diff options
Diffstat (limited to 'Eigen/src/Eigenvalues/GeneralizedEigenSolver.h')
| -rw-r--r-- | Eigen/src/Eigenvalues/GeneralizedEigenSolver.h | 197 |
1 files changed, 133 insertions, 64 deletions
diff --git a/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h b/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h index 956e80d9e..36a91dffc 100644 --- a/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h +++ b/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h @@ -1,8 +1,9 @@ // This file is part of Eigen, a lightweight C++ template library // for linear algebra. // -// Copyright (C) 2012 Gael Guennebaud <gael.guennebaud@inria.fr> +// Copyright (C) 2012-2016 Gael Guennebaud <gael.guennebaud@inria.fr> // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk> +// Copyright (C) 2016 Tobias Wood <tobias@spinicist.org.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 @@ -72,7 +73,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver /** \brief Scalar type for matrices of type #MatrixType. */ typedef typename MatrixType::Scalar Scalar; typedef typename NumTraits<Scalar>::Real RealScalar; - typedef typename MatrixType::Index Index; + typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 /** \brief Complex scalar type for #MatrixType. * @@ -89,7 +90,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver */ typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> VectorType; - /** \brief Type for vector of complex scalar values eigenvalues as returned by betas(). + /** \brief Type for vector of complex scalar values eigenvalues as returned by alphas(). * * This is a column vector with entries of type #ComplexScalar. * The length of the vector is the size of #MatrixType. @@ -114,7 +115,14 @@ template<typename _MatrixType> class GeneralizedEigenSolver * * \sa compute() for an example. */ - GeneralizedEigenSolver() : m_eivec(), m_alphas(), m_betas(), m_isInitialized(false), m_realQZ(), m_matS(), m_tmp() {} + GeneralizedEigenSolver() + : m_eivec(), + m_alphas(), + m_betas(), + m_valuesOkay(false), + m_vectorsOkay(false), + m_realQZ() + {} /** \brief Default constructor with memory preallocation * @@ -122,14 +130,13 @@ template<typename _MatrixType> class GeneralizedEigenSolver * according to the specified problem \a size. * \sa GeneralizedEigenSolver() */ - GeneralizedEigenSolver(Index size) + explicit GeneralizedEigenSolver(Index size) : m_eivec(size, size), m_alphas(size), m_betas(size), - m_isInitialized(false), - m_eigenvectorsOk(false), + m_valuesOkay(false), + m_vectorsOkay(false), m_realQZ(size), - m_matS(size, size), m_tmp(size) {} @@ -149,10 +156,9 @@ template<typename _MatrixType> class GeneralizedEigenSolver : m_eivec(A.rows(), A.cols()), m_alphas(A.cols()), m_betas(A.cols()), - m_isInitialized(false), - m_eigenvectorsOk(false), + m_valuesOkay(false), + m_vectorsOkay(false), m_realQZ(A.cols()), - m_matS(A.rows(), A.cols()), m_tmp(A.cols()) { compute(A, B, computeEigenvectors); @@ -160,22 +166,20 @@ template<typename _MatrixType> class GeneralizedEigenSolver /* \brief Returns the computed generalized eigenvectors. * - * \returns %Matrix whose columns are the (possibly complex) eigenvectors. + * \returns %Matrix whose columns are the (possibly complex) right eigenvectors. + * i.e. the eigenvectors that solve (A - l*B)x = 0. The ordering matches the eigenvalues. * * \pre Either the constructor * GeneralizedEigenSolver(const MatrixType&,const MatrixType&, bool) or the member function * compute(const MatrixType&, 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 - * generalized eigendecomposition \f$ A = B V D V^{-1} \f$, if it exists. - * * \sa eigenvalues() */ -// EigenvectorsType eigenvectors() const; + EigenvectorsType