diff options
Diffstat (limited to 'Eigen/src/Cholesky/LDLT.h')
| -rw-r--r-- | Eigen/src/Cholesky/LDLT.h | 122 |
1 files changed, 67 insertions, 55 deletions
diff --git a/Eigen/src/Cholesky/LDLT.h b/Eigen/src/Cholesky/LDLT.h index 68e54b1d4..c52b7d1a6 100644 --- a/Eigen/src/Cholesky/LDLT.h +++ b/Eigen/src/Cholesky/LDLT.h @@ -16,7 +16,10 @@ namespace Eigen { namespace internal { -template<typename MatrixType, int UpLo> struct LDLT_Traits; + template<typename MatrixType, int UpLo> struct LDLT_Traits; + + // PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef + enum SignMatrix { PositiveSemiDef, NegativeSemiDef, ZeroSign, Indefinite }; } /** \ingroup Cholesky_Module @@ -69,7 +72,12 @@ template<typename _MatrixType, int _UpLo> class LDLT * The default constructor is useful in cases in which the user intends to * perform decompositions via LDLT::compute(const MatrixType&). */ - LDLT() : m_matrix(), m_transpositions(), m_isInitialized(false) {} + LDLT() + : m_matrix(), + m_transpositions(), + m_sign(internal::ZeroSign), + m_isInitialized(false) + {} /** \brief Default Constructor with memory preallocation * @@ -81,6 +89,7 @@ template<typename _MatrixType, int _UpLo> class LDLT : m_matrix(size, size), m_transpositions(size), m_temporary(size), + m_sign(internal::ZeroSign), m_isInitialized(false) {} @@ -93,6 +102,7 @@ template<typename _MatrixType, int _UpLo> class LDLT : m_matrix(matrix.rows(), matrix.cols()), m_transpositions(matrix.rows()), m_temporary(matrix.rows()), + m_sign(internal::ZeroSign), m_isInitialized(false) { compute(matrix); @@ -139,7 +149,7 @@ template<typename _MatrixType, int _UpLo> class LDLT inline bool isPositive() const { eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_sign == 1; + return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign; } #ifdef EIGEN2_SUPPORT @@ -153,7 +163,7 @@ template<typename _MatrixType, int _UpLo> class LDLT inline bool isNegative(void) const { eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_sign == -1; + return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign; } /** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A. @@ -196,7 +206,7 @@ template<typename _MatrixType, int _UpLo> class LDLT LDLT& compute(const MatrixType& matrix); template <typename Derived> - LDLT& rankUpdate(const MatrixBase<Derived>& w,RealScalar alpha=1); + LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha=1); /** \returns the internal LDLT decomposition matrix * @@ -235,7 +245,7 @@ template<typename _MatrixType, int _UpLo> class LDLT MatrixType m_matrix; TranspositionType m_transpositions; TmpMatrixType m_temporary; - int m_sign; + internal::SignMatrix m_sign; bool m_isInitialized; }; @@ -246,8 +256,9 @@ template<int UpLo> struct ldlt_inplace; template<> struct ldlt_inplace<Lower> { template<typename MatrixType, typename TranspositionType, typename Workspace> - static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, int* sign=0) + static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign) { + using std::abs; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; @@ -257,38 +268,19 @@ template<> struct ldlt_inplace<Lower> if (size <= 1) { transpositions.setIdentity(); - if(sign) - *sign = real(mat.coeff(0,0))>0 ? 1:-1; + if (numext::real(mat.coeff(0,0)) > 0) sign = PositiveSemiDef; + else if (numext::real(mat.coeff(0,0)) < 0) sign = NegativeSemiDef; + else sign = ZeroSign; return true; } - RealScalar cutoff(0), biggest_in_corner; - for (Index k = 0; k < size; ++k) { // Find largest diagonal element Index index_of_biggest_in_corner; - biggest_in_corner = mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner); + mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner); index_of_biggest_in_corner += k; - if(k == 0) - { - // The biggest overall is the point of reference to which further diagonals - // are compared; if any diagonal is negligible compared - // to the largest overall, the algorithm bails. - cutoff = abs(NumTraits<Scalar>::epsilon() * biggest_in_corner); - - if(sign) - *sign = real(mat.diagonal().coeff(index_of_biggest_in_corner)) > 0 ? 1 : -1; - } - - // Finish early if the matrix is not full rank. - if(biggest_in_corner < cutoff) - { - for(Index i = k; i < size; i++) transpositions.coeffRef(i) = i; - break; - } - transpositions.coeffRef(k) = index_of_biggest_in_corner; if(k != index_of_biggest_in_corner) { @@ -301,11 +293,11 @@ template<> struct ldlt_inplace<Lower> for(int i=k+1;i<index_of_biggest_in_corner;++i) { Scalar tmp = mat.coeffRef(i,k); - mat.coeffRef(i,k) = conj(mat.coeffRef(index_of_biggest_in_corner,i)); - mat.coeffRef(index_of_biggest_in_corner,i) = conj(tmp); + mat.coeffRef(i,k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner,i)); + mat.coeffRef(index_of_biggest_in_corner,i) = numext::conj(tmp); } if(NumTraits<Scalar>::IsComplex) - mat.coeffRef(index_of_biggest_in_corner,k) = conj(mat.coeff(index_of_biggest_in_corner,k)); + mat.coeffRef(index_of_biggest_in_corner,k) = numext::conj(mat.coeff(index_of_biggest_in_corner,k)); } // partition the matrix: @@ -319,13 +311,28 @@ template<> struct ldlt_inplace<Lower> if(k>0) { - temp.head(k) = mat.diagonal().head(k).asDiagonal() * A10.adjoint(); + temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint(); mat.coeffRef(k,k) -= (A10 * temp.head(k)).value(); if(rs>0) A21.noalias() -= A20 * temp.head(k); } - if((rs>0) && (abs(mat.coeffRef(k,k)) > cutoff)) - A21 /= mat.coeffRef(k,k); + + // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot + // was smaller than the cutoff value. However, soince LDLT is not rank-revealing + // we should only make sure we do not introduce INF or NaN values. + // LAPACK also