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++ // 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 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_TRIDIAGONALIZATION_H
++ #define EIGEN_TRIDIAGONALIZATION_H
++
++ // IWYU pragma: private
++ #include "./InternalHeaderCheck.h"
++
++ namespace Eigen {
++
++ namespace internal {
++
++ template <typename MatrixType>
++ struct TridiagonalizationMatrixTReturnType;
++ template <typename MatrixType>
++ struct traits<TridiagonalizationMatrixTReturnType<MatrixType>> : public traits<typename MatrixType::PlainObject> {
++ typedef typename MatrixType::PlainObject ReturnType; // FIXME shall it be a BandMatrix?
++ enum { Flags = 0 };
++ };
++
++ template <typename MatrixType, typename CoeffVectorType>
++ EIGEN_DEVICE_FUNC void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
++ } // namespace internal
++
++ /** \eigenvalues_module \ingroup Eigenvalues_Module
++ *
++ *
++ * \class Tridiagonalization
++ *
++ * \brief Tridiagonal decomposition of a selfadjoint matrix
++ *
++ * \tparam MatrixType_ the type of the matrix of which we are computing the
++ * tridiagonal decomposition; this is expected to be an instantiation of the
++ * Matrix class template.
++ *
++ * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
++ * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
++ *
++ * A tridiagonal matrix is a matrix which has nonzero elements only on the
++ * main diagonal and the first diagonal below and above it. The Hessenberg
++ * decomposition of a selfadjoint matrix is in fact a tridiagonal
++ * decomposition. This class is used in SelfAdjointEigenSolver to compute the
++ * eigenvalues and eigenvectors of a selfadjoint matrix.
++ *
++ * Call the function compute() to compute the tridiagonal decomposition of a
++ * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
++ * constructor which computes the tridiagonal Schur decomposition at
++ * construction time. Once the decomposition is computed, you can use the
++ * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
++ * decomposition.
++ *
++ * The documentation of Tridiagonalization(const MatrixType&) contains an
++ * example of the typical use of this class.
++ *
++ * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
++ */
++ template <typename MatrixType_>
++ class Tridiagonalization {
++ public:
++ /** \brief Synonym for the template parameter \p MatrixType_. */
++ typedef MatrixType_ MatrixType;
++
++ typedef typename MatrixType::Scalar Scalar;
++ typedef typename NumTraits<Scalar>::Real RealScalar;
++ typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
++
++ enum {
++ Size = MatrixType::RowsAtCompileTime,
++ SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
++ Options = internal::traits<MatrixType>::Options,
++ MaxSize = MatrixType::MaxRowsAtCompileTime,
++ MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
++ };
++
++ typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
++ typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
++ typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
++ typedef internal::remove_all_t<typename MatrixType::RealReturnType> MatrixTypeRealView;
++ typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
++
++ typedef std::conditional_t<NumTraits<Scalar>::IsComplex,
++ internal::add_const_on_value_type_t<typename Diagonal<const MatrixType>::RealReturnType>,
++ const Diagonal<const MatrixType>>
++ DiagonalReturnType;
++
++ typedef std::conditional_t<
++ NumTraits<Scalar>::IsComplex,
++ internal::add_const_on_value_type_t<typename Diagonal<const MatrixType, -1>::RealReturnType>,
++ const Diagonal<const MatrixType, -1>>
++ SubDiagonalReturnType;
++
++ /** \brief Return type of matrixQ() */
++ typedef HouseholderSequence<MatrixType, internal::remove_all_t<typename CoeffVectorType::ConjugateReturnType>>
++ HouseholderSequenceType;
++
++ /** \brief Default constructor.
++ *
++ * \param [in] size Positive integer, size of the matrix whose tridiagonal
++ * 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 Tridiagonalization(Index size = Size == Dynamic ? 2 : Size)
++ : m_matrix(size, size), m_hCoeffs(size > 1 ? size - 1 : 1), m_isInitialized(false) {}
++
++ /** \brief Constructor; computes tridiagonal decomposition of given matrix.