eigenvectors() const { + eigen_assert(m_vectorsOkay && "Eigenvectors for GeneralizedEigenSolver were not calculated."); + return m_eivec; + } /** \brief Returns an expression of the computed generalized eigenvalues. * @@ -197,7 +201,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver */ EigenvalueType eigenvalues() const { - eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized."); + eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized."); return EigenvalueType(m_alphas,m_betas); } @@ -208,7 +212,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver * \sa betas(), eigenvalues() */ ComplexVectorType alphas() const { - eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized."); + eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized."); return m_alphas; } @@ -219,7 +223,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver * \sa alphas(), eigenvalues() */ VectorType betas() const { - eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized."); + eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized."); return m_betas; } @@ -250,7 +254,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver ComputationInfo info() const { - eigen_assert(m_isInitialized && "EigenSolver is not initialized."); + eigen_assert(m_valuesOkay && "EigenSolver is not initialized."); return m_realQZ.info(); } @@ -270,29 +274,14 @@ template<typename _MatrixType> class GeneralizedEigenSolver EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL); } - MatrixType m_eivec; + EigenvectorsType m_eivec; ComplexVectorType m_alphas; VectorType m_betas; - bool m_isInitialized; - bool m_eigenvectorsOk; + bool m_valuesOkay, m_vectorsOkay; RealQZ<MatrixType> m_realQZ; - MatrixType m_matS; - - typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType; - ColumnVectorType m_tmp; + ComplexVectorType m_tmp; }; -//template<typename MatrixType> -//typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType GeneralizedEigenSolver<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."); -// Index n = m_eivec.cols(); -// EigenvectorsType matV(n,n); -// // TODO -// return matV; -//} - template<typename MatrixType> GeneralizedEigenSolver<MatrixType>& GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors) @@ -302,46 +291,126 @@ GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixTyp using std::sqrt; using std::abs; eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows()); - + Index size = A.cols(); + m_valuesOkay = false; + m_vectorsOkay = false; // Reduce to generalized real Schur form: // A = Q S Z and B = Q T Z m_realQZ.compute(A, B, computeEigenvectors); - if (m_realQZ.info() == Success) { - m_matS = m_realQZ.matrixS(); + // Resize storage + m_alphas.resize(size); + m_betas.resize(size); if (computeEigenvectors) - m_eivec = m_realQZ.matrixZ().transpose(); - - // Compute eigenvalues from matS - m_alphas.resize(A.cols()); - m_betas.resize(A.cols()); + { + m_eivec.resize(size,size); + m_tmp.resize(size); + } + + // Aliases: + Map<VectorType> v(reinterpret_cast<Scalar*>(m_tmp.data()), size); + ComplexVectorType &cv = m_tmp; + const MatrixType &mZ = m_realQZ.matrixZ(); + const MatrixType &mS = m_realQZ.matrixS(); + const MatrixType &mT = m_realQZ.matrixT(); + Index i = 0; - while (i < A.cols()) + while (i < size) { - if (i == A.cols() - 1 || m_matS.coeff(i+1, i) == Scalar(0)) + if (i == size - 1 || mS.coeff(i+1, i) == Scalar(0)) { - m_alphas.coeffRef(i) = m_matS.coeff(i, i); - m_betas.coeffRef(i) = m_realQZ.matrixT().coeff(i,i); + // Real eigenvalue + m_alphas.coeffRef(i) = mS.diagonal().coeff(i); + m_betas.coeffRef(i) = mT.diagonal().coeff(i); + if (computeEigenvectors) + { + v.setConstant(Scalar(0.0)); + v.coeffRef(i) = Scalar(1.0); + // For