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Diffstat (limited to 'unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h')
| -rw-r--r-- | unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h | 708 |
1 files changed, 349 insertions, 359 deletions
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h b/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h index 7d426640c..db2449d02 100644 --- a/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h +++ b/unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h @@ -1,7 +1,7 @@ // This file is part of Eigen, a lightweight C++ template library // for linear algebra. // -// Copyright (C) 2009-2011 Jitse Niesen <jitse@maths.leeds.ac.uk> +// Copyright (C) 2009-2011, 2013 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 @@ -11,398 +11,245 @@ #define EIGEN_MATRIX_FUNCTION #include "StemFunction.h" -#include "MatrixFunctionAtomic.h" namespace Eigen { +namespace internal { + +/** \brief Maximum distance allowed between eigenvalues to be considered "close". */ +static const float matrix_function_separation = 0.1f; + /** \ingroup MatrixFunctions_Module - * \brief Class for computing matrix functions. - * \tparam MatrixType type of the argument of the matrix function, - * expected to be an instantiation of the Matrix class template. - * \tparam AtomicType type for computing matrix function of atomic blocks. - * \tparam IsComplex used internally to select correct specialization. + * \class MatrixFunctionAtomic + * \brief Helper class for computing matrix functions of atomic matrices. * - * This class implements the Schur-Parlett algorithm for computing matrix functions. The spectrum of the - * matrix is divided in clustered of eigenvalues that lies close together. This class delegates the - * computation of the matrix function on every block corresponding to these clusters to an object of type - * \p AtomicType and uses these results to compute the matrix function of the whole matrix. The class - * \p AtomicType should have a \p compute() member function for computing the matrix function of a block. - * - * \sa class MatrixFunctionAtomic, class MatrixLogarithmAtomic + * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other. */ -template <typename MatrixType, - typename AtomicType, - int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex> -class MatrixFunction -{ +template <typename MatrixType> +class MatrixFunctionAtomic +{ public: - /** \brief Constructor. - * - * \param[in] A argument of matrix function, should be a square matrix. - * \param[in] atomic class for computing matrix function of atomic blocks. - * - * The class stores references to \p A and \p atomic, so they should not be - * changed (or destroyed) before compute() is called. - */ - MatrixFunction(const MatrixType& A, AtomicType& atomic); - - /** \brief Compute the matrix function. - * - * \param[out] result the function \p f applied to \p A, as - * specified in the constructor. - * - * See MatrixBase::matrixFunction() for details on how this computation - * is implemented. - */ - template <typename ResultType> - void compute(ResultType &result); -}; - - -/** \internal \ingroup MatrixFunctions_Module - * \brief Partial specialization of MatrixFunction for real matrices - */ -template <typename MatrixType, typename AtomicType> -class MatrixFunction<MatrixType, AtomicType, 0> -{ - private: - - typedef internal::traits<MatrixType> Traits; - typedef typename Traits::Scalar Scalar; - static const int Rows = Traits::RowsAtCompileTime; - static const int Cols = Traits::ColsAtCompileTime; - static const int Options = MatrixType::Options; - static const int MaxRows = Traits::MaxRowsAtCompileTime; - static const int MaxCols = Traits::MaxColsAtCompileTime; - - typedef std::complex<Scalar> ComplexScalar; - typedef Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols> ComplexMatrix; - - public: + typedef typename MatrixType::Scalar Scalar; + typedef typename stem_function<Scalar>::type StemFunction; - /** \brief Constructor. - * - * \param[in] A argument of matrix function, should be a square matrix. - * \param[in] atomic class for computing matrix function of atomic blocks. + /** \brief Constructor + * \param[in] f matrix function to compute. */ - MatrixFunction(const MatrixType& A, AtomicType& atomic) : m_A(A), m_atomic(atomic) { } + MatrixFunctionAtomic(StemFunction f) : m_f(f) { } - /** \brief Compute the matrix function. - * - * \param[out] result the function \p f applied to \p A, as - * specified in the constructor. - * - * This function converts the real matrix \c A to a complex matrix, - * uses MatrixFunction<MatrixType,1> and then converts the result back to - * a real matrix. + /** \brief Compute matrix function of atomic matrix + * \param[in] A argument of matrix function, should be upper triangular and atomic + * \returns f(A), the matrix function evaluated at the given matrix */ - template <typename ResultType> - void compute(ResultType& result) - { - ComplexMatrix CA = m_A.template cast<ComplexScalar>(); - ComplexMatrix Cresult; - MatrixFunction<ComplexMatrix, AtomicType> mf(CA, m_atomic); - mf.compute(Cresult); - result = Cresult.real(); - } - - private: - typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */ - AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */ - - MatrixFunction& operator=(const MatrixFunction&); -}; - - -/** \internal \ingroup MatrixFunctions_Module - * \brief Partial specialization of MatrixFunction for complex matrices - */ -template <typename MatrixType, typename AtomicType> -class MatrixFunction<MatrixType, AtomicType, 1> -{ - private: - - typedef internal::traits<MatrixType> Traits; - typedef typename MatrixType::Scalar Scalar; - typedef typename MatrixType::Index Index; - static const int RowsAtCompileTime = Traits::RowsAtCompileTime; - static const int ColsAtCompileTime = Traits::ColsAtCompileTime; - static const int Options = MatrixType::Options; - typedef typename NumTraits<Scalar>::Real RealScalar; - typedef Matrix<Scalar, Traits::RowsAtCompileTime, 1> VectorType; - typedef Matrix<Index, Traits::RowsAtCompileTime, 1> IntVectorType; - typedef Matrix<Index, Dynamic, 1> DynamicIntVectorType; - typedef std::list<Scalar> Cluster; - typedef std::list<Cluster> ListOfClusters; - typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; - - public: - - MatrixFunction(const MatrixType& A, AtomicType& atomic); - template <typename ResultType> void compute(ResultType& result); + MatrixType compute(const MatrixType& A); private: - - void computeSchurDecomposition(); - void partitionEigenvalues(); - typename ListOfClusters::iterator findCluster(Scalar key); - void computeClusterSize(); - void computeBlockStart(); - void constructPermutation(); - void permuteSchur(); - void swapEntriesInSchur(Index index); - void computeBlockAtomic(); - Block<MatrixType> block(MatrixType& A, Index i, Index j); - void computeOffDiagonal(); - DynMatrixType solveTriangularSylvester(const DynMatrixType& A, const DynMatrixType& B, const DynMatrixType& C); - - typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */ - AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */ - MatrixType m_T; /**< \brief Triangular part of Schur decomposition */ - MatrixType m_U; /**< \brief Unitary part of Schur decomposition */ - MatrixType m_fT; /**< \brief %Matrix function applied to #m_T */ - ListOfClusters m_clusters; /**< \brief Partition of eigenvalues into clusters of ei'vals "close" to each other */ - DynamicIntVectorType m_eivalToCluster; /**< \brief m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster */ - DynamicIntVectorType m_clusterSize; /**< \brief Number of eigenvalues in each clusters */ - DynamicIntVectorType m_blockStart; /**< \brief Row index at which block corresponding to i-th cluster starts */ - IntVectorType m_permutation; /**< \brief Permutation which groups ei'vals in the same cluster together */ - - /** \brief Maximum distance allowed between eigenvalues to be considered "close". - * - * This is morally a \c static \c const \c Scalar, but only - * integers can be static constant class members in C++. The - * separation constant is set to 0.1, a value taken from the - * paper by Davies and Higham. */ - static const RealScalar separation() { return static_cast<RealScalar>(0.1); } - - MatrixFunction& operator=(const MatrixFunction&); + StemFunction* m_f; }; -/** \brief Constructor. - * - * \param[in] A argument of matrix function, should be a square matrix. - * \param[in] atomic class for computing matrix function of atomic blocks. - */ -template <typename MatrixType, typename AtomicType> -MatrixFunction<MatrixType,AtomicType,1>::MatrixFunction(const MatrixType& A, AtomicType& atomic) - : m_A(A), m_atomic(atomic) +template <typename MatrixType> +typename NumTraits<typename MatrixType::Scalar>::Real matrix_function_compute_mu(const MatrixType& A) { - /* empty body */ + typedef typename plain_col_type<MatrixType>::type VectorType; + typename MatrixType::Index rows = A.rows(); + const MatrixType N = MatrixType::Identity(rows, rows) - A; + VectorType e = VectorType::Ones(rows); + N.template triangularView<Upper>().solveInPlace(e); + return e.cwiseAbs().maxCoeff(); } -/** \brief Compute the matrix function. - * - * \param[out] result the function \p f applied to \p A, as - * specified in the constructor. - */ -template <typename MatrixType, typename AtomicType> -template <typename ResultType> -void MatrixFunction<MatrixType,AtomicType,1>::compute(ResultType& result) +template <typename MatrixType> +MatrixType MatrixFunctionAtomic<MatrixType>::compute(const MatrixType& A) { - computeSchurDecomposition(); - partitionEigenvalues(); - computeClusterSize(); - computeBlockStart(); - constructPermutation(); - permuteSchur(); - computeBlockAtomic(); - computeOffDiagonal(); - result = m_U * (m_fT.template triangularView<Upper>() * m_U.adjoint()); + // TODO: Use that A is upper triangular + typedef typename NumTraits<Scalar>::Real RealScalar; + typedef typename MatrixType::Index Index; + Index rows = A.rows(); + Scalar avgEival = A.trace() / Scalar(RealScalar(rows)); + MatrixType Ashifted = A - avgEival * MatrixType::Identity(rows, rows); + RealScalar mu = matrix_function_compute_mu(Ashifted); + MatrixType F = m_f(avgEival, 0) * MatrixType::Identity(rows, rows); + MatrixType P = Ashifted; + MatrixType Fincr; + for (Index s = 1; s < 1.1 * rows + 10; s++) { // upper limit is fairly arbitrary + Fincr = m_f(avgEival, static_cast<int>(s)) * P; + F += Fincr; + P = Scalar(RealScalar(1.0/(s + 1))) * P * Ashifted; + + // test whether Taylor series converged + const RealScalar F_norm = F.cwiseAbs().rowwise().sum().maxCoeff(); + const RealScalar Fincr_norm = Fincr.cwiseAbs().rowwise().sum().maxCoeff(); + if (Fincr_norm < NumTraits<Scalar>::epsilon() * F_norm) { + RealScalar delta = 0; + RealScalar rfactorial = 1; + for (Index r = 0; r < rows; r++) { + RealScalar mx = 0; + for (Index i = 0; i < rows; i++) + mx = (std::max)(mx, std::abs(m_f(Ashifted(i, i) + avgEival, static_cast<int>(s+r)))); + if (r != 0) + rfactorial *= RealScalar(r); + delta = (std::max)(delta, mx / rfactorial); + } + const RealScalar P_norm = P.cwiseAbs().rowwise().sum().maxCoeff(); + if (mu * delta * P_norm < NumTraits<Scalar>::epsilon() * F_norm) // series converged + break; + } + } + return F; } -/** \brief Store the Schur decomposition of #m_A in #m_T and #m_U */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::computeSchurDecomposition() +/** \brief Find cluster in \p clusters containing some value + * \param[in] key Value to find + * \returns Iterator to cluster containing \p key, or \c clusters.end() if no cluster in \p m_clusters + * contains \p key. + */ +template <typename Index, typename ListOfClusters> +typename ListOfClusters::iterator matrix_function_find_cluster(Index key, ListOfClusters& clusters) { - const ComplexSchur<MatrixType> schurOfA(m_A); - m_T = schurOfA.matrixT(); - m_U = schurOfA.matrixU(); + typename std::list<Index>::iterator j; + for (typename ListOfClusters::iterator i = clusters.begin(); i != clusters.end(); ++i) { + j = std::find(i->begin(), i->end(), key); + if (j != i->end()) + return i; + } + return clusters.end(); } /** \brief Partition eigenvalues in clusters of ei'vals close to each other * - * This function computes #m_clusters. This is a partition of the - * eigenvalues of #m_T in clusters, such that - * # Any eigenvalue in a certain cluster is at most separation() away - * from another eigenvalue in the same cluster. - * # The distance between two eigenvalues in different clusters is - * more than separation(). - * The implementation follows Algorithm 4.1 in the paper of Davies - * and Higham. + * \param[in] eivals Eigenvalues + * \param[out] clusters Resulting partition of eigenvalues + * + * The partition satisfies the following two properties: + * # Any eigenvalue in a certain cluster is at most matrix_function_separation() away from another eigenvalue + * in the same cluster. + * # The distance between two eigenvalues in different clusters is more than matrix_function_separation(). + * The implementation follows Algorithm 4.1 in the paper of Davies and Higham. */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::partitionEigenvalues() +template <typename EivalsType, typename Cluster> +void matrix_function_partition_eigenvalues(const EivalsType& eivals, std::list<Cluster>& clusters) { - using std::abs; - const Index rows = m_T.rows(); - VectorType diag = m_T.diagonal(); // contains eigenvalues of A - - for (Index i=0; i<rows; ++i) { - // Find set containing diag(i), adding a new set if necessary - typename ListOfClusters::iterator qi = findCluster(diag(i)); - if (qi == m_clusters.end()) { + typedef typename EivalsType::Index Index; + typedef typename EivalsType::RealScalar RealScalar; + for (Index i=0; i<eivals.rows(); ++i) { + // Find cluster containing i-th ei'val, adding a new cluster if necessary + typename std::list<Cluster>::iterator qi = matrix_function_find_cluster(i, clusters); + if (qi == clusters.end()) { Cluster l; - l.push_back(diag(i)); - m_clusters.push_back(l); - qi = m_clusters.end(); + l.push_back(i); + clusters.push_back(l); + qi = clusters.end(); --qi; } // Look for other element to add to the set - for (Index j=i+1; j<rows; ++j) { - if (abs(diag(j) - diag(i)) <= separation() && std::find(qi->begin(), qi->end(), diag(j)) == qi->end()) { - typename ListOfClusters::iterator qj = findCluster(diag(j)); - if (qj == m_clusters.end()) { - qi->push_back(diag(j)); + for (Index j=i+1; j<eivals.rows(); ++j) { + if (abs(eivals(j) - eivals(i)) <= RealScalar(matrix_function_separation) + && std::find(qi->begin(), qi->end(), j) == qi->end()) { + typename std::list<Cluster>::iterator qj = matrix_function_find_cluster(j, clusters); + if (qj == clusters.end()) { + qi->push_back(j); } else { qi->insert(qi->end(), qj->begin(), qj->end()); - m_clusters.erase(qj); + clusters.erase(qj); } } } } } -/** \brief Find cluster in #m_clusters containing some value - * \param[in] key Value to find - * \returns Iterator to cluster containing \c key, or - * \c m_clusters.end() if no cluster in m_clusters contains \c key. - */ -template <typename MatrixType, typename AtomicType> -typename MatrixFunction<MatrixType,AtomicType,1>::ListOfClusters::iterator MatrixFunction<MatrixType,AtomicType,1>::findCluster(Scalar key) +/** \brief Compute size of each cluster given a partitioning */ +template <typename ListOfClusters, typename Index> +void matrix_function_compute_cluster_size(const ListOfClusters& clusters, Matrix<Index, Dynamic, 1>& clusterSize) { - typename Cluster::iterator j; - for (typename ListOfClusters::iterator i = m_clusters.begin(); i != m_clusters.end(); ++i) { - j = std::find(i->begin(), i->end(), key); - if (j != i->end()) - return i; + const Index numClusters = static_cast<Index>(clusters.size()); + clusterSize.setZero(numClusters); + Index clusterIndex = 0; + for (typename ListOfClusters::const_iterator cluster = clusters.begin(); cluster != clusters.end(); ++cluster) { + clusterSize[clusterIndex] = cluster->size(); + ++clusterIndex; } - return m_clusters.end(); } -/** \brief Compute #m_clusterSize and #m_eivalToCluster using #m_clusters */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::computeClusterSize() +/** \brief Compute start of each block using clusterSize */ +template <typename VectorType> +void matrix_function_compute_block_start(const VectorType& clusterSize, VectorType& blockStart) { - const Index rows = m_T.rows(); - VectorType diag = m_T.diagonal(); - const Index numClusters = static_cast<Index>(m_clusters.size()); + blockStart.resize(clusterSize.rows()); + blockStart(0) = 0; + for (typename VectorType::Index i = 1; i < clusterSize.rows(); i++) { + blockStart(i) = blockStart(i-1) + clusterSize(i-1); + } +} - m_clusterSize.setZero(numClusters); - m_eivalToCluster.resize(rows); +/** \brief Compute mapping of eigenvalue indices to cluster indices */ +template <typename EivalsType, typename ListOfClusters, typename VectorType> +void matrix_function_compute_map(const EivalsType& eivals, const ListOfClusters& clusters, VectorType& eivalToCluster) +{ + typedef typename EivalsType::Index Index; + eivalToCluster.resize(eivals.rows()); Index clusterIndex = 0; - for (typename ListOfClusters::const_iterator cluster = m_clusters.begin(); cluster != m_clusters.end(); ++cluster) { - for (Index i = 0; i < diag.rows(); ++i) { - if (std::find(cluster->begin(), cluster->end(), diag(i)) != cluster->end()) { - ++m_clusterSize[clusterIndex]; - m_eivalToCluster[i] = clusterIndex; + for (typename ListOfClusters::const_iterator cluster = clusters.begin(); cluster != clusters.end(); ++cluster) { + for (Index i = 0; i < eivals.rows(); ++i) { + if (std::find(cluster->begin(), cluster->end(), i) != cluster->end()) { + eivalToCluster[i] = clusterIndex; } } ++clusterIndex; } } -/** \brief Compute #m_blockStart using #m_clusterSize */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::computeBlockStart() -{ - m_blockStart.resize(m_clusterSize.rows()); - m_blockStart(0) = 0; - for (Index i = 1; i < m_clusterSize.rows(); i++) { - m_blockStart(i) = m_blockStart(i-1) + m_clusterSize(i-1); - } -} - -/** \brief Compute #m_permutation using #m_eivalToCluster and #m_blockStart */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::constructPermutation() +/** \brief Compute permutation which groups ei'vals in same cluster together */ +template <typename DynVectorType, typename VectorType> +void matrix_function_compute_permutation(const DynVectorType& blockStart, const DynVectorType& eivalToCluster, VectorType& permutation) { - DynamicIntVectorType indexNextEntry = m_blockStart; - m_permutation.resize(m_T.rows()); - for (Index i = 0; i < m_T.rows(); i++) { - Index cluster = m_eivalToCluster[i]; - m_permutation[i] = indexNextEntry[cluster]; + typedef typename VectorType::Index Index; + DynVectorType indexNextEntry = blockStart; + permutation.resize(eivalToCluster.rows()); + for (Index i = 0; i < eivalToCluster.rows(); i++) { + Index cluster = eivalToCluster[i]; + permutation[i] = indexNextEntry[cluster]; ++indexNextEntry[cluster]; } } -/** \brief Permute Schur decomposition in #m_U and #m_T according to #m_permutation */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::permuteSchur() +/** \brief Permute Schur decomposition in U and T according to permutation */ +template <typename VectorType, typename MatrixType> +void matrix_function_permute_schur(VectorType& permutation, MatrixType& U, MatrixType& T) { - IntVectorType p = m_permutation; - for (Index i = 0; i < p.rows() - 1; i++) { + typedef typename VectorType::Index Index; + for (Index i = 0; i < permutation.rows() - 1; i++) { Index j; - for (j = i; j < p.rows(); j++) { - if (p(j) == i) break; + for (j = i; j < permutation.rows(); j++) { + if (permutation(j) == i) break; } - eigen_assert(p(j) == i); + eigen_assert(permutation(j) == i); for (Index k = j-1; k >= i; k--) { - swapEntriesInSchur(k); - std::swap(p.coeffRef(k), p.coeffRef(k+1)); + JacobiRotation<typename MatrixType::Scalar> rotation; + rotation.makeGivens(T(k, k+1), T(k+1, k+1) - T(k, k)); + T.applyOnTheLeft(k, k+1, rotation.adjoint()); + T.applyOnTheRight(k, k+1, rotation); + U.applyOnTheRight(k, k+1, rotation); + std::swap(permutation.coeffRef(k), permutation.coeffRef(k+1)); } } } -/** \brief Swap rows \a index and \a index+1 in Schur decomposition in #m_U and #m_T */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::swapEntriesInSchur(Index index) -{ - JacobiRotation<Scalar> rotation; - rotation.makeGivens(m_T(index, index+1), m_T(index+1, index+1) - m_T(index, index)); - m_T.applyOnTheLeft(index, index+1, rotation.adjoint()); - m_T.applyOnTheRight(index, index+1, rotation); - m_U.applyOnTheRight(index, index+1, rotation); -} - -/** \brief Compute block diagonal part of #m_fT. - * - * This routine computes the matrix function applied to the block diagonal part of #m_T, with the blocking - * given by #m_blockStart. The matrix function of each diagonal block is computed by #m_atomic. The - * off-diagonal parts of #m_fT are set to zero. - */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::computeBlockAtomic() -{ - m_fT.resize(m_T.rows(), m_T.cols()); - m_fT.setZero(); - for (Index i = 0; i < m_clusterSize.rows(); ++i) { - block(m_fT, i, i) = m_atomic.compute(block(m_T, i, i)); - } -} - -/** \brief Return block of matrix according to blocking given by #m_blockStart */ -template <typename MatrixType, typename AtomicType> -Block<MatrixType> MatrixFunction<MatrixType,AtomicType,1>::block(MatrixType& A, Index i, Index j) -{ - return A.block(m_blockStart(i), m_blockStart(j), m_clusterSize(i), m_clusterSize(j)); -} - -/** \brief Compute part of #m_fT above block diagonal. +/** \brief Compute block diagonal part of matrix function. * - * This routine assumes that the block diagonal part of #m_fT (which - * equals the matrix function applied to #m_T) has already been computed and computes - * the part above the block diagonal. The part below the diagonal is - * zero, because #m_T is upper triangular. + * This routine computes the matrix function applied to the block diagonal part of \p T (which should be + * upper triangular), with the blocking given by \p blockStart and \p clusterSize. The matrix function of + * each diagonal block is computed by \p atomic. The off-diagonal parts of \p fT are set to zero. */ -template <typename MatrixType, typename AtomicType> -void MatrixFunction<MatrixType,AtomicType,1>::computeOffDiagonal() +template <typename MatrixType, typename AtomicType, typename VectorType> +void matrix_function_compute_block_atomic(const MatrixType& T, AtomicType& atomic, const VectorType& blockStart, const VectorType& clusterSize, MatrixType& fT) { - for (Index diagIndex = 1; diagIndex < m_clusterSize.rows(); diagIndex++) { - for (Index blockIndex = 0; blockIndex < m_clusterSize.rows() - diagIndex; blockIndex++) { - // compute (blockIndex, blockIndex+diagIndex) block - DynMatrixType A = block(m_T, blockIndex, blockIndex); - DynMatrixType B = -block(m_T, blockIndex+diagIndex, blockIndex+diagIndex); - DynMatrixType C = block(m_fT, blockIndex, blockIndex) * block(m_T, blockIndex, blockIndex+diagIndex); - C -= block(m_T, blockIndex, blockIndex+diagIndex) * block(m_fT, blockIndex+diagIndex, blockIndex+diagIndex); - for (Index k = blockIndex + 1; k < blockIndex + diagIndex; k++) { - C += block(m_fT, blockIndex, k) * block(m_T, k, blockIndex+diagIndex); - C -= block(m_T, blockIndex, k) * block(m_fT, k, blockIndex+diagIndex); - } - block(m_fT, blockIndex, blockIndex+diagIndex) = solveTriangularSylvester(A, B, C); - } + fT.setZero(T.rows(), T.cols()); + for (typename VectorType::Index i = 0; i < clusterSize.rows(); ++i) { + fT.