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* Make HouseholderSequence::evalTo works in place
* Clean a bit the Triadiagonalization making sure it the inplace function really works inplace ;), and that only the lower triangular part of the matrix is referenced. * Remove the Tridiagonalization member object of SelfAdjointEigenSolver exploiting the in place capability of HouseholdeSequence. * Update unit test to check SelfAdjointEigenSolver only consider the lower triangular part.
This commit is contained in:
Gael Guennebaud
@@ -154,7 +154,7 @@ template<typename _MatrixType> class Tridiagonalization
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{
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m_matrix = matrix;
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m_hCoeffs.resize(matrix.rows()-1, 1);
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_compute(m_matrix, m_hCoeffs);
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ei_tridiagonalization_inplace(m_matrix, m_hCoeffs);
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m_isInitialized = true;
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return *this;
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}
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@@ -285,43 +285,6 @@ template<typename _MatrixType> class Tridiagonalization
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*/
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const SubDiagonalReturnType subDiagonal() const;
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/** \brief Performs a full decomposition in place
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*
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* \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
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* decomposition is to be computed. On output, the orthogonal matrix Q
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* in the decomposition if \p extractQ is true.
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* \param[out] diag The diagonal of the tridiagonal matrix T in the
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* decomposition.
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* \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
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* the decomposition.
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* \param[in] extractQ If true, the orthogonal matrix Q in the
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* decomposition is computed and stored in \p mat.
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*
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* Compute the tridiagonal matrix of \p mat in place. The tridiagonal
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* matrix T is passed to the output parameters \p diag and \p subdiag. If
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* \p extractQ is true, then the orthogonal matrix Q is passed to \p mat.
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*
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* The vectors \p diag and \p subdiag are not resized. The function
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* assumes that they are already of the correct size. The length of the
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* vector \p diag should equal the number of rows in \p mat, and the
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* length of the vector \p subdiag should be one left.
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*
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* This implementation contains an optimized path for real 3-by-3 matrices
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* which is especially useful for plane fitting.
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*
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* \note Notwithstanding the name, the current implementation copies
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* \p mat to a temporary matrix and uses that matrix to compute the
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* decomposition.
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*
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* Example (this uses the same matrix as the example in
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* Tridiagonalization(const MatrixType&)):
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* \include Tridiagonalization_decomposeInPlace.cpp
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* Output: \verbinclude Tridiagonalization_decomposeInPlace.out
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*
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* \sa Tridiagonalization(const MatrixType&), compute()
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*/
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static void decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);
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protected:
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static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
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@@ -368,21 +331,36 @@ Tridiagonalization<MatrixType>::matrixT() const
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}
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/** \internal
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* Performs a tridiagonal decomposition of \a matA in place.
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* Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
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*
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* \param matA the input selfadjoint matrix
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* \param hCoeffs returned Householder coefficients
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* \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
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* On output, the strict upper part is left unchanged, and the lower triangular part
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* represents the T and Q matrices in packed format has detailed below.
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* \param[out] hCoeffs returned Householder coefficients (see below)
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*
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* The result is written in the lower triangular part of \a matA.
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* On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
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* and lower sub-diagonal of the matrix \a matA.
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* The unitary matrix Q is represented in a compact way as a product of
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* Householder reflectors \f$ H_i \f$ such that:
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* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
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* The Householder reflectors are defined as
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* \f$ H_i = (I - h_i v_i v_i^T) \f$
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* where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
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* \f$ v_i \f$ is the Householder vector defined by
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* \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
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*
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* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
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*
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* \sa packedMatrix()
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* \sa Tridiagonalization::packedMatrix()
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*/
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template<typename MatrixType>
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void Tridiagonalization<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
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template<typename MatrixType, typename CoeffVectorType>
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void ei_tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
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{
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assert(matA.rows()==matA.cols());
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ei_assert(matA.rows()==matA.cols());
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ei_assert(matA.rows()==hCoeffs.size()+1);
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typedef typename MatrixType::Index Index;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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Index n = matA.rows();
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for (Index i = 0; i<n-1; ++i)
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{
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@@ -408,67 +386,139 @@ void Tridiagonalization<MatrixType>::_compute(MatrixType& matA, CoeffVectorType&
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}
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}
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template<typename MatrixType>
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void Tridiagonalization<MatrixType>::decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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// forward declaration, implementation at the end of this file
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template<typename MatrixType, int Size=MatrixType::ColsAtCompileTime>
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struct ei_tridiagonalization_inplace_selector;
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/** \brief Performs a full tridiagonalization in place
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*
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* \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
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* decomposition is to be computed. Only the lower triangular part referenced.
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* The rest is left unchanged. On output, the orthogonal matrix Q
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* in the decomposition if \p extractQ is true.
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* \param[out] diag The diagonal of the tridiagonal matrix T in the
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* decomposition.
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* \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
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* the decomposition.
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* \param[in] extractQ If true, the orthogonal matrix Q in the
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* decomposition is computed and stored in \p mat.
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*
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* Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
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* such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
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* symmetric tridiagonal matrix.
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*
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* The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
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* \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
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* part of the matrix \p mat is destroyed.
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*
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* The vectors \p diag and \p subdiag are not resized. The function
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* assumes that they are already of the correct size. The length of the
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* vector \p diag should equal the number of rows in \p mat, and the
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* length of the vector \p subdiag should be one left.
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*
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* This implementation contains an optimized path for 3-by-3 matrices
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* which is especially useful for plane fitting.
