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

#include <PartialPivLU.h>

Inheritance diagram for Eigen::PartialPivLU< _MatrixType >:

Collaboration diagram for Eigen::PartialPivLU< _MatrixType >:

Public Types
enum	{ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef _MatrixType	MatrixType
typedef SolverBase< PartialPivLU >	Base
typedef PermutationMatrix < RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType
typedef Transpositions < RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType
typedef MatrixType::PlainObject	PlainObject
Public Member Functions
	PartialPivLU ()
	Default Constructor.
	PartialPivLU (Index size)
	Default Constructor with memory preallocation.
template<typename InputType >
	PartialPivLU (const EigenBase< InputType > &matrix)
template<typename InputType >
PartialPivLU &	compute (const EigenBase< InputType > &matrix)
const MatrixType &	matrixLU () const
const PermutationType &	permutationP () const
template<typename Rhs >
const Solve< PartialPivLU, Rhs >	solve (const MatrixBase< Rhs > &b) const
const Inverse< PartialPivLU >	inverse () const
internal::traits< MatrixType > ::Scalar	determinant () const
MatrixType	reconstructedMatrix () const
Index	rows () const
Index	cols () const
template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const
template<bool Conjugate, typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl_transposed (const RhsType &rhs, DstType &dst) const
Static Protected Member Functions
static void	check_template_parameters ()
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.

Template 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 62 of file PartialPivLU.h.

Member Typedef Documentation

template<typename _MatrixType >

typedef SolverBase<PartialPivLU> Eigen::PartialPivLU< _MatrixType >::Base

Reimplemented from Eigen::SolverBase< PartialPivLU< _MatrixType > >.

Definition at line 68 of file PartialPivLU.h.

template<typename _MatrixType >

typedef _MatrixType Eigen::PartialPivLU< _MatrixType >::MatrixType

Definition at line 67 of file PartialPivLU.h.

template<typename _MatrixType >

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

Definition at line 75 of file PartialPivLU.h.

template<typename _MatrixType >

typedef MatrixType::PlainObject Eigen::PartialPivLU< _MatrixType >::PlainObject

Definition at line 77 of file PartialPivLU.h.

template<typename _MatrixType >

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

Definition at line 76 of file PartialPivLU.h.

Member Enumeration Documentation

template<typename _MatrixType >

anonymous enum

Enumerator:

MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 71 of file PartialPivLU.h.

         {
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

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 254 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 ) [explicit]

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 264 of file PartialPivLU.h.

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

template<typename MatrixType >

template<typename InputType >

Eigen::PartialPivLU< MatrixType >::PartialPivLU ( const EigenBase< InputType > & matrix ) [explicit]

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 275 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.derived());
}

Member Function Documentation

template<typename _MatrixType >

template<typename RhsType , typename DstType >

EIGEN_DEVICE_FUNC void Eigen::PartialPivLU< _MatrixType >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const `[inline]`

Definition at line 191 of file PartialPivLU.h.

                                                             {
     /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
      * So we proceed as follows:
      * Step 1: compute c = Pb.
      * Step 2: replace c by the solution x to Lx = c.
      * Step 3: replace c by the solution x to Ux = c.
      */

      eigen_assert(rhs.rows() == m_lu.rows());

      // Step 1
      dst = permutationP() * rhs;

      // Step 2
      m_lu.template triangularView<UnitLower>().solveInPlace(dst);

      // Step 3
      m_lu.template triangularView<Upper>().solveInPlace(dst);
    }

template<typename _MatrixType >

template<bool Conjugate, typename RhsType , typename DstType >

EIGEN_DEVICE_FUNC void Eigen::PartialPivLU< _MatrixType >::_solve_impl_transposed	(	const RhsType &	rhs,
		DstType &	dst
	)		const `[inline]`

Definition at line 213 of file PartialPivLU.h.

