// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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_LLT_H
#define EIGEN_LLT_H
namespace Eigen {
namespace internal{
template<typename MatrixType, int UpLo> struct LLT_Traits;
}
/** \ingroup Cholesky_Module
*
* \class LLT
*
* \brief Standard Cholesky decomposition (LL^T) of a matrix and associated features
*
* \param MatrixType the type of the matrix of which we are computing the LL^T Cholesky decomposition
* \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
* The other triangular part won't be read.
*
* This class performs a LL^T Cholesky decomposition of a symmetric, positive definite
* matrix A such that A = LL^* = U^*U, where L is lower triangular.
*
* While the Cholesky decomposition is particularly useful to solve selfadjoint problems like D^*D x = b,
* for that purpose, we recommend the Cholesky decomposition without square root which is more stable
* and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other
* situations like generalised eigen problems with hermitian matrices.
*
* Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices,
* use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations
* has a solution.
*
* Example: \include LLT_example.cpp
* Output: \verbinclude LLT_example.out
*
* \sa MatrixBase::llt(), class LDLT
*/
/* HEY THIS DOX IS DISABLED BECAUSE THERE's A BUG EITHER HERE OR IN LDLT ABOUT THAT (OR BOTH)
* Note that during the decomposition, only the upper triangular part of A is considered. Therefore,
* the strict lower part does not have to store correct values.
*/
template<typename _MatrixType, int _UpLo> class LLT
{
public:
typedef _MatrixType MatrixType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef typename MatrixType::Index Index;
enum {
PacketSize = internal::packet_traits<Scalar>::size,
AlignmentMask = int(PacketSize)-1,
UpLo = _UpLo
};
typedef internal::LLT_Traits<MatrixType,UpLo> Traits;
/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via LLT::compute(const MatrixType&).
*/
LLT() : m_matrix(), m_isInitialized(false) {}
/** \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 LLT()
*/
LLT(Index size) : m_matrix(size, size),
m_isInitialized(false) {}
LLT(const MatrixType& matrix)
: m_matrix(matrix.rows(), matrix.cols()),
m_isInitialized(false)
{
compute(matrix);
}
/** \returns a view of the upper triangular matrix U */
inline typename Traits::MatrixU matrixU() const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
return Traits::getU(m_matrix);
}
/** \returns a view of the lower triangular matrix L */
inline typename Traits::MatrixL matrixL() const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
return Traits::getL(m_matrix);
}
/** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
*
* Since this LLT class assumes anyway that the matrix A is invertible, the solution
* theoretically exists and is unique regardless of b.
*
* Example: \include LLT_solve.cpp
* Output: \verbinclude LLT_solve.out
*
* \sa solveInPlace(), MatrixBase::llt()
*/
template<typename Rhs>
inline const internal::solve_retval<LLT, Rhs>
solve(const MatrixBase<Rhs>& b) const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
eigen_assert(m_matrix.rows()==b.rows()
&& "LLT::solve(): invalid number of rows of the right hand side matrix b");
return internal::solve_retval<LLT, Rhs>(*this, b.derived());
}
#ifdef EIGEN2_SUPPORT
template<typename OtherDerived, typename ResultType>
bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
{
*result = this->solve(b);
return true;
}
bool isPositiveDefinite() const { return true; }
#endif
template<typename Derived>
void solveInPlace(MatrixBase<Derived> &bAndX) const;
LLT& compute(const MatrixType& matrix);
/** \returns the LLT decomposition matrix
*
* TODO: document the storage layout
*/
inline const MatrixType& matrixLLT() const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
return m_matrix;
}
MatrixType reconstructedMatrix() const;
/** \brief Reports whether previous computation was successful.
*
* \returns \c Success if computation was succesful,
* \c NumericalIssue if the matrix.appears to be negative.
*/
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
return m_info;
}
inline Index rows() const { return m_matrix.rows(); }
inline Index cols() const { return m_matrix.cols(); }
template<typename VectorType>
LLT rankUpdate(const VectorType& vec, const RealScalar& sigma = 1);
protected:
/** \internal
* Used to compute and store L
* The strict upper part is not used and even not initialized.
*/
MatrixType m_matrix;
bool m_isInitialized;
ComputationInfo m_info;
};
namespace internal {
template<typename Scalar, int UpLo> struct llt_inplace;
template<typename MatrixType, typename VectorType>
static typename MatrixType::Index llt_rank_update_lower(MatrixType& mat, const VectorType& vec, const typename MatrixType::RealScalar& sigma)
{
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef typename MatrixType::Index Index;
typedef typename MatrixType::ColXpr ColXpr;
typedef typename internal::remove_all<ColXpr>::type ColXprCleaned;
typedef typename ColXprCleaned::SegmentReturnType ColXprSegment;
typedef Matrix<Scalar,Dynamic,1> TempVectorType;
typedef typename TempVectorType::SegmentReturnType TempVecSegment;
int n = mat.cols();
eigen_assert(mat.rows()==n && vec.size()==n);
TempVectorType temp;
if(sigma>0)
{
// This version is based on Givens rotations.
