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Diffstat (limited to 'runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/EigenSolver.h')
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diff --git a/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/EigenSolver.h b/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/EigenSolver.h deleted file mode 100644 index f205b185d..000000000 --- a/runtimes/nn/depend/external/eigen/Eigen/src/Eigenvalues/EigenSolver.h +++ /dev/null @@ -1,622 +0,0 @@ -// 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_EIGENSOLVER_H -#define EIGEN_EIGENSOLVER_H - -#include "./RealSchur.h" - -namespace Eigen { - -/** \eigenvalues_module \ingroup Eigenvalues_Module - * - * - * \class EigenSolver - * - * \brief Computes eigenvalues and eigenvectors of general matrices - * - * \tparam _MatrixType the type of the matrix of which we are computing the - * eigendecomposition; this is expected to be an instantiation of the Matrix - * class template. Currently, only real matrices are supported. - * - * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars - * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$. If - * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and - * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V = - * V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we - * have \f$ A = V D V^{-1} \f$. This is called the eigendecomposition. - * - * The eigenvalues and eigenvectors of a matrix may be complex, even when the - * matrix is real. However, we can choose real matrices \f$ V \f$ and \f$ D - * \f$ satisfying \f$ A V = V D \f$, just like the eigendecomposition, if the - * matrix \f$ D \f$ is not required to be diagonal, but if it is allowed to - * have blocks of the form - * \f[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f] - * (where \f$ u \f$ and \f$ v \f$ are real numbers) on the diagonal. These - * blocks correspond to complex eigenvalue pairs \f$ u \pm iv \f$. We call - * this variant of the eigendecomposition the pseudo-eigendecomposition. - * - * Call the function compute() to compute the eigenvalues and eigenvectors of - * a given matrix. Alternatively, you can use the - * EigenSolver(const MatrixType&, bool) constructor which computes the - * eigenvalues and eigenvectors at construction time. Once the eigenvalue and - * eigenvectors are computed, they can be retrieved with the eigenvalues() and - * eigenvectors() functions. The pseudoEigenvalueMatrix() and - * pseudoEigenvectors() methods allow the construction of the - * pseudo-eigendecomposition. - * - * The documentation for EigenSolver(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 MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver - */ -template<typename _MatrixType> class EigenSolver -{ - public: - - /** \brief Synonym for the template parameter \p _MatrixType. */ - typedef _MatrixType MatrixType; - - enum { - RowsAtCompileTime = MatrixType::RowsAtCompileTime, - ColsAtCompileTime = MatrixType::ColsAtCompileTime, - Options = MatrixType::Options, - MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, - MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime - }; - - /** \brief Scalar type for matrices of type #MatrixType. */ - typedef typename MatrixType::Scalar Scalar; - typedef typename NumTraits<Scalar>::Real RealScalar; - typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 - - /** \brief Complex scalar type for #MatrixType. - * - * This is \c std::complex<Scalar> if #Scalar is real (e.g., - * \c float or \c double) and just \c Scalar if #Scalar is - * complex. - */ - typedef std::complex<RealScalar> ComplexScalar; - - /** \brief Type for vector of eigenvalues as returned by eigenvalues(). - * - * This is a column vector with entries of type #ComplexScalar. - * The length of the vector is the size of #MatrixType. - */ - typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType; - - /** \brief Type for matrix of eigenvectors as returned by eigenvectors(). - * - * This