++ *
++ * \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
++ * is to be computed.
++ *
++ * This constructor calls compute() to compute the tridiagonal decomposition.
++ *
++ * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
++ * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
++ */
++ template <typename InputType>
++ explicit Tridiagonalization(const EigenBase<InputType>& matrix)
++ : m_matrix(matrix.derived()), m_hCoeffs(matrix.cols() > 1 ? matrix.cols() - 1 : 1), m_isInitialized(false) {
++ internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
++ m_isInitialized = true;
++ }
++
++ /** \brief Computes tridiagonal decomposition of given matrix.
++ *
++ * \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
++ * is to be computed.
++ * \returns Reference to \c *this
++ *
++ * The tridiagonal decomposition is computed by bringing the columns of
++ * the matrix successively in the required form using Householder
++ * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
++ * the size of the given matrix.
++ *
++ * This method reuses of the allocated data in the Tridiagonalization
++ * object, if the size of the matrix does not change.
++ *
++ * Example: \include Tridiagonalization_compute.cpp
++ * Output: \verbinclude Tridiagonalization_compute.out
++ */
++ template <typename InputType>
++ Tridiagonalization& compute(const EigenBase<InputType>& matrix) {
++ m_matrix = matrix.derived();
++ m_hCoeffs.resize(matrix.rows() - 1, 1);
++ internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
++ m_isInitialized = true;
++ return *this;
++ }
++
++ /** \brief Returns the Householder coefficients.
++ *
++ * \returns a const reference to the vector of Householder coefficients
++ *
++ * \pre Either the constructor Tridiagonalization(const MatrixType&) or
++ * the member function compute(const MatrixType&) has been called before
++ * to compute the tridiagonal decomposition of a matrix.
++ *
++ * The Householder coefficients allow the reconstruction of the matrix
++ * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
++ *
++ * Example: \include Tridiagonalization_householderCoefficients.cpp
++ * Output: \verbinclude Tridiagonalization_householderCoefficients.out
++ *
++ * \sa packedMatrix(), \ref Householder_Module "Householder module"
++ */
++ inline CoeffVectorType householderCoefficients() const {
++ eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
++ return m_hCoeffs;
++ }
++
++ /** \brief Returns the internal representation of the decomposition
++ *
++ * \returns a const reference to a matrix with the internal representation
++ * of the decomposition.
++ *
++ * \pre Either the constructor Tridiagonalization(const MatrixType&) or
++ * the member function compute(const MatrixType&) has been called before
++ * to compute the tridiagonal decomposition of a matrix.
++ *
++ * The returned matrix contains the following information:
++ * - the strict upper triangular part is equal to the input matrix A.
++ * - the diagonal and lower sub-diagonal represent the real tridiagonal
++ * symmetric matrix T.
++ * - the rest of the lower part contains the Householder vectors that,
++ * combined with Householder coefficients returned by
++ * householderCoefficients(), allows to reconstruct the matrix Q as
++ * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
++ * Here, the matrices \f$ H_i \f$ are the Householder transformations
++ * \f$ H_i = (I - h_i v_i v_i^T) \f$
++ * where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
++ * \f$ v_i \f$ is the Householder vector defined by
++ * \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
++ * with M the matrix returned by this function.
++ *
++ * See LAPACK for further details on this packed storage.
++ *
++ * Example: \include Tridiagonalization_packedMatrix.cpp
++ * Output: \verbinclude Tridiagonalization_packedMatrix.out
++ *
++ * \sa householderCoefficients()
++ */
++ inline const MatrixType& packedMatrix() const {
++ eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
++ return m_matrix;
++ }
++
++ /** \brief Returns the unitary matrix Q in the decomposition
++ *
++ * \returns object representing the matrix Q
++ *
++ * \pre Either the constructor Tridiagonalization(const MatrixType&) or
++ * the member function compute(const MatrixType&) has been called before
++ * to compute the tridiagonal decomposition of a matrix.