uses 0 as the cutoff value. + RealScalar realAkk = numext::real(mat.coeffRef(k,k)); + if((rs>0) && (abs(realAkk) > RealScalar(0))) + A21 /= realAkk; + + if (sign == PositiveSemiDef) { + if (realAkk < 0) sign = Indefinite; + } else if (sign == NegativeSemiDef) { + if (realAkk > 0) sign = Indefinite; + } else if (sign == ZeroSign) { + if (realAkk > 0) sign = PositiveSemiDef; + else if (realAkk < 0) sign = NegativeSemiDef; + } } return true; @@ -339,9 +346,9 @@ template<> struct ldlt_inplace<Lower> // Here only rank-1 updates are implemented, to reduce the // requirement for intermediate storage and improve accuracy template<typename MatrixType, typename WDerived> - static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, typename MatrixType::RealScalar sigma=1) + static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma=1) { - using internal::isfinite; + using numext::isfinite; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; @@ -359,9 +366,9 @@ template<> struct ldlt_inplace<Lower> break; // Update the diagonal terms - RealScalar dj = real(mat.coeff(j,j)); + RealScalar dj = numext::real(mat.coeff(j,j)); Scalar wj = w.coeff(j); - RealScalar swj2 = sigma*abs2(wj); + RealScalar swj2 = sigma*numext::abs2(wj); RealScalar gamma = dj*alpha + swj2; mat.coeffRef(j,j) += swj2/alpha; @@ -372,13 +379,13 @@ template<> struct ldlt_inplace<Lower> Index rs = size-j-1; w.tail(rs) -= wj * mat.col(j).tail(rs); if(gamma != 0) - mat.col(j).tail(rs) += (sigma*conj(wj)/gamma)*w.tail(rs); + mat.col(j).tail(rs) += (sigma*numext::conj(wj)/gamma)*w.tail(rs); } return true; } template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType> - static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, typename MatrixType::RealScalar sigma=1) + static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma=1) { // Apply the permutation to the input w tmp = transpositions * w; @@ -390,14 +397,14 @@ template<> struct ldlt_inplace<Lower> template<> struct ldlt_inplace<Upper> { template<typename MatrixType, typename TranspositionType, typename Workspace> - static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, int* sign=0) + static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign) { Transpose<MatrixType> matt(mat); return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign); } template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType> - static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, typename MatrixType::RealScalar sigma=1) + static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma=1) { Transpose<MatrixType> matt(mat); return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma); @@ -436,7 +443,7 @@ LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const MatrixType& a) m_isInitialized = false; m_temporary.resize(size); - internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, &m_sign); + internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign); m_isInitialized = true; return *this; @@ -449,7 +456,7 @@ LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const MatrixType& a) */ template<typename MatrixType, int _UpLo> template<typename Derived> -LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Derived>& w,typename NumTraits<typename MatrixType::Scalar>::Real sigma) +LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Derived>& w, const typename NumTraits<typename MatrixType::Scalar>::Real& sigma) { const Index size = w.rows(); if (m_isInitialized) @@ -464,7 +471,7 @@ LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Deri for (Index i = 0; i < size; i++) m_transpositions.coeffRef(i) = i; m_temporary.resize(size); - m_sign = sigma>=0 ? 1 : -1; + m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef; m_isInitialized = true; } @@ -497,14 +504,20 @@ struct solve_retval<LDLT<_MatrixType,_UpLo>, Rhs> typedef typename LDLTType::MatrixType MatrixType; typedef typename LDLTType::Scalar Scalar; typedef typename LDLTType::RealScalar RealScalar; - const Diagonal<const MatrixType> vectorD = dec().vectorD(); - RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() * NumTraits<Scalar>::epsilon(), - RealScalar(1) / NumTraits<RealScalar>::highest()); // motivated by LAPACK's xGELSS + const typename Diagonal<const MatrixType>::RealReturnType vectorD(dec().vectorD()); + // In some previous versions, tolerance was set to the max of 1/highest and the maximal diagonal entry * epsilon + // as motivated by LAPACK's xGELSS: + // RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() *NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest()); + // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest + // diagonal element is not well justified and to numerical issues in some cases. + // Moreover, Lapack's xSYTRS routines use 0 for the tolerance. + RealScalar tolerance = RealScalar(1) / NumTraits<RealScalar>::highest(); + for (Index i = 0; i < vectorD.size(); ++i) { if(abs(vectorD(i)) > tolerance) - dst.row(i) /= vectorD(i); + dst.row(i) /= vectorD(i); else - dst.row(i).setZero(); + dst.row(i).setZero(); } // dst = L^-T (D^-1 L^-1 P b) @@ -534,8 +547,7 @@ template<typename Derived> bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const { eigen_assert(m_isInitialized && "LDLT is not initialized."); - const Index size = m_matrix.rows(); - eigen_assert(size == bAndX.rows()); + eigen_assert(m_matrix.rows() == bAndX.rows()); bAndX = this->solve(bAndX); @@ -558,7 +570,7 @@ MatrixType LDLT<MatrixType,_UpLo>::reconstructedMatrix() const // L^* P res = matrixU() * res; // D(L^*P) - res = vectorD().asDiagonal() * res; + res = vectorD().real().asDiagonal() * res; // L(DL^*P) res = matrixL() * res; // P^T (LDL^*P) |