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().

This class supports the inplace decomposition mechanism.

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

Inheritance diagram for Eigen::PartialPivLU< MatrixType_ >:

Public Member Functions
Scalar	determinant () const

const Inverse< PartialPivLU >	inverse () const

const MatrixType &	matrixLU () const

	PartialPivLU ()
	Default Constructor. More...

template<typename InputType >
	PartialPivLU (const EigenBase< InputType > &matrix)

template<typename InputType >
	PartialPivLU (EigenBase< InputType > &matrix)

	PartialPivLU (Index size)
	Default Constructor with memory preallocation. More...

const PermutationType &	permutationP () const

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

template<typename Rhs >
const Solve< PartialPivLU, Rhs >	solve (const MatrixBase< Rhs > &b) const

Public Member Functions inherited from Eigen::SolverBase< PartialPivLU< MatrixType_ > >
const AdjointReturnType	adjoint () const

PartialPivLU< MatrixType_ > &	derived ()

const PartialPivLU< MatrixType_ > &	derived () const

const Solve< PartialPivLU< MatrixType_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()

const ConstTransposeReturnType	transpose () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

Derived &	derived ()

const Derived &	derived () const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

Additional Inherited Members
Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index	Index
	The interface type of indices. More...

Constructor & Destructor Documentation

◆ PartialPivLU() [1/4]

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&).

◆ PartialPivLU() [2/4]

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()

◆ PartialPivLU() [3/4]

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.

◆ PartialPivLU() [4/4]

template<typename MatrixType >

template<typename InputType >

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

explicit

Constructor for inplace decomposition

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.

Member Function Documentation

◆ determinant()

template<typename MatrixType >

PartialPivLU< MatrixType >::Scalar Eigen::PartialPivLU< MatrixType >::determinant

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()

◆ inverse()

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()

◆ matrixLU()

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()

◆ permutationP()

template<typename MatrixType_ >

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

inline

Returns: the permutation matrix P.

◆ rcond()

template<typename MatrixType_ >

RealScalar Eigen::PartialPivLU< MatrixType_ >::rcond ( ) const

inline

Returns: an estimate of the reciprocal condition number of the matrix of which *this is the LU decomposition.

◆ reconstructedMatrix()

template<typename MatrixType >

MatrixType Eigen::PartialPivLU< MatrixType >::reconstructedMatrix

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

◆ solve()

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:

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:

Here is the invertible matrix A:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Here is the matrix B:
  0.108   -0.27
-0.0452  0.0268
  0.258   0.904
Here is the (unique) solution X to the equation AX=B:
 0.609   2.68
-0.231  -1.57
  0.51   3.51
Relative error: 3.28e-16

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()

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

PartialPivLU.h

Detailed Description

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

Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ PartialPivLU() [1/4]

◆ PartialPivLU() [2/4]

◆ PartialPivLU() [3/4]

◆ PartialPivLU() [4/4]

Member Function Documentation

◆ determinant()

◆ inverse()

◆ matrixLU()

◆ permutationP()

◆ rcond()

◆ reconstructedMatrix()

◆ solve()

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