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)) + = atomic.compute(T.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i))); } } @@ -414,8 +261,8 @@ void MatrixFunction<MatrixType,AtomicType,1>::computeOffDiagonal() * * \returns the solution X. * - * If A is m-by-m and B is n-by-n, then both C and X are m-by-n. - * The (i,j)-th component of the Sylvester equation is + * If A is m-by-m and B is n-by-n, then both C and X are m-by-n. The (i,j)-th component of the Sylvester + * equation is * \f[ * \sum_{k=i}^m A_{ik} X_{kj} + \sum_{k=1}^j X_{ik} B_{kj} = C_{ij}. * \f] @@ -424,16 +271,12 @@ void MatrixFunction<MatrixType,AtomicType,1>::computeOffDiagonal() * X_{ij} = \frac{1}{A_{ii} + B_{jj}} \Bigl( C_{ij} * - \sum_{k=i+1}^m A_{ik} X_{kj} - \sum_{k=1}^{j-1} X_{ik} B_{kj} \Bigr). * \f] - * It is assumed that A and B are such that the numerator is never - * zero (otherwise the Sylvester equation does not have a unique - * solution). In that case, these equations can be evaluated in the - * order \f$ i=m,\ldots,1 \f$ and \f$ j=1,\ldots,n \f$. + * It is assumed that A and B are such that the numerator is never zero (otherwise the Sylvester equation + * does not have a unique solution). In that case, these equations can be evaluated in the order + * \f$ i=m,\ldots,1 \f$ and \f$ j=1,\ldots,n \f$. */ -template <typename MatrixType, typename AtomicType> -typename MatrixFunction<MatrixType,AtomicType,1>::DynMatrixType MatrixFunction<MatrixType,AtomicType,1>::solveTriangularSylvester( - const DynMatrixType& A, - const DynMatrixType& B, - const DynMatrixType& C) +template <typename MatrixType> +MatrixType matrix_function_solve_triangular_sylvester(const MatrixType& A, const MatrixType& B, const MatrixType& C) { eigen_assert(A.rows() == A.cols()); eigen_assert(A.isUpperTriangular()); @@ -442,9 +285,12 @@ typename MatrixFunction<MatrixType,AtomicType,1>::DynMatrixType MatrixFunction<M eigen_assert(C.rows() == A.rows()); eigen_assert(C.cols() == B.rows()); + typedef typename MatrixType::Index Index; + typedef typename MatrixType::Scalar Scalar; + Index m = A.rows(); Index n = B.rows(); - DynMatrixType X(m, n); + MatrixType X(m, n); for (Index i = m - 1; i >= 0; --i) { for (Index j = 0; j < n; ++j) { @@ -473,66 +319,210 @@ typename MatrixFunction<MatrixType,AtomicType,1>::DynMatrixType MatrixFunction<M return X; } +/** \brief Compute part of matrix function above block diagonal. + * + * This routine completes the computation of \p fT, denoting a matrix function applied to the triangular + * matrix \p T. It assumes that the block diagonal part of \p fT has already been computed. The part below + * the diagonal is zero, because \p T is upper triangular. + */ +template <typename MatrixType, typename VectorType> +void matrix_function_compute_above_diagonal(const MatrixType& T, const VectorType& blockStart, const VectorType& clusterSize, MatrixType& fT) +{ + typedef internal::traits<MatrixType> Traits; + typedef typename MatrixType::Scalar Scalar; + typedef typename MatrixType::Index Index; + static const int RowsAtCompileTime = Traits::RowsAtCompileTime; + static const int ColsAtCompileTime = Traits::ColsAtCompileTime; + static const int Options = MatrixType::Options; + typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; + + for (Index k = 1; k < clusterSize.rows(); k++) { + for (Index i = 0; i < clusterSize.rows() - k; i++) { + // compute (i, i+k) block + DynMatrixType A = T.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)); + DynMatrixType B = -T.block(blockStart(i+k), blockStart(i+k), clusterSize(i+k), clusterSize(i+k)); + DynMatrixType C = fT.