// It is faster than the other one below, but only works for updates,
// i.e., for sigma > 0
temp = sqrt(sigma) * vec;
for(int i=0; i<n; ++i)
{
JacobiRotation<Scalar> g;
g.makeGivens(mat(i,i), -temp(i), &mat(i,i));
int rs = n-i-1;
if(rs>0)
{
ColXprSegment x(mat.col(i).tail(rs));
TempVecSegment y(temp.tail(rs));
apply_rotation_in_the_plane(x, y, g);
}
}
}
else
{
temp = vec;
RealScalar beta = 1;
for(int j=0; j<n; ++j)
{
RealScalar Ljj = real(mat.coeff(j,j));
RealScalar dj = abs2(Ljj);
Scalar wj = temp.coeff(j);
RealScalar swj2 = sigma*abs2(wj);
RealScalar gamma = dj*beta + swj2;
RealScalar x = dj + swj2/beta;
if (x<=RealScalar(0))
return j;
RealScalar nLjj = sqrt(x);
mat.coeffRef(j,j) = nLjj;
beta += swj2/dj;
// Update the terms of L
Index rs = n-j-1;
if(rs)
{
temp.tail(rs) -= (wj/Ljj) * mat.col(j).tail(rs);
if(gamma != 0)
mat.col(j).tail(rs) = (nLjj/Ljj) * mat.col(j).tail(rs) + (nLjj * sigma*conj(wj)/gamma)*temp.tail(rs);
}
}
}
return -1;
}
template<typename Scalar> struct llt_inplace<Scalar, Lower>
{
typedef typename NumTraits<Scalar>::Real RealScalar;
template<typename MatrixType>
static typename MatrixType::Index unblocked(MatrixType& mat)
{
typedef typename MatrixType::Index Index;
eigen_assert(mat.rows()==mat.cols());
const Index size = mat.rows();
for(Index k = 0; k < size; ++k)
{
Index rs = size-k-1; // remaining size
Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
RealScalar x = real(mat.coeff(k,k));
if (k>0) x -= A10.squaredNorm();
if (x<=RealScalar(0))
return k;
mat.coeffRef(k,k) = x = sqrt(x);
if (k>0 && rs>0) A21.noalias() -= A20 * A10.adjoint();
if (rs>0) A21 *= RealScalar(1)/x;
}
return -1;
}
template<typename MatrixType>
static typename MatrixType::Index blocked(MatrixType& m)
{
typedef typename MatrixType::Index Index;
eigen_assert(m.rows()==m.cols());
Index size = m.rows();
if(size<32)
return unblocked(m);
Index blockSize = size/8;
blockSize = (blockSize/16)*16;
blockSize = (std::min)((std::max)(blockSize,Index(8)), Index(128));
for (Index k=0; k<size; k+=blockSize)
{
// partition the matrix:
// A00 | - | -
// lu = A10 | A11 | -
// A20 | A21 | A22
Index bs = (std::min)(blockSize, size-k);
Index rs = size - k - bs;
Block<MatrixType,Dynamic,Dynamic> A11(m,k, k, bs,bs);
Block<MatrixType,Dynamic,Dynamic> A21(m,k+bs,k, rs,bs);
Block<MatrixType,Dynamic,Dynamic> A22(m,k+bs,k+bs,rs,rs);
Index ret;
if((ret=unblocked(A11))>=0) return k+ret;
if(rs>0) A11.adjoint().template triangularView<Upper>().template solveInPlace<OnTheRight>(A21);
if(rs>0) A22.template selfadjointView<Lower>().rankUpdate(A21,-1); // bottleneck
}
return -1;
}
template<typename MatrixType, typename VectorType>
static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
{
return Eigen::internal::llt_rank_update_lower(mat, vec, sigma);
}
};
template<typename Scalar> struct llt_inplace<Scalar, Upper>
{
typedef typename NumTraits<Scalar>::Real RealScalar;
template<typename MatrixType>
static EIGEN_STRONG_INLINE typename MatrixType::Index unblocked(MatrixType& mat)
{
Transpose<MatrixType> matt(mat);
return llt_inplace<Scalar, Lower>::unblocked(matt);
}
template<typename MatrixType>
static EIGEN_STRONG_INLINE typename MatrixType::Index blocked(MatrixType& mat)
{
Transpose<MatrixType> matt(mat);
return llt_inplace<Scalar, Lower>::blocked(matt);
}
template<typename MatrixType, typename VectorType>
static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
{
Transpose<MatrixType> matt(mat);
return llt_inplace<Scalar, Lower>::rankUpdate(matt, vec.conjugate(), sigma);
}
};
template<typename MatrixType> struct LLT_Traits<MatrixType,Lower>
{
typedef const TriangularView<const MatrixType, Lower> MatrixL;
typedef const TriangularView<const typename MatrixType::AdjointReturnType, Upper> MatrixU;
static inline MatrixL getL(const MatrixType& m) { return m; }
