LU decomposition of a matrix with partial pivoting, and related features. More...

#include <PartialPivLU.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef MatrixType::Index	Index
typedef _MatrixType	MatrixType
typedef PermutationMatrix < RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType
typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar
typedef MatrixType::Scalar	Scalar
typedef internal::traits < MatrixType >::StorageKind	StorageKind
typedef Transpositions < RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType
Public Member Functions
Index	cols () const
PartialPivLU &	compute (const MatrixType &matrix)
internal::traits< MatrixType > ::Scalar	determinant () const
const internal::solve_retval < PartialPivLU, typename MatrixType::IdentityReturnType >	inverse () const
const MatrixType &	matrixLU () const
	PartialPivLU ()
	Default Constructor.
	PartialPivLU (Index size)
	Default Constructor with memory preallocation.
	PartialPivLU (const MatrixType &matrix)
const PermutationType &	permutationP () const
MatrixType	reconstructedMatrix () const
Index	rows () const
template<typename Rhs >
const internal::solve_retval < PartialPivLU, Rhs >	solve (const MatrixBase< Rhs > &b) const
Protected Attributes
Index	m_det_p
bool	m_isInitialized
MatrixType	m_lu
PermutationType	m_p
TranspositionType	m_rowsTranspositions

Detailed Description

template<typename _MatrixType>
class PartialPivLU< _MatrixType >

LU decomposition of a matrix with partial pivoting, and related features.

Parameters:

MatrixType the type of the matrix of which we are computing the LU decomposition

This class represents a LU decomposition of a square invertible matrix, with partial pivoting: the matrix A is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P is a permutation matrix.

Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.

The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided by class FullPivLU.

This is not a rank-revealing LU decomposition. Many features are intentionally absent from this class, such as rank computation. If you need these features, use class FullPivLU.

This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses in the general case. On the other hand, it is not suitable to determine whether a given matrix is invertible.

The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().

See also:: MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU

Definition at line 60 of file PartialPivLU.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef MatrixType::Index PartialPivLU< _MatrixType >::Index

Definition at line 75 of file PartialPivLU.h.

template<typename _MatrixType>

typedef _MatrixType PartialPivLU< _MatrixType >::MatrixType

Definition at line 64 of file PartialPivLU.h.

template<typename _MatrixType>

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PartialPivLU< _MatrixType >::PermutationType

Definition at line 76 of file PartialPivLU.h.

template<typename _MatrixType>

typedef NumTraits<typename MatrixType::Scalar>::Real PartialPivLU< _MatrixType >::RealScalar

Definition at line 73 of file PartialPivLU.h.

template<typename _MatrixType>

typedef MatrixType::Scalar PartialPivLU< _MatrixType >::Scalar

Definition at line 72 of file PartialPivLU.h.

template<typename _MatrixType>

typedef internal::traits<MatrixType>::StorageKind PartialPivLU< _MatrixType >::StorageKind

Definition at line 74 of file PartialPivLU.h.

template<typename _MatrixType>

typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> PartialPivLU< _MatrixType >::TranspositionType

Definition at line 77 of file PartialPivLU.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 65 of file PartialPivLU.h.

Constructor & Destructor Documentation

template<typename MatrixType >

PartialPivLU< MatrixType >::PartialPivLU ( )

Default Constructor.

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

Definition at line 195 of file PartialPivLU.h.

template<typename MatrixType >

PartialPivLU< MatrixType >::PartialPivLU ( Index size )

Default Constructor with memory preallocation.

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

See also:: PartialPivLU()

Definition at line 205 of file PartialPivLU.h.

template<typename MatrixType >

PartialPivLU< MatrixType >::PartialPivLU ( const MatrixType & matrix )

Constructor.

Parameters:

matrix the matrix of which to compute the LU decomposition.

Warning:: The matrix should have full rank (e.g. if it's square, it should be invertible). If you need to deal with non-full rank, use class FullPivLU instead.

Definition at line 215 of file PartialPivLU.h.