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)) + * T.block(blockStart(i), blockStart(i+k), clusterSize(i), clusterSize(i+k)); + C -= T.block(blockStart(i), blockStart(i+k), clusterSize(i), clusterSize(i+k)) + * fT.block(blockStart(i+k), blockStart(i+k), clusterSize(i+k), clusterSize(i+k)); + for (Index m = i + 1; m < i + k; m++) { + C += fT.block(blockStart(i), blockStart(m), clusterSize(i), clusterSize(m)) + * T.block(blockStart(m), blockStart(i+k), clusterSize(m), clusterSize(i+k)); + C -= T.block(blockStart(i), blockStart(m), clusterSize(i), clusterSize(m)) + * fT.block(blockStart(m), blockStart(i+k), clusterSize(m), clusterSize(i+k)); + } + fT.block(blockStart(i), blockStart(i+k), clusterSize(i), clusterSize(i+k)) + = matrix_function_solve_triangular_sylvester(A, B, C); + } + } +} + +/** \ingroup MatrixFunctions_Module + * \brief Class for computing matrix functions. + * \tparam MatrixType type of the argument of the matrix function, + * expected to be an instantiation of the Matrix class template. + * \tparam AtomicType type for computing matrix function of atomic blocks. + * \tparam IsComplex used internally to select correct specialization. + * + * This class implements the Schur-Parlett algorithm for computing matrix functions. The spectrum of the + * matrix is divided in clustered of eigenvalues that lies close together. This class delegates the + * computation of the matrix function on every block corresponding to these clusters to an object of type + * \p AtomicType and uses these results to compute the matrix function of the whole matrix. The class + * \p AtomicType should have a \p compute() member function for computing the matrix function of a block. + * + * \sa class MatrixFunctionAtomic, class MatrixLogarithmAtomic + */ +template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex> +struct matrix_function_compute +{ + /** \brief Compute the matrix function. + * + * \param[in] A argument of matrix function, should be a square matrix. + * \param[in] atomic class for computing matrix function of atomic blocks. + * \param[out] result the function \p f applied to \p A, as + * specified in the constructor. + * + * See MatrixBase::matrixFunction() for details on how this computation + * is implemented. + */ + template <typename AtomicType, typename ResultType> + static void run(const MatrixType& A, AtomicType& atomic, ResultType &result); +}; + +/** \internal \ingroup MatrixFunctions_Module + * \brief Partial specialization of MatrixFunction for real matrices + * + * This converts the real matrix to a complex matrix, compute the matrix function of that matrix, and then + * converts the result back to a real matrix. + */ +template <typename MatrixType> +struct matrix_function_compute<MatrixType, 0> +{ + template <typename AtomicType, typename ResultType> + static void run(const MatrixType& A, AtomicType& atomic, ResultType &result) + { + typedef internal::traits<MatrixType> Traits; + typedef typename Traits::Scalar Scalar; + static const int Rows = Traits::RowsAtCompileTime, Cols = Traits::ColsAtCompileTime; + static const int MaxRows = Traits::MaxRowsAtCompileTime, MaxCols = Traits::MaxColsAtCompileTime; + + typedef std::complex<Scalar> ComplexScalar; + typedef Matrix<ComplexScalar, Rows, Cols, 0, MaxRows, MaxCols> ComplexMatrix; + + ComplexMatrix CA = A.template cast<ComplexScalar>(); + ComplexMatrix Cresult; + matrix_function_compute<ComplexMatrix>::run(CA, atomic, Cresult); + result = Cresult.real(); + } +}; + +/** \internal \ingroup MatrixFunctions_Module + * \brief Partial specialization of MatrixFunction for complex matrices + */ +template <typename MatrixType> +struct matrix_function_compute<MatrixType, 1> +{ + template <typename AtomicType, typename ResultType> + static void run(const MatrixType& A, AtomicType& atomic, ResultType &result) + { + typedef internal::traits<MatrixType> Traits; + typedef typename MatrixType::Index Index; + + // compute Schur decomposition of A + const ComplexSchur<MatrixType> schurOfA(A); + MatrixType T = schurOfA.matrixT(); + MatrixType U = schurOfA.matrixU(); + + // partition eigenvalues into clusters of ei'vals "close" to each other + std::list<std::list<Index> > clusters; + matrix_function_partition_eigenvalues(T.diagonal(), clusters); + + // compute size of each cluster + Matrix<Index, Dynamic, 1> clusterSize; + matrix_function_compute_cluster_size(clusters, clusterSize); + + // blockStart[i] is row index at which block corresponding to i-th cluster starts + Matrix<Index, Dynamic, 1> blockStart; + matrix_function_compute_block_start(clusterSize, blockStart); + + // compute map so that eivalToCluster[i] = j means that i-th ei'val is in j-th cluster + Matrix<Index, Dynamic, 1> eivalToCluster; + matrix_function_compute_map(T.diagonal(), clusters, eivalToCluster); + + // compute permutation which groups ei'vals in same cluster together + Matrix<Index, Traits::RowsAtCompileTime, 1> permutation; + matrix_function_compute_permutation(blockStart, eivalToCluster, permutation); + + // permute Schur decomposition + matrix_function_permute_schur(permutation, U, T); + + // compute result + MatrixType fT; // matrix function applied to T + matrix_function_compute_block_atomic(T, atomic, blockStart, clusterSize, fT); + matrix_function_compute_above_diagonal(T, blockStart, clusterSize, fT); + result = U * (fT.template