                                                                        {
     /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
      * So we proceed as follows:
      * Step 1: compute c = Pb.
      * Step 2: replace c by the solution x to Lx = c.
      * Step 3: replace c by the solution x to Ux = c.
      */

      eigen_assert(rhs.rows() == m_lu.cols());

      if (Conjugate) {
        // Step 1
        dst = m_lu.template triangularView<Upper>().adjoint().solve(rhs);
        // Step 2
        m_lu.template triangularView<UnitLower>().adjoint().solveInPlace(dst);
      } else {
        // Step 1
        dst = m_lu.template triangularView<Upper>().transpose().solve(rhs);
        // Step 2
        m_lu.template triangularView<UnitLower>().transpose().solveInPlace(dst);
      }
      // Step 3
      dst = permutationP().transpose() * dst;
    }

template<typename _MatrixType >

static void Eigen::PartialPivLU< _MatrixType >::check_template_parameters ( ) [inline, static, protected]

Definition at line 241 of file PartialPivLU.h.

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

template<typename _MatrixType >

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

Returns:: the number of columns.

See also:: rows(), ColsAtCompileTime

Reimplemented from Eigen::EigenBase< PartialPivLU< _MatrixType > >.

Definition at line 186 of file PartialPivLU.h.

{ return m_lu.cols(); }

template<typename MatrixType >

template<typename InputType >

PartialPivLU< MatrixType > & Eigen::PartialPivLU< MatrixType >::compute ( const EigenBase< InputType > & matrix )

Definition at line 462 of file PartialPivLU.h.

{
  check_template_parameters();

  // the row permutation is stored as int indices, so just to be sure:
  eigen_assert(matrix.rows()<NumTraits<int>::highest());

  m_lu = matrix.derived();

  eigen_assert(matrix.rows() == matrix.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
  const Index size = matrix.rows();

  m_rowsTranspositions.resize(size);

  typename TranspositionType::StorageIndex nb_transpositions;
  internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
  m_det_p = (nb_transpositions%2) ? -1 : 1;

  m_p = m_rowsTranspositions;

  m_isInitialized = true;
  return *this;
}

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 487 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 Inverse<PartialPivLU> Eigen::PartialPivLU< _MatrixType >::inverse ( ) 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 162 of file PartialPivLU.h.

    {
      eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
      return Inverse<PartialPivLU>(*this);
    }

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 115 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 123 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 497 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 >

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

Returns:: the number of rows.

See also:: cols(), RowsAtCompileTime

Reimplemented from Eigen::EigenBase< PartialPivLU< _MatrixType > >.

Definition at line 185 of file PartialPivLU.h.

{ return m_lu.rows(); }

template<typename _MatrixType >

template<typename Rhs >

const Solve<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()

Reimplemented from Eigen::SolverBase< PartialPivLU< _MatrixType > >.

Definition at line 149 of file PartialPivLU.h.

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

Member Data Documentation

template<typename _MatrixType >

Index Eigen::PartialPivLU< _MatrixType >::m_det_p [protected]

Definition at line 249 of file PartialPivLU.h.

template<typename _MatrixType >

bool Eigen::PartialPivLU< _MatrixType >::m_isInitialized [protected]

Definition at line 250 of file PartialPivLU.h.

template<typename _MatrixType >

MatrixType Eigen::PartialPivLU< _MatrixType >::m_lu [protected]

Definition at line 246 of file PartialPivLU.h.

template<typename _MatrixType >

PermutationType Eigen::PartialPivLU< _MatrixType >::m_p [protected]

Definition at line 247 of file PartialPivLU.h.

template<typename _MatrixType >

TranspositionType Eigen::PartialPivLU< _MatrixType >::m_rowsTranspositions [protected]

Definition at line 248 of file PartialPivLU.h.

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

PartialPivLU.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

template<typename _MatrixType> class Eigen::PartialPivLU< _MatrixType >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename _MatrixType>
class Eigen::PartialPivLU< _MatrixType >