Eigen::PartialPivLU< _MatrixType > Class Template Reference

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

#include <PartialPivLU.h>

Collaboration diagram for Eigen::PartialPivLU< _MatrixType >:

Public Types
enum	{   RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,   MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef _MatrixType	MatrixType
typedef MatrixType::Scalar	Scalar
typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar
typedef internal::traits< MatrixType >::StorageKind	StorageKind
typedef MatrixType::Index	Index
typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType
typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType

Public Member Functions
	PartialPivLU ()
	Default Constructor. More...
	PartialPivLU (Index size)
	Default Constructor with memory preallocation. More...
	PartialPivLU (const MatrixType &matrix)
PartialPivLU &	compute (const MatrixType &matrix)
const MatrixType &	matrixLU () const
const PermutationType &	permutationP () const
template<typename Rhs >
const internal::solve_retval< PartialPivLU, Rhs >	solve (const MatrixBase< Rhs > &b) const
const internal::solve_retval< PartialPivLU, typename MatrixType::IdentityReturnType >	inverse () const
internal::traits< MatrixType >::Scalar	determinant () const
MatrixType	reconstructedMatrix () const
Index	rows () const
Index	cols () const

Protected Attributes
MatrixType	m_lu
PermutationType	m_p
TranspositionType	m_rowsTranspositions
Index	m_det_p
bool	m_isInitialized

Detailed Description

template<typename _MatrixType>
class Eigen::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 217 of file ForwardDeclarations.h.

Constructor & Destructor Documentation

template<typename MatrixType >

Eigen::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 182 of file PartialPivLU.h.

  : m_lu(),
    m_p(),
    m_rowsTranspositions(),
    m_det_p(0),
    m_isInitialized(false)
{
}

template<typename MatrixType >

Eigen::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 192 of file PartialPivLU.h.

  : m_lu(size, size),
    m_p(size),
    m_rowsTranspositions(size),
    m_det_p(0),
    m_isInitialized(false)
{
}

template<typename MatrixType >

Eigen::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 202 of file PartialPivLU.h.

  : m_lu(matrix.rows(), matrix.rows()),
    m_p(matrix.rows()),
    m_rowsTranspositions(matrix.rows()),
    m_det_p(0),
    m_isInitialized(false)
{
  compute(matrix);
}

Member Function Documentation

template<typename MatrixType >

internal::traits< MatrixType >::Scalar Eigen::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 410 of file PartialPivLU.h.

{
  eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
  return Scalar(m_det_p) * m_lu.diagonal().prod();
}

template<typename _MatrixType>

const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> Eigen::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 146 of file PartialPivLU.h.

    {
      eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
      return internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType>
               (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
    }

template<typename _MatrixType>

const MatrixType& Eigen::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 100 of file PartialPivLU.h.

    {
      eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
      return m_lu;
    }

template<typename _MatrixType>

const PermutationType& Eigen::PartialPivLU< _MatrixType >::permutationP ( ) const

inline

Returns

the permutation matrix P.

Definition at line 108 of file PartialPivLU.h.

    {
      eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
      return m_p;
    }

template<typename MatrixType >

MatrixType Eigen::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 420 of file PartialPivLU.h.

{
  eigen_assert(m_isInitialized && "LU is not initialized.");
  // LU
  MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
                 * m_lu.template triangularView<Upper>();

  // P^{-1}(LU)
  res = m_p.inverse() * res;

  return res;
}

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<PartialPivLU, Rhs> Eigen::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:

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 133 of file PartialPivLU.h.

    {
      eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
      return internal::solve_retval<PartialPivLU, Rhs>(*this, b.derived());
    }

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

• C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Core/util/ForwardDeclarations.h
• C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/LU/PartialPivLU.h