++ *
++ * This function returns a light-weight object of template class
++ * HouseholderSequence. You can either apply it directly to a matrix or
++ * you can convert it to a matrix of type #MatrixType.
++ *
++ * \sa Tridiagonalization(const MatrixType&) for an example,
++ * matrixT(), class HouseholderSequence
++ */
++ HouseholderSequenceType matrixQ() const {
++ eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
++ return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate()).setLength(m_matrix.rows() - 1).setShift(1);
++ }
++
++ /** \brief Returns an expression of the tridiagonal matrix T in the decomposition
++ *
++ * \returns expression object representing the matrix T
++ *
++ * \pre Either the constructor Tridiagonalization(const MatrixType&) or
++ * the member function compute(const MatrixType&) has been called before
++ * to compute the tridiagonal decomposition of a matrix.
++ *
++ * Currently, this function can be used to extract the matrix T from internal
++ * data and copy it to a dense matrix object. In most cases, it may be
++ * sufficient to directly use the packed matrix or the vector expressions
++ * returned by diagonal() and subDiagonal() instead of creating a new
++ * dense copy matrix with this function.
++ *
++ * \sa Tridiagonalization(const MatrixType&) for an example,
++ * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
++ */
++ MatrixTReturnType matrixT() const {
++ eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
++ return MatrixTReturnType(m_matrix.real());
++ }
++
++ /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
++ *
++ * \returns expression representing the diagonal of T
++ *
++ * \pre Either the constructor Tridiagonalization(const MatrixType&) or
++ * the member function compute(const MatrixType&) has been called before
++ * to compute the tridiagonal decomposition of a matrix.
++ *
++ * Example: \include Tridiagonalization_diagonal.cpp
++ * Output: \verbinclude Tridiagonalization_diagonal.out
++ *
++ * \sa matrixT(), subDiagonal()
++ */
++ DiagonalReturnType diagonal() const;
++
++ /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
++ *
++ * \returns expression representing the subdiagonal of T
++ *
++ * \pre Either the constructor Tridiagonalization(const MatrixType&) or
++ * the member function compute(const MatrixType&) has been called before
++ * to compute the tridiagonal decomposition of a matrix.
++ *
++ * \sa diagonal() for an example, matrixT()
++ */
++ SubDiagonalReturnType subDiagonal() const;
++
++ protected:
++ MatrixType m_matrix;
++ CoeffVectorType m_hCoeffs;
++ bool m_isInitialized;
++ };
++
++ template <typename MatrixType>
++ typename Tridiagonalization<MatrixType>::DiagonalReturnType Tridiagonalization<MatrixType>::diagonal() const {
++ eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
++ return m_matrix.diagonal().real();
++ }
++
++ template <typename MatrixType>
++ typename Tridiagonalization<MatrixType>::SubDiagonalReturnType Tridiagonalization<MatrixType>::subDiagonal() const {
++ eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
++ return m_matrix.template diagonal<-1>().real();
++ }
++
++ namespace internal {
++
++ /** \internal
++ * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
++ *
++ * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
++ * On output, the strict upper part is left unchanged, and the lower triangular part
++ * represents the T and Q matrices in packed format has detailed below.
++ * \param[out] hCoeffs returned Householder coefficients (see below)
++ *
++ * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
++ * and lower sub-diagonal of the matrix \a matA.