is a square matrix with entries of type #ComplexScalar. - * The size is the same as the size of #MatrixType. - */ - typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType; - - /** \brief Default constructor. - * - * The default constructor is useful in cases in which the user intends to - * perform decompositions via EigenSolver::compute(const MatrixType&, bool). - * - * \sa compute() for an example. - */ - EigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false), m_realSchur(), m_matT(), m_tmp() {} - - /** \brief Default constructor with memory preallocation - * - * Like the default constructor but with preallocation of the internal data - * according to the specified problem \a size. - * \sa EigenSolver() - */ - explicit EigenSolver(Index size) - : m_eivec(size, size), - m_eivalues(size), - m_isInitialized(false), - m_eigenvectorsOk(false), - m_realSchur(size), - m_matT(size, size), - m_tmp(size) - {} - - /** \brief Constructor; computes eigendecomposition of given matrix. - * - * \param[in] matrix Square matrix whose eigendecomposition is to be computed. - * \param[in] computeEigenvectors If true, both the eigenvectors and the - * eigenvalues are computed; if false, only the eigenvalues are - * computed. - * - * This constructor calls compute() to compute the eigenvalues - * and eigenvectors. - * - * Example: \include EigenSolver_EigenSolver_MatrixType.cpp - * Output: \verbinclude EigenSolver_EigenSolver_MatrixType.out - * - * \sa compute() - */ - template<typename InputType> - explicit EigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true) - : m_eivec(matrix.rows(), matrix.cols()), - m_eivalues(matrix.cols()), - m_isInitialized(false), - m_eigenvectorsOk(false), - m_realSchur(matrix.cols()), - m_matT(matrix.rows(), matrix.cols()), - m_tmp(matrix.cols()) - { - compute(matrix.derived(), computeEigenvectors); - } - - /** \brief Returns the eigenvectors of given matrix. - * - * \returns %Matrix whose columns are the (possibly complex) eigenvectors. - * - * \pre Either the constructor - * EigenSolver(const MatrixType&,bool) or the member function - * compute(const MatrixType&, bool) has been called before, and - * \p computeEigenvectors was set to true (the default). - * - * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding - * to eigenvalue number \f$ k \f$ as returned by eigenvalues(). The - * eigenvectors are normalized to have (Euclidean) norm equal to one. The - * matrix returned by this function is the matrix \f$ V \f$ in the - * eigendecomposition \f$ A = V D V^{-1} \f$, if it exists. - * - * Example: \include EigenSolver_eigenvectors.cpp - * Output: \verbinclude EigenSolver_eigenvectors.out - * - * \sa eigenvalues(), pseudoEigenvectors() - */ - EigenvectorsType eigenvectors() const; - - /** \brief Returns the pseudo-eigenvectors of given matrix. - * - * \returns Const reference to matrix whose columns are the pseudo-eigenvectors. - * - * \pre Either the constructor - * EigenSolver(const MatrixType&,bool) or the member function - * compute(const MatrixType&, bool) has been called before, and - * \p computeEigenvectors was set to true (the default). - * - * The real matrix \f$ V \f$ returned by this function and the - * block-diagonal matrix \f$ D \f$ returned by pseudoEigenvalueMatrix() - * satisfy \f$ AV = VD \f$. - * - * Example: \include EigenSolver_pseudoEigenvectors.cpp - * Output: \verbinclude EigenSolver_pseudoEigenvectors.out - * - * \sa pseudoEigenvalueMatrix(), eigenvectors() - */ - const MatrixType& pseudoEigenvectors() const - { - eigen_assert(m_isInitialized && "EigenSolver is not initialized."); - eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues."); - return m_eivec; - } - - /** \brief Returns the block-diagonal matrix in the pseudo-eigendecomposition. - * - * \returns A block-diagonal matrix. - * - * \pre Either the constructor - * EigenSolver(const MatrixType&,bool) or the member function - * compute(const MatrixType&, bool) has been called before. - * - * The