static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
static bool inplace_decomposition(MatrixType& m)
{ return llt_inplace<typename MatrixType::Scalar, Lower>::blocked(m)==-1; }
};
template<typename MatrixType> struct LLT_Traits<MatrixType,Upper>
{
typedef const TriangularView<const typename MatrixType::AdjointReturnType, Lower> MatrixL;
typedef const TriangularView<const MatrixType, Upper> MatrixU;
static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
static inline MatrixU getU(const MatrixType& m) { return m; }
static bool inplace_decomposition(MatrixType& m)
{ return llt_inplace<typename MatrixType::Scalar, Upper>::blocked(m)==-1; }
};
} // end namespace internal
/** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
*
* \returns a reference to *this
*
* Example: \include TutorialLinAlgComputeTwice.cpp
* Output: \verbinclude TutorialLinAlgComputeTwice.out
*/
template<typename MatrixType, int _UpLo>
LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const MatrixType& a)
{
eigen_assert(a.rows()==a.cols());
const Index size = a.rows();
m_matrix.resize(size, size);
m_matrix = a;
m_isInitialized = true;
bool ok = Traits::inplace_decomposition(m_matrix);
m_info = ok ? Success : NumericalIssue;
return *this;
}
/** Performs a rank one update (or dowdate) of the current decomposition.
* If A = LL^* before the rank one update,
* then after it we have LL^* = A + sigma * v v^* where \a v must be a vector
* of same dimension.
*/
template<typename _MatrixType, int _UpLo>
template<typename VectorType>
LLT<_MatrixType,_UpLo> LLT<_MatrixType,_UpLo>::rankUpdate(const VectorType& v, const RealScalar& sigma)
{
EIGEN_STATIC_ASSERT_VECTOR_ONLY(VectorType);
eigen_assert(v.size()==m_matrix.cols());
eigen_assert(m_isInitialized);
if(internal::llt_inplace<typename MatrixType::Scalar, UpLo>::rankUpdate(m_matrix,v,sigma)>=0)
m_info = NumericalIssue;
else
m_info = Success;
return *this;
}
namespace internal {
template<typename _MatrixType, int UpLo, typename Rhs>
struct solve_retval<LLT<_MatrixType, UpLo>, Rhs>
: solve_retval_base<LLT<_MatrixType, UpLo>, Rhs>
{
typedef LLT<_MatrixType,UpLo> LLTType;
EIGEN_MAKE_SOLVE_HELPERS(LLTType,Rhs)
template<typename Dest> void evalTo(Dest& dst) const
{
dst = rhs();
dec().solveInPlace(dst);
}
};
}
/** \internal use x = llt_object.solve(x);
*
* This is the \em in-place version of solve().
*
* \param bAndX represents both the right-hand side matrix b and result x.
*
* \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
*
* This version avoids a copy when the right hand side matrix b is not
* needed anymore.
*
* \sa LLT::solve(), MatrixBase::llt()
*/
template<typename MatrixType, int _UpLo>
template<typename Derived>
void LLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
eigen_assert(m_matrix.rows()==bAndX.rows());
matrixL().solveInPlace(bAndX);
matrixU().solveInPlace(bAndX);
}
/** \returns the matrix represented by the decomposition,
* i.e., it returns the product: L L^*.
* This function is provided for debug purpose. */
template<typename MatrixType, int _UpLo>
MatrixType LLT<MatrixType,_UpLo>::reconstructedMatrix() const
{
eigen_assert(m_isInitialized && "LLT is not initialized.");
return matrixL() * matrixL().adjoint().toDenseMatrix();
}
/** \cholesky_module
* \returns the LLT decomposition of \c *this
*/
template<typename Derived>
inline const LLT<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::llt() const
{
return LLT<PlainObject>(derived());
}
/** \cholesky_module
* \returns the LLT decomposition of \c *this
*/
template<typename MatrixType, unsigned int UpLo>
inline const LLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
SelfAdjointView<MatrixType, UpLo>::llt() const
{
return LLT<PlainObject,UpLo>(m_matrix);
}
} // end namespace Eigen
#endif // EIGEN_LLT_H