++ * The unitary matrix Q is represented in a compact way as a product of
++ * Householder reflectors \f$ H_i \f$ such that:
++ * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
++ * The Householder reflectors are defined as
++ * \f$ H_i = (I - h_i v_i v_i^T) \f$
++ * where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
++ * \f$ v_i \f$ is the Householder vector defined by
++ * \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
++ *
++ * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
++ *
++ * \sa Tridiagonalization::packedMatrix()
++ */
++ template <typename MatrixType, typename CoeffVectorType>
++ EIGEN_DEVICE_FUNC void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs) {
++ using numext::conj;
++ typedef typename MatrixType::Scalar Scalar;
++ typedef typename MatrixType::RealScalar RealScalar;
++ Index n = matA.rows();
++ eigen_assert(n == matA.cols());
++ eigen_assert(n == hCoeffs.size() + 1 || n == 1);
++
++ for (Index i = 0; i < n - 1; ++i) {
++ Index remainingSize = n - i - 1;
++ RealScalar beta;
++ Scalar h;
++ matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
++
++ // Apply similarity transformation to remaining columns,
++ // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
++ matA.col(i).coeffRef(i + 1) = Scalar(1);
++
++ hCoeffs.tail(n - i - 1).noalias() =
++ (matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>() *
++ (conj(h) * matA.col(i).tail(remainingSize)));
++
++ hCoeffs.tail(n - i - 1) +=
++ (conj(h) * RealScalar(-0.5) * (hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) *
++ matA.col(i).tail(n - i - 1);
++
++ matA.bottomRightCorner(remainingSize, remainingSize)
++ .template selfadjointView<Lower>()
++ .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), Scalar(-1));
++
++ matA.col(i).coeffRef(i + 1) = beta;
++ hCoeffs.coeffRef(i) = h;
++ }
++ }
++
++ // forward declaration, implementation at the end of this file
++ template <typename MatrixType, int Size = MatrixType::ColsAtCompileTime,
++ bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
++ struct tridiagonalization_inplace_selector;
++
++ /** \brief Performs a full tridiagonalization in place
++ *
++ * \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
++ * decomposition is to be computed. Only the lower triangular part referenced.
++ * The rest is left unchanged. On output, the orthogonal matrix Q
++ * in the decomposition if \p extractQ is true.
++ * \param[out] diag The diagonal of the tridiagonal matrix T in the
++ * decomposition.
++ * \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
++ * the decomposition.
++ * \param[out] hcoeffs
++ * \param[out] workspace
++ * \param[in] extractQ If true, the orthogonal matrix Q in the
++ * decomposition is computed and stored in \p mat.
++ *
++ * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
++ * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
++ * symmetric tridiagonal matrix.
++ *
++ * The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
++ * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
++ * part of the matrix \p mat is destroyed.
++ *
++ * The vectors \p diag and \p subdiag are not resized. The function
++ * assumes that they are already of the correct size. The length of the
++ * vector \p diag should equal the number of rows in \p mat, and the
++ * length of the vector \p subdiag should be one left.
++ *
++ * This implementation contains an optimized path for 3-by-3 matrices
++ * which is especially useful for plane fitting.
++ *
++ * \note Currently, it requires two temporary vectors to hold the intermediate
++ * Householder coefficients, and to reconstruct the matrix Q from the Householder
++ * reflectors.
++ *
++ * Example (this uses the same matrix as the example in
++ * Tridiagonalization::Tridiagonalization(const MatrixType&)):
++ * \include Tridiagonalization_decomposeInPlace.cpp
++ * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
++ *
++ * \sa class Tridiagonalization
++ */
++ template <typename MatrixType, typename DiagonalType, typename SubDiagonalType, typename CoeffVectorType,
++ typename WorkSpaceType>
++ EIGEN_DEVICE_FUNC void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag,
++ CoeffVectorType& hcoeffs, WorkSpaceType& workspace, bool extractQ) {
++ eigen_assert(mat.cols() == mat.rows() && diag.size() == mat.rows() && subdiag.size() == mat.rows() - 1);
++ tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, hcoeffs, workspace, extractQ);
++ }
++
++ /** \internal
++ * General full tridiagonalization
++ */
++ template <typename MatrixType, int Size, bool IsComplex>
++ struct tridiagonalization_inplace_selector {
++ typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
++ template <typename DiagonalType, typename SubDiagonalType, typename CoeffVectorType, typename WorkSpaceType>
++ static EIGEN_DEVICE_FUNC void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag,
++ CoeffVectorType& hCoeffs, WorkSpaceType& workspace, bool extractQ) {
++ tridiagonalization_inplace(mat, hCoeffs);
++ diag = mat.diagonal().real();
++ subdiag = mat.template diagonal<-1>().real();
++ if (extractQ) {
++ HouseholderSequenceType(mat, hCoeffs.conjugate()).setLength(mat.rows() - 1).setShift(1).evalTo(mat, workspace);
++ }
++ }
++ };
++
++ /** \internal
++ * Specialization for 3x3 real matrices.