matrix \f$ D \f$ returned by this function is real and - * block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 - * blocks of the form - * \f$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f$. - * These blocks are not sorted in any particular order. - * The matrix \f$ D \f$ and the matrix \f$ V \f$ returned by - * pseudoEigenvectors() satisfy \f$ AV = VD \f$. - * - * \sa pseudoEigenvectors() for an example, eigenvalues() - */ - MatrixType pseudoEigenvalueMatrix() const; - - /** \brief Returns the eigenvalues of given matrix. - * - * \returns A const reference to the column vector containing the eigenvalues. - * - * \pre Either the constructor - * EigenSolver(const MatrixType&,bool) or the member function - * compute(const MatrixType&, bool) has been called before. - * - * The eigenvalues are repeated according to their algebraic multiplicity, - * so there are as many eigenvalues as rows in the matrix. The eigenvalues - * are not sorted in any particular order. - * - * Example: \include EigenSolver_eigenvalues.cpp - * Output: \verbinclude EigenSolver_eigenvalues.out - * - * \sa eigenvectors(), pseudoEigenvalueMatrix(), - * MatrixBase::eigenvalues() - */ - const EigenvalueType& eigenvalues() const - { - eigen_assert(m_isInitialized && "EigenSolver is not initialized."); - return m_eivalues; - } - - /** \brief Computes eigendecomposition of given matrix. - * - * \param[in] matrix Square matrix whose eigendecomposition is to be computed. - * \param[in] computeEigenvectors If true, both the eigenvectors and the - * eigenvalues are computed; if false, only the eigenvalues are - * computed. - * \returns Reference to \c *this - * - * This function computes the eigenvalues of the real matrix \p matrix. - * The eigenvalues() function can be used to retrieve them. If - * \p computeEigenvectors is true, then the eigenvectors are also computed - * and can be retrieved by calling eigenvectors(). - * - * The matrix is first reduced to real Schur form using the RealSchur - * class. The Schur decomposition is then used to compute the eigenvalues - * and eigenvectors. - * - * The cost of the computation is dominated by the cost of the - * Schur decomposition, which is very approximately \f$ 25n^3 \f$ - * (where \f$ n \f$ is the size of the matrix) if \p computeEigenvectors - * is true, and \f$ 10n^3 \f$ if \p computeEigenvectors is false. - * - * This method reuses of the allocated data in the EigenSolver object. - * - * Example: \include EigenSolver_compute.cpp - * Output: \verbinclude EigenSolver_compute.out - */ - template<typename InputType> - EigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true); - - /** \returns NumericalIssue if the input contains INF or NaN values or overflow occured. Returns Success otherwise. */ - ComputationInfo info() const - { - eigen_assert(m_isInitialized && "EigenSolver is not initialized."); - return m_info; - } - - /** \brief Sets the maximum number of iterations allowed. */ - EigenSolver& setMaxIterations(Index maxIters) - { - m_realSchur.setMaxIterations(maxIters); - return *this; - } - - /** \brief Returns the maximum number of iterations. */ - Index getMaxIterations() - { - return m_realSchur.getMaxIterations(); - } - - private: - void doComputeEigenvectors(); - - protected: - - static void check_template_parameters() - { - EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); - EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL); - } - - MatrixType m_eivec; - EigenvalueType m_eivalues; - bool m_isInitialized; - bool m_eigenvectorsOk; - ComputationInfo m_info; - RealSchur<MatrixType> m_realSchur; - MatrixType m_matT; - - typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType; - ColumnVectorType m_tmp; -}; - -template<typename MatrixType> -MatrixType EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const -{ - eigen_assert(m_isInitialized && "EigenSolver is not initialized."); - const RealScalar precision = RealScalar(2)*NumTraits<RealScalar>::epsilon(); - Index