Public Types | Public Member Functions | Protected Attributes
LLT< _MatrixType, _UpLo > Class Template Reference

Standard Cholesky decomposition (LL^T) of a matrix and associated features. More...

#include <LLT.h>

List of all members.

Public Types

enum  { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
enum  { PacketSize = internal::packet_traits<Scalar>::size, AlignmentMask = int(PacketSize)-1, UpLo = _UpLo }
typedef MatrixType::Index Index
typedef _MatrixType MatrixType
typedef NumTraits< typename
MatrixType::Scalar >::Real 
RealScalar
typedef MatrixType::Scalar Scalar
typedef internal::LLT_Traits
< MatrixType, UpLo
Traits

Public Member Functions

Index cols () const
LLTcompute (const MatrixType &matrix)
ComputationInfo info () const
 Reports whether previous computation was successful.
 LLT ()
 Default Constructor.
 LLT (Index size)
 Default Constructor with memory preallocation.
 LLT (const MatrixType &matrix)
Traits::MatrixL matrixL () const
const MatrixTypematrixLLT () const
Traits::MatrixU matrixU () const
MatrixType reconstructedMatrix () const
Index rows () const
template<typename Rhs >
const internal::solve_retval
< LLT, Rhs > 
solve (const MatrixBase< Rhs > &b) const
template<typename Derived >
void solveInPlace (MatrixBase< Derived > &bAndX) const

Protected Attributes

ComputationInfo m_info
bool m_isInitialized
MatrixType m_matrix

Detailed Description

template<typename _MatrixType, int _UpLo>
class LLT< _MatrixType, _UpLo >

Standard Cholesky decomposition (LL^T) of a matrix and associated features.

Parameters:
MatrixTypethe type of the matrix of which we are computing the LL^T Cholesky decomposition

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.

See also:
MatrixBase::llt(), class LDLT

Definition at line 58 of file LLT.h.


Member Typedef Documentation

template<typename _MatrixType, int _UpLo>
typedef MatrixType::Index LLT< _MatrixType, _UpLo >::Index

Definition at line 70 of file LLT.h.

template<typename _MatrixType, int _UpLo>
typedef _MatrixType LLT< _MatrixType, _UpLo >::MatrixType

Definition at line 61 of file LLT.h.

template<typename _MatrixType, int _UpLo>
typedef NumTraits<typename MatrixType::Scalar>::Real LLT< _MatrixType, _UpLo >::RealScalar

Definition at line 69 of file LLT.h.

template<typename _MatrixType, int _UpLo>
typedef MatrixType::Scalar LLT< _MatrixType, _UpLo >::Scalar

Definition at line 68 of file LLT.h.

template<typename _MatrixType, int _UpLo>
typedef internal::LLT_Traits<MatrixType,UpLo> LLT< _MatrixType, _UpLo >::Traits

Definition at line 78 of file LLT.h.


Member Enumeration Documentation

template<typename _MatrixType, int _UpLo>
anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxColsAtCompileTime 

Definition at line 62 of file LLT.h.

template<typename _MatrixType, int _UpLo>
anonymous enum
Enumerator:
PacketSize 
AlignmentMask 
UpLo 

Definition at line 72 of file LLT.h.


Constructor & Destructor Documentation

template<typename _MatrixType, int _UpLo>
LLT< _MatrixType, _UpLo >::LLT ( ) [inline]

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via LLT::compute(const MatrixType&).

Definition at line 86 of file LLT.h.

template<typename _MatrixType, int _UpLo>
LLT< _MatrixType, _UpLo >::LLT ( Index  size) [inline]

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also:
LLT()

Definition at line 94 of file LLT.h.

template<typename _MatrixType, int _UpLo>
LLT< _MatrixType, _UpLo >::LLT ( const MatrixType matrix) [inline]

Definition at line 97 of file LLT.h.


Member Function Documentation

template<typename _MatrixType, int _UpLo>
Index LLT< _MatrixType, _UpLo >::cols ( ) const [inline]

Definition at line 179 of file LLT.h.

template<typename MatrixType , int _UpLo>
LLT< MatrixType, _UpLo > & LLT< MatrixType, _UpLo >::compute ( const MatrixType a)

Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix

Returns:
a reference to *this

Definition at line 303 of file LLT.h.

template<typename _MatrixType, int _UpLo>
ComputationInfo LLT< _MatrixType, _UpLo >::info ( ) const [inline]

Reports whether previous computation was successful.

Returns:
Success if computation was succesful, NumericalIssue if the matrix.appears to be negative.

Definition at line 172 of file LLT.h.

template<typename _MatrixType, int _UpLo>
Traits::MatrixL LLT< _MatrixType, _UpLo >::matrixL ( ) const [inline]
Returns:
a view of the lower triangular matrix L

Definition at line 112 of file LLT.h.

template<typename _MatrixType, int _UpLo>
const MatrixType& LLT< _MatrixType, _UpLo >::matrixLLT ( ) const [inline]
Returns:
the LLT decomposition matrix

TODO: document the storage layout

Definition at line 158 of file LLT.h.

template<typename _MatrixType, int _UpLo>
Traits::MatrixU LLT< _MatrixType, _UpLo >::matrixU ( ) const [inline]
Returns:
a view of the upper triangular matrix U

Definition at line 105 of file LLT.h.

template<typename MatrixType , int _UpLo>
MatrixType LLT< MatrixType, _UpLo >::reconstructedMatrix ( ) const
Returns:
the matrix represented by the decomposition, i.e., it returns the product: L L^*. This function is provided for debug purpose.

Definition at line 360 of file LLT.h.

template<typename _MatrixType, int _UpLo>
Index LLT< _MatrixType, _UpLo >::rows ( ) const [inline]

Definition at line 178 of file LLT.h.

template<typename _MatrixType, int _UpLo>
template<typename Rhs >
const internal::solve_retval<LLT, Rhs> LLT< _MatrixType, _UpLo >::solve ( const MatrixBase< Rhs > &  b) const [inline]
Returns:
the solution x of $ A x = b $ 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:

typedef Matrix<float,Dynamic,2> DataMatrix;
// let's generate some samples on the 3D plane of equation z = 2x+3y (with some noise)
DataMatrix samples = DataMatrix::Random(12,2);
VectorXf elevations = 2*samples.col(0) + 3*samples.col(1) + VectorXf::Random(12)*0.1;
// and let's solve samples * [x y]^T = elevations in least square sense:
Matrix<float,2,1> xy
 = (samples.adjoint() * samples).llt().solve((samples.adjoint()*elevations));
cout << xy << endl;

Output:

See also:
solveInPlace(), MatrixBase::llt()

Definition at line 130 of file LLT.h.

template<typename MatrixType , int _UpLo>
template<typename Derived >
void LLT< MatrixType, _UpLo >::solveInPlace ( MatrixBase< Derived > &  bAndX) const

Definition at line 348 of file LLT.h.


Member Data Documentation

template<typename _MatrixType, int _UpLo>
ComputationInfo LLT< _MatrixType, _UpLo >::m_info [protected]

Definition at line 188 of file LLT.h.

template<typename _MatrixType, int _UpLo>
bool LLT< _MatrixType, _UpLo >::m_isInitialized [protected]

Definition at line 187 of file LLT.h.

template<typename _MatrixType, int _UpLo>
MatrixType LLT< _MatrixType, _UpLo >::m_matrix [protected]

Definition at line 186 of file LLT.h.


The documentation for this class was generated from the following file:


libicr
Author(s): Robert Krug
autogenerated on Mon Jan 6 2014 11:34:21