triangularView<Upper>() * U.adjoint()); + } +}; + +} // end of namespace internal + /** \ingroup MatrixFunctions_Module * * \brief Proxy for the matrix function of some matrix (expression). * * \tparam Derived Type of the argument to the matrix function. * - * This class holds the argument to the matrix function until it is - * assigned or evaluated for some other reason (so the argument - * should not be changed in the meantime). It is the return type of - * matrixBase::matrixFunction() and related functions and most of the - * time this is the only way it is used. + * This class holds the argument to the matrix function until it is assigned or evaluated for some other + * reason (so the argument should not be changed in the meantime). It is the return type of + * matrixBase::matrixFunction() and related functions and most of the time this is the only way it is used. */ template<typename Derived> class MatrixFunctionReturnValue : public ReturnByValue<MatrixFunctionReturnValue<Derived> > { public: - typedef typename Derived::Scalar Scalar; typedef typename Derived::Index Index; typedef typename internal::stem_function<Scalar>::type StemFunction; - /** \brief Constructor. + protected: + typedef typename internal::ref_selector<Derived>::type DerivedNested; + + public: + + /** \brief Constructor. * - * \param[in] A %Matrix (expression) forming the argument of the - * matrix function. + * \param[in] A %Matrix (expression) forming the argument of the matrix function. * \param[in] f Stem function for matrix function under consideration. */ MatrixFunctionReturnValue(const Derived& A, StemFunction f) : m_A(A), m_f(f) { } /** \brief Compute the matrix function. * - * \param[out] result \p f applied to \p A, where \p f and \p A - * are as in the constructor. + * \param[out] result \p f applied to \p A, where \p f and \p A are as in the constructor. */ template <typename ResultType> inline void evalTo(ResultType& result) const { - typedef typename Derived::PlainObject PlainObject; - typedef internal::traits<PlainObject> Traits; + typedef typename internal::nested_eval<Derived, 10>::type NestedEvalType; + typedef typename internal::remove_all<NestedEvalType>::type NestedEvalTypeClean; + typedef internal::traits<NestedEvalTypeClean> Traits; static const int RowsAtCompileTime = Traits::RowsAtCompileTime; static const int ColsAtCompileTime = Traits::ColsAtCompileTime; - static const int Options = PlainObject::Options; typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; - typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; - typedef MatrixFunctionAtomic<DynMatrixType> AtomicType; + typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; + + typedef internal::MatrixFunctionAtomic<DynMatrixType> AtomicType; AtomicType atomic(m_f); - const PlainObject Aevaluated = m_A.eval(); - MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic); - mf.compute(result); + internal::matrix_function_compute<NestedEvalTypeClean>::run(m_A, atomic, result); } Index rows() const { return m_A.rows(); } Index cols() const { return m_A.cols(); } private: - typename internal::nested<Derived>::type m_A; + const DerivedNested m_A; StemFunction *m_f; - - MatrixFunctionReturnValue& operator=(const MatrixFunctionReturnValue&); }; namespace internal { @@ -559,7 +549,7 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const { eigen_assert(rows() == cols()); typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar; - return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sin); + return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_sin<ComplexScalar>); } template <typename Derived> @@ -567,7 +557,7 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const { eigen_assert(rows() == cols()); typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar; - return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cos); + return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_cos<ComplexScalar>); } template <typename Derived> @@ -575,7 +565,7 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const { eigen_assert(rows() == cols()); typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar; - return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sinh); + return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_sinh<ComplexScalar>); } template <typename Derived> @@ -583,7 +573,7 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const { eigen_assert(rows() == cols()); typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar; - return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cosh); + return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_cosh<ComplexScalar>); } } // end namespace Eigen |