++ * Especially useful for plane fitting.
++ */
++ template <typename MatrixType>
++ struct tridiagonalization_inplace_selector<MatrixType, 3, false> {
++ typedef typename MatrixType::Scalar Scalar;
++ typedef typename MatrixType::RealScalar RealScalar;
++
++ template <typename DiagonalType, typename SubDiagonalType, typename CoeffVectorType, typename WorkSpaceType>
++ static EIGEN_DEVICE_FUNC void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, CoeffVectorType&,
++ WorkSpaceType&, bool extractQ) {
++ using std::sqrt;
++ const RealScalar tol = (std::numeric_limits<RealScalar>::min)();
++ diag[0] = mat(0, 0);
++ RealScalar v1norm2 = numext::abs2(mat(2, 0));
++ if (v1norm2 <= tol) {
++ diag[1] = mat(1, 1);
++ diag[2] = mat(2, 2);
++ subdiag[0] = mat(1, 0);
++ subdiag[1] = mat(2, 1);
++ if (extractQ) mat.setIdentity();
++ } else {
++ RealScalar beta = sqrt(numext::abs2(mat(1, 0)) + v1norm2);
++ RealScalar invBeta = RealScalar(1) / beta;
++ Scalar m01 = mat(1, 0) * invBeta;
++ Scalar m02 = mat(2, 0) * invBeta;
++ Scalar q = RealScalar(2) * m01 * mat(2, 1) + m02 * (mat(2, 2) - mat(1, 1));
++ diag[1] = mat(1, 1) + m02 * q;
++ diag[2] = mat(2, 2) - m02 * q;
++ subdiag[0] = beta;
++ subdiag[1] = mat(2, 1) - m01 * q;
++ if (extractQ) {
++ mat << 1, 0, 0, 0, m01, m02, 0, m02, -m01;
++ }
++ }
++ }
++ };
++
++ /** \internal
++ * Trivial specialization for 1x1 matrices
++ */
++ template <typename MatrixType, bool IsComplex>
++ struct tridiagonalization_inplace_selector<MatrixType, 1, IsComplex> {
++ typedef typename MatrixType::Scalar Scalar;
++
++ template <typename DiagonalType, typename SubDiagonalType, typename CoeffVectorType, typename WorkSpaceType>
++ static EIGEN_DEVICE_FUNC void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, CoeffVectorType&,
++ WorkSpaceType&, bool extractQ) {
++ diag(0, 0) = numext::real(mat(0, 0));
++ if (extractQ) mat(0, 0) = Scalar(1);
++ }
++ };
++
++ /** \internal
++ * \eigenvalues_module \ingroup Eigenvalues_Module
++ *
++ * \brief Expression type for return value of Tridiagonalization::matrixT()
++ *
++ * \tparam MatrixType type of underlying dense matrix
++ */
++ template <typename MatrixType>
++ struct TridiagonalizationMatrixTReturnType : public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType>> {
++ public:
++ /** \brief Constructor.
++ *
++ * \param[in] mat The underlying dense matrix
++ */
++ TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) {}
++
++ template <typename ResultType>
++ inline void evalTo(ResultType& result) const {
++ result.setZero();
++ result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
++ result.diagonal() = m_matrix.diagonal();
++ result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
++ }
++
++ constexpr Index rows() const noexcept { return m_matrix.rows(); }
++ constexpr Index cols() const noexcept { return m_matrix.cols(); }
++
++ protected:
++ typename MatrixType::Nested m_matrix;
++ };
++
++ } // end namespace internal
++
++ } // end namespace Eigen
++
++ #endif // EIGEN_TRIDIAGONALIZATION_H