Member Function Documentation

template<typename _MatrixType>

Index PartialPivLU< _MatrixType >::cols ( void ) const [inline]

Definition at line 184 of file PartialPivLU.h.

template<typename MatrixType >

PartialPivLU< MatrixType > & PartialPivLU< MatrixType >::compute ( const MatrixType & matrix )

Definition at line 400 of file PartialPivLU.h.

template<typename MatrixType >

internal::traits< MatrixType >::Scalar PartialPivLU< MatrixType >::determinant ( ) const

Returns:: the determinant of the matrix of which *this is the LU decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the LU decomposition has already been computed.

Note:: For fixed-size matrices of size up to 4, MatrixBase::determinant() offers optimized paths.

Warning:: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow.

See also:: MatrixBase::determinant()

Definition at line 420 of file PartialPivLU.h.

template<typename _MatrixType>

const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> PartialPivLU< _MatrixType >::inverse ( void ) const [inline]

Returns:: the inverse of the matrix of which *this is the LU decomposition.

Warning:: The matrix being decomposed here is assumed to be invertible. If you need to check for invertibility, use class FullPivLU instead.

See also:: MatrixBase::inverse(), LU::inverse()

Definition at line 159 of file PartialPivLU.h.

template<typename _MatrixType>

const MatrixType& PartialPivLU< _MatrixType >::matrixLU ( ) const [inline]

Returns:: the LU decomposition matrix: the upper-triangular part is U, the unit-lower-triangular part is L (at least for square matrices; in the non-square case, special care is needed, see the documentation of class FullPivLU).

See also:: matrixL(), matrixU()

Definition at line 113 of file PartialPivLU.h.

template<typename _MatrixType>

const PermutationType& PartialPivLU< _MatrixType >::permutationP ( ) const [inline]

Returns:: the permutation matrix P.

Definition at line 121 of file PartialPivLU.h.

template<typename MatrixType >

MatrixType PartialPivLU< MatrixType >::reconstructedMatrix ( ) const

Returns:: the matrix represented by the decomposition, i.e., it returns the product: P^{-1} L U. This function is provided for debug purpose.

Definition at line 430 of file PartialPivLU.h.

template<typename _MatrixType>

Index PartialPivLU< _MatrixType >::rows ( void ) const [inline]

Definition at line 183 of file PartialPivLU.h.

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<PartialPivLU, Rhs> PartialPivLU< _MatrixType >::solve ( const MatrixBase< Rhs > & b ) const [inline]

This method returns the solution x to the equation Ax=b, where A is the matrix of which *this is the LU decomposition.

Parameters:

b	the right-hand-side of the equation to solve. Can be a vector or a matrix, the only requirement in order for the equation to make sense is that b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.

Returns:: the solution.

Example:

MatrixXd A = MatrixXd::Random(3,3);
MatrixXd B = MatrixXd::Random(3,2);
cout << "Here is the invertible matrix A:" << endl << A << endl;
cout << "Here is the matrix B:" << endl << B << endl;
MatrixXd X = A.lu().solve(B);
cout << "Here is the (unique) solution X to the equation AX=B:" << endl << X << endl;
cout << "Relative error: " << (A*X-B).norm() / B.norm() << endl;

Output:

Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.

See also:: TriangularView::solve(), inverse(), computeInverse()

Definition at line 146 of file PartialPivLU.h.

Member Data Documentation

template<typename _MatrixType>

Index PartialPivLU< _MatrixType >::m_det_p [protected]

Definition at line 190 of file PartialPivLU.h.

template<typename _MatrixType>

bool PartialPivLU< _MatrixType >::m_isInitialized [protected]

Definition at line 191 of file PartialPivLU.h.

template<typename _MatrixType>

MatrixType PartialPivLU< _MatrixType >::m_lu [protected]

Definition at line 187 of file PartialPivLU.h.

template<typename _MatrixType>

PermutationType PartialPivLU< _MatrixType >::m_p [protected]

Definition at line 188 of file PartialPivLU.h.

template<typename _MatrixType>

TranspositionType PartialPivLU< _MatrixType >::m_rowsTranspositions [protected]

Definition at line 189 of file PartialPivLU.h.

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

PartialPivLU.h

Public Types

Public Member Functions

Protected Attributes

Detailed Description

template<typename _MatrixType> class PartialPivLU< _MatrixType >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename _MatrixType>
class PartialPivLU< _MatrixType >