singular eigenvalues do nothing more + if(abs(m_betas.coeffRef(i)) >= (std::numeric_limits<RealScalar>::min)()) + { + // Non-singular eigenvalue + const Scalar alpha = real(m_alphas.coeffRef(i)); + const Scalar beta = m_betas.coeffRef(i); + for (Index j = i-1; j >= 0; j--) + { + const Index st = j+1; + const Index sz = i-j; + if (j > 0 && mS.coeff(j, j-1) != Scalar(0)) + { + // 2x2 block + Matrix<Scalar, 2, 1> rhs = (alpha*mT.template block<2,Dynamic>(j-1,st,2,sz) - beta*mS.template block<2,Dynamic>(j-1,st,2,sz)) .lazyProduct( v.segment(st,sz) ); + Matrix<Scalar, 2, 2> lhs = beta * mS.template block<2,2>(j-1,j-1) - alpha * mT.template block<2,2>(j-1,j-1); + v.template segment<2>(j-1) = lhs.partialPivLu().solve(rhs); + j--; + } + else + { + v.coeffRef(j) = -v.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum() / (beta*mS.coeffRef(j,j) - alpha*mT.coeffRef(j,j)); + } + } + } + m_eivec.col(i).real().noalias() = mZ.transpose() * v; + m_eivec.col(i).real().normalize(); + m_eivec.col(i).imag().setConstant(0); + } ++i; } else { - Scalar p = Scalar(0.5) * (m_matS.coeff(i, i) - m_matS.coeff(i+1, i+1)); - Scalar z = sqrt(abs(p * p + m_matS.coeff(i+1, i) * m_matS.coeff(i, i+1))); - m_alphas.coeffRef(i) = ComplexScalar(m_matS.coeff(i+1, i+1) + p, z); - m_alphas.coeffRef(i+1) = ComplexScalar(m_matS.coeff(i+1, i+1) + p, -z); - - m_betas.coeffRef(i) = m_realQZ.matrixT().coeff(i,i); - m_betas.coeffRef(i+1) = m_realQZ.matrixT().coeff(i,i); + // We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a positive diagonal 2x2 block T + // Then taking beta=T_00*T_11, we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00): + + // T = [a 0] + // [0 b] + RealScalar a = mT.diagonal().coeff(i), + b = mT.diagonal().coeff(i+1); + const RealScalar beta = m_betas.coeffRef(i) = m_betas.coeffRef(i+1) = a*b; + + // ^^ NOTE: using diagonal()(i) instead of coeff(i,i) workarounds a MSVC bug. + Matrix<RealScalar,2,2> S2 = mS.template block<2,2>(i,i) * Matrix<Scalar,2,1>(b,a).asDiagonal(); + + Scalar p = Scalar(0.5) * (S2.coeff(0,0) - S2.coeff(1,1)); + Scalar z = sqrt(abs(p * p + S2.coeff(1,0) * S2.coeff(0,1))); + const ComplexScalar alpha = ComplexScalar(S2.coeff(1,1) + p, (beta > 0) ? z : -z); + m_alphas.coeffRef(i) = conj(alpha); + m_alphas.coeffRef(i+1) = alpha; + + if (computeEigenvectors) { + // Compute eigenvector in position (i+1) and then position (i) is just the conjugate + cv.setZero(); + cv.coeffRef(i+1) = Scalar(1.0); + // here, the "static_cast" workaound expression template issues. + cv.coeffRef(i) = -(static_cast<Scalar>(beta*mS.coeffRef(i,i+1)) - alpha*mT.coeffRef(i,i+1)) + / (static_cast<Scalar>(beta*mS.coeffRef(i,i)) - alpha*mT.coeffRef(i,i)); + for (Index j = i-1; j >= 0; j--) + { + const Index st = j+1; + const Index sz = i+1-j; + if (j > 0 && mS.coeff(j, j-1) != Scalar(0)) + { + // 2x2 block + Matrix<ComplexScalar, 2, 1> rhs = (alpha*mT.template block<2,Dynamic>(j-1,st,2,sz) - beta*mS.template block<2,Dynamic>(j-1,st,2,sz)) .lazyProduct( cv.segment(st,sz) ); + Matrix<ComplexScalar, 2, 2> lhs = beta * mS.template block<2,2>(j-1,j-1) - alpha * mT.template block<2,2>(j-1,j-1); + cv.template segment<2>(j-1) = lhs.partialPivLu().solve(rhs); + j--; + } else { + cv.coeffRef(j) = cv.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum() + / (alpha*mT.coeffRef(j,j) - static_cast<Scalar>(beta*mS.coeffRef(j,j))); + } + } + m_eivec.col(i+1).noalias() = (mZ.transpose() * cv); + m_eivec.col(i+1).normalize(); + m_eivec.col(i) = m_eivec.col(i+1).conjugate(); + } i += 2; } } - } - - m_isInitialized = true; - m_eigenvectorsOk = false;//computeEigenvectors; + m_valuesOkay = true; + m_vectorsOkay = computeEigenvectors; + } return *this; } |