l,l-1) - G.makeGivens(m_T.coeff(l,l),m_T.coeff(l,l-1)); - m_S.applyOnTheRight(l,l-1,G); - m_T.applyOnTheRight(l,l-1,G); - if (m_computeQZ) - m_Z.applyOnTheLeft(l,l-1,G.adjoint()); - m_T.coeffRef(l,l-1) = Scalar(0.0); - } - - template<typename MatrixType> - RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ) - { - - const Index dim = A_in.cols(); - - eigen_assert (A_in.rows()==dim && A_in.cols()==dim - && B_in.rows()==dim && B_in.cols()==dim - && "Need square matrices of the same dimension"); - - m_isInitialized = true; - m_computeQZ = computeQZ; - m_S = A_in; m_T = B_in; - m_workspace.resize(dim*2); - m_global_iter = 0; - - // entrance point: hessenberg triangular decomposition - hessenbergTriangular(); - // compute L1 vector norms of T, S into m_normOfS, m_normOfT - computeNorms(); - - Index l = dim-1, - f, - local_iter = 0; - - while (l>0 && local_iter<m_maxIters) - { - f = findSmallSubdiagEntry(l); - // now rows and columns f..l (including) decouple from the rest of the problem - if (f>0) m_S.coeffRef(f,f-1) = Scalar(0.0); - if (f == l) // One root found - { - l--; - local_iter = 0; - } - else if (f == l-1) // Two roots found - { - splitOffTwoRows(f); - l -= 2; - local_iter = 0; - } - else // No convergence yet - { - // if there's zero on diagonal of T, we can isolate an eigenvalue with Givens rotations - Index z = findSmallDiagEntry(f,l); - if (z>=f) - { - // zero found - pushDownZero(z,f,l); - } - else - { - // We are sure now that S.block(f,f, l-f+1,l-f+1) is underuced upper-Hessenberg - // and T.block(f,f, l-f+1,l-f+1) is invertible uper-triangular, which allows to - // apply a QR-like iteration to rows and columns f..l. - step(f,l, local_iter); - local_iter++; - m_global_iter++; - } - } - } - // check if we converged before reaching iterations limit - m_info = (local_iter<m_maxIters) ? Success : NoConvergence; - - // For each non triangular 2x2 diagonal block of S, - // reduce the respective 2x2 diagonal block of T to positive diagonal form using 2x2 SVD. - // This step is not mandatory for QZ, but it does help further extraction of eigenvalues/eigenvectors, - // and is in par with Lapack/Matlab QZ. - if(m_info==Success) - { - for(Index i=0; i<dim-1; ++i) - { - if(m_S.coeff(i+1, i) != Scalar(0)) - { - JacobiRotation<Scalar> j_left, j_right; - internal::real_2x2_jacobi_svd(m_T, i, i+1, &j_left, &j_right); - - // Apply resulting Jacobi rotations - m_S.applyOnTheLeft(i,i+1,j_left); - m_S.applyOnTheRight(i,i+1,j_right); - m_T.applyOnTheLeft(i,i+1,j_left); - m_T.applyOnTheRight(i,i+1,j_right); - m_T(i+1,i) = m_T(i,i+1) = Scalar(0); - - if(m_computeQZ) { - m_Q.applyOnTheRight(i,i+1,j_left.transpose()); - m_Z.applyOnTheLeft(i,i+1,j_right.transpose()); - } - - i++; - } - } - } - - return *this; - } // end compute - -} // end namespace Eigen - -#endif //EIGEN_REAL_QZ |