n = m_eivalues.rows(); - MatrixType matD = MatrixType::Zero(n,n); - for (Index i=0; i<n; ++i) - { - if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)), precision)) - matD.coeffRef(i,i) = numext::real(m_eivalues.coeff(i)); - else - { - matD.template block<2,2>(i,i) << numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)), - -numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)); - ++i; - } - } - return matD; -} - -template<typename MatrixType> -typename EigenSolver<MatrixType>::EigenvectorsType EigenSolver<MatrixType>::eigenvectors() const -{ - eigen_assert(m_isInitialized && "EigenSolver is not initialized."); - eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues."); - const RealScalar precision = RealScalar(2)*NumTraits<RealScalar>::epsilon(); - Index n = m_eivec.cols(); - EigenvectorsType matV(n,n); - for (Index j=0; j<n; ++j) - { - if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j)), precision) || j+1==n) - { - // we have a real eigen value - matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>(); - matV.col(j).normalize(); - } - else - { - // we have a pair of complex eigen values - for (Index i=0; i<n; ++i) - { - matV.coeffRef(i,j) = ComplexScalar(m_eivec.coeff(i,j), m_eivec.coeff(i,j+1)); - matV.coeffRef(i,j+1) = ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1)); - } - matV.col(j).normalize(); - matV.col(j+1).normalize(); - ++j; - } - } - return matV; -} - -template<typename MatrixType> -template<typename InputType> -EigenSolver<MatrixType>& -EigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeEigenvectors) -{ - check_template_parameters(); - - using std::sqrt; - using std::abs; - using numext::isfinite; - eigen_assert(matrix.cols() == matrix.rows()); - - // Reduce to real Schur form. - m_realSchur.compute(matrix.derived(), computeEigenvectors); - - m_info = m_realSchur.info(); - - if (m_info == Success) - { - m_matT = m_realSchur.matrixT(); - if (computeEigenvectors) - m_eivec = m_realSchur.matrixU(); - - // Compute eigenvalues from matT - m_eivalues.resize(matrix.cols()); - Index i = 0; - while (i < matrix.cols()) - { - if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) == Scalar(0)) - { - m_eivalues.coeffRef(i) = m_matT.coeff(i, i); - if(!(isfinite)(m_eivalues.coeffRef(i))) - { - m_isInitialized = true; - m_eigenvectorsOk = false; - m_info = NumericalIssue; - return *this; - } - ++i; - } - else - { - Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1)); - Scalar z; - // Compute z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1))); - // without overflow - { - Scalar t0 = m_matT.coeff(i+1, i); - Scalar t1 = m_matT.coeff(i, i+1); - Scalar maxval = numext::maxi<Scalar>(abs(p),numext::maxi<Scalar>(abs(t0),abs(t1))); - t0 /= maxval; - t1 /= maxval; - Scalar p0 = p/maxval; - z = maxval * sqrt(abs(p0 * p0 + t0 * t1)); - } - - m_eivalues.coeffRef(i) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, z); - m_eivalues.coeffRef(i+1) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z); - if(!((isfinite)(m_eivalues.coeffRef(i)) && (isfinite)(m_eivalues.coeffRef(i+1)))) - { - m_isInitialized = true; - m_eigenvectorsOk = false; - m_info = NumericalIssue; - return *this; - } - i += 2; - } - } - - // Compute eigenvectors. - if (computeEigenvectors) - doComputeEigenvectors(); - } - - m_isInitialized = true; - m_eigenvectorsOk = computeEigenvectors; - - return *this; -} - - -template<typename MatrixType> -void EigenSolver<MatrixType>::doComputeEigenvectors() -{ - using std::abs; - const Index size = m_eivec.cols(); - const Scalar eps = NumTraits<Scalar>::epsilon(); - - // inefficient! this is already computed in RealSchur - Scalar norm(0); - for (Index j = 0; j < size; ++j) - { - norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum(); - } - - // Backsubstitute to find vectors of upper triangular form - if (norm == Scalar(0)) - { - return; - } - - for (Index n = size-1; n >= 0; n--) - { - Scalar p = m_eivalues.coeff(n).real(); - Scalar q = m_eivalues.coeff(n).imag(); - - // Scalar vector - if (q == Scalar(0)) - { - Scalar lastr(0), lastw(0); - Index l = n; - - m_matT.coeffRef(n,n) = Scalar(1); - for (Index i = n-1; i >= 0; i--) - { - Scalar w = m_matT.coeff(i,i) - p; - Scalar r = m_matT.row(i).segment(l,n-l+1).dot(m_matT.col(n).segment(l, n-l+1)); - - if (m_eivalues.coeff(i).imag() < Scalar(0)) - { - lastw = w; - lastr = r; - } - else - { - l = i; - if (m_eivalues.coeff(i).imag() == Scalar(0)) - { - if (w != Scalar(0)) - m_matT.coeffRef(i,n) = -r / w; - else - m_matT.coeffRef(i,n) = -r / (eps * norm); - } - else // Solve real equations - { - Scalar x = m_matT.coeff(i,i+1); - Scalar y = m_matT.coeff(i+1,i); - Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag(); - Scalar t = (x * lastr - lastw * r) / denom; - m_matT.coeffRef(i,n) = t; - if (abs(x) > abs(lastw)) - m_matT.coeffRef(i+1,n) = (-r - w * t) / x; - else - m_matT.coeffRef(i+1,n) = (-lastr - y * t) / lastw; - } - - // Overflow control - Scalar t = abs(m_matT.coeff(i,n)); - if ((eps * t) * t > Scalar(1)) - m_matT.col(n).tail(size-i) /= t; - } - } - } - else if (q < Scalar(0) && n > 0) // Complex vector - { - Scalar lastra(0), lastsa(0), lastw(0); - Index l = n-1; - - // Last vector component imaginary so matrix is triangular - if (abs(m_matT.coeff(n,n-1)) > abs(m_matT.coeff(n-1,n))) - { - m_matT.coeffRef(n-1,n-1) = q / m_matT.coeff(n,n-1); - m_matT.coeffRef(n-1,n) = -(m_matT.coeff(n,n) - p) / m_matT.coeff(n,n-1); - } - else - { - ComplexScalar cc = ComplexScalar(Scalar(0),-m_matT.coeff(n-1,n)) / ComplexScalar(m_matT.coeff(n-1,n-1)-p,q); - m_matT.coeffRef(n-1,n-1) = numext::real(cc); - m_matT.coeffRef(n-1,n) = numext::imag(cc); - } - m_matT.coeffRef(n,n-1) = Scalar(0); - m_matT.coeffRef(n,n) = Scalar(1); - for (Index i = n-2; i >= 0; i--) - { - Scalar ra = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n-1).segment(l, n-l+1)); - Scalar sa = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n).segment(l, n-l+1)); - Scalar w = m_matT.coeff(i,i) - p; - - if (m_eivalues.coeff(i).imag() < Scalar(0)) - { - lastw = w; - lastra = ra; - lastsa = sa; - } - else - { - l = i; - if (m_eivalues.coeff(i).imag() == RealScalar(0)) - { - ComplexScalar cc = ComplexScalar(-ra,-sa) / ComplexScalar(w,q); - m_matT.coeffRef(i,n-1) = numext::real(cc); - m_matT.coeffRef(i,n) = numext::imag(cc); - } - else - { - // Solve complex equations - Scalar x = m_matT.coeff(i,i+1); - Scalar y = m_matT.coeff(i+1,i); - Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q; - Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q; - if ((vr == Scalar(0)) && (vi == Scalar(0))) - vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw)); - - ComplexScalar cc = ComplexScalar(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra) / ComplexScalar(vr,vi); - m_matT.coeffRef(i,n-1) = numext::real(cc); - m_matT.coeffRef(i,n) = numext::imag(cc); - if (abs(x) > (abs(lastw) + abs(q))) - { - m_matT.coeffRef(i+1,n-1) = (-ra - w * m_matT.coeff(i,n-1) + q * m_matT.coeff(i,n)) / x; - m_matT.coeffRef(i+1,n) = (-sa - w * m_matT.coeff(i,n) - q * m_matT.coeff(i,n-1)) / x; - } - else - { - cc = ComplexScalar(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n)) / ComplexScalar(lastw,q); - m_matT.coeffRef(i+1,n-1) = numext::real(cc); - m_matT.coeffRef(i+1,n) = numext::imag(cc); - } - } - - // Overflow control - Scalar t = numext::maxi<Scalar>(abs(m_matT.coeff(i,n-1)),abs(m_matT.coeff(i,n))); - if ((eps * t) * t > Scalar(1)) - m_matT.block(i, n-1, size-i, 2) /= t; - - } - } - - // We handled a pair of complex conjugate eigenvalues, so need to skip them both - n--; - } - else - { - eigen_assert(0 && "Internal bug in EigenSolver (INF or NaN has not been detected)"); // this should not happen - } - } - - // Back transformation to get eigenvectors of original matrix - for (Index j = size-1; j >= 0; j--) - { - m_tmp.noalias() = m_eivec.leftCols(j+1) * m_matT.col(j).segment(0, j+1); - m_eivec.col(j) = m_tmp; - } -} - -} // end namespace Eigen - -#endif // EIGEN_EIGENSOLVER_H |