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diff --git a/src/3rdparty/eigen/Eigen/src/Eigenvalues/RealQZ.h b/src/3rdparty/eigen/Eigen/src/Eigenvalues/RealQZ.h new file mode 100644 index 000000000..509130184 --- /dev/null +++ b/src/3rdparty/eigen/Eigen/src/Eigenvalues/RealQZ.h @@ -0,0 +1,657 @@ +// 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), + m_computeQZ(true) + {} + + /** \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), + m_computeQZ(true) + { + 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 successful, \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 |