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*
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* \note Currently, it requires two temporary vectors to hold the intermediate
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* Householder coefficients, and to reconstruct the matrix Q from the Householder
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* reflectors.
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*
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* Example (this uses the same matrix as the example in
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* Tridiagonalization::Tridiagonalization(const MatrixType&)):
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* \include Tridiagonalization_decomposeInPlace.cpp
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* Output: \verbinclude Tridiagonalization_decomposeInPlace.out
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*
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* \sa class Tridiagonalization
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*/
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template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
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void ei_tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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typedef typename MatrixType::Index Index;
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Index n = mat.rows();
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ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1);
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if (n==3 && (!NumTraits<Scalar>::IsComplex) )
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{
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_decomposeInPlace3x3(mat, diag, subdiag, extractQ);
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}
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else
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{
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Tridiagonalization tridiag(mat);
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diag = tridiag.diagonal();
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subdiag = tridiag.subDiagonal();
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if (extractQ)
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mat = tridiag.matrixQ();
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}
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ei_tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
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}
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/** \internal
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* Optimized path for 3x3 matrices.
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* General full tridiagonalization
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*/
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template<typename MatrixType, int Size>
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struct ei_tridiagonalization_inplace_selector
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{
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typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
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typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
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typedef typename MatrixType::Index Index;
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template<typename DiagonalType, typename SubDiagonalType>
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static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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CoeffVectorType hCoeffs(mat.cols()-1);
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ei_tridiagonalization_inplace(mat,hCoeffs);
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diag = mat.diagonal().real();
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subdiag = mat.template diagonal<-1>().real();
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if(extractQ)
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mat = HouseholderSequenceType(mat, hCoeffs.conjugate(), false, mat.rows() - 1, 1);
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}
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};
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/** \internal
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* Specialization for 3x3 matrices.
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* Especially useful for plane fitting.
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*/
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template<typename MatrixType>
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void Tridiagonalization<MatrixType>::_decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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struct ei_tridiagonalization_inplace_selector<MatrixType,3>
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{
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diag[0] = ei_real(mat(0,0));
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RealScalar v1norm2 = ei_abs2(mat(0,2));
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if (ei_isMuchSmallerThan(v1norm2, RealScalar(1)))
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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template<typename DiagonalType, typename SubDiagonalType>
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static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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diag[1] = ei_real(mat(1,1));
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diag[2] = ei_real(mat(2,2));
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subdiag[0] = ei_real(mat(0,1));
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subdiag[1] = ei_real(mat(1,2));
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if (extractQ)
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mat.setIdentity();
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}
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else
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{
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RealScalar beta = ei_sqrt(ei_abs2(mat(0,1))+v1norm2);
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RealScalar invBeta = RealScalar(1)/beta;
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Scalar m01 = mat(0,1) * invBeta;
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Scalar m02 = mat(0,2) * invBeta;
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Scalar q = RealScalar(2)*m01*mat(1,2) + m02*(mat(2,2) - mat(1,1));
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diag[1] = ei_real(mat(1,1) + m02*q);
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diag[2] = ei_real(mat(2,2) - m02*q);
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subdiag[0] = beta;
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subdiag[1] = ei_real(mat(1,2) - m01 * q);
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if (extractQ)
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diag[0] = ei_real(mat(0,0));
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RealScalar v1norm2 = ei_abs2(mat(2,0));
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if (ei_isMuchSmallerThan(v1norm2, RealScalar(1)))
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{
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mat(0,0) = 1;
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mat(0,1) = 0;
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mat(0,2) = 0;
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mat(1,0) = 0;
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mat(1,1) = m01;
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mat(1,2) = m02;
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mat(2,0) = 0;
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mat(2,1) = m02;
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mat(2,2) = -m01;
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diag[1] = ei_real(mat(1,1));
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diag[2] = ei_real(mat(2,2));
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subdiag[0] = ei_real(mat(1,0));
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subdiag[1] = ei_real(mat(2,1));
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if (extractQ)
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mat.setIdentity();
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}
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else
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{
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RealScalar beta = ei_sqrt(ei_abs2(mat(1,0)) + v1norm2);
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RealScalar invBeta = RealScalar(1)/beta;
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Scalar m01 = ei_conj(mat(1,0)) * invBeta;
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Scalar m02 = ei_conj(mat(2,0)) * invBeta;
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Scalar q = RealScalar(2)*m01*ei_conj(mat(2,1)) + m02*(mat(2,2) - mat(1,1));
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diag[1] = ei_real(mat(1,1) + m02*q);
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diag[2] = ei_real(mat(2,2) - m02*q);
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subdiag[0] = beta;
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subdiag[1] = ei_real(ei_conj(mat(2,1)) - m01 * q);
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if (extractQ)
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{
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mat << 1, 0, 0,
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0, m01, m02,
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0, m02, -m01;
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}
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}
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}
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}
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};
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/** \internal
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* Trivial specialization for 1x1 matrices
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*/
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template<typename MatrixType>
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struct ei_tridiagonalization_inplace_selector<MatrixType,1>
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{
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typedef typename MatrixType::Scalar Scalar;
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template<typename DiagonalType, typename SubDiagonalType>
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static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
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{
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diag(0,0) = ei_real(mat(0,0));
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if(extractQ)
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mat(0,0) = Scalar(1);
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}
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};
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#endif // EIGEN_TRIDIAGONALIZATION_H
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