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

#include <FullPivLU.h>

Collaboration diagram for Eigen::FullPivLU< _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 internal::plain_row_type< MatrixType, Index >::type	IntRowVectorType

typedef internal::plain_col_type< MatrixType, Index >::type	IntColVectorType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationQType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationPType

Public Member Functions
	FullPivLU ()
	Default Constructor. More...

	FullPivLU (Index rows, Index cols)
	Default Constructor with memory preallocation. More...

	FullPivLU (const MatrixType &matrix)

FullPivLU &	compute (const MatrixType &matrix)

const MatrixType &	matrixLU () const

Index	nonzeroPivots () const

RealScalar	maxPivot () const

const PermutationPType &	permutationP () const

const PermutationQType &	permutationQ () const

const internal::kernel_retval< FullPivLU >	kernel () const

const internal::image_retval< FullPivLU >	image (const MatrixType &originalMatrix) const

template<typename Rhs >
const internal::solve_retval< FullPivLU, Rhs >	solve (const MatrixBase< Rhs > &b) const

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

FullPivLU &	setThreshold (const RealScalar &threshold)

FullPivLU &	setThreshold (Default_t)

RealScalar	threshold () const

Index	rank () const

Index	dimensionOfKernel () const

bool	isInjective () const

bool	isSurjective () const

bool	isInvertible () const

const internal::solve_retval< FullPivLU, typename MatrixType::IdentityReturnType >	inverse () const

MatrixType	reconstructedMatrix () const

Index	rows () const

Index	cols () const

Protected Attributes
MatrixType	m_lu

PermutationPType	m_p

PermutationQType	m_q

IntColVectorType	m_rowsTranspositions

IntRowVectorType	m_colsTranspositions

Index	m_det_pq

Index	m_nonzero_pivots

RealScalar	m_maxpivot

RealScalar	m_prescribedThreshold

bool	m_isInitialized

bool	m_usePrescribedThreshold

Detailed Description

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

LU decomposition of a matrix with complete 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 any matrix, with complete pivoting: the matrix A is decomposed as A = PLUQ where L is unit-lower-triangular, U is upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any zeros are at the end.

This decomposition provides the generic approach to solving systems of linear equations, computing the rank, invertibility, inverse, kernel, and determinant.

This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, working with the SVD allows to select the smallest singular values of the matrix, something that the LU decomposition doesn't see.

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

As an exemple, here is how the original matrix can be retrieved:

Output:

See also: MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()

Definition at line 216 of file ForwardDeclarations.h.

Constructor & Destructor Documentation

template<typename MatrixType >

Eigen::FullPivLU< MatrixType >::FullPivLU ( )

Default Constructor.

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

Definition at line 387 of file FullPivLU.h.

   : m_isInitialized(false), m_usePrescribedThreshold(false)
 {
 }

template<typename MatrixType >

Eigen::FullPivLU< MatrixType >::FullPivLU	(	Index	rows,
		Index	cols
	)

Default Constructor with memory preallocation.

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

See also: FullPivLU()

Definition at line 393 of file FullPivLU.h.

   : m_lu(rows, cols),
     m_p(rows),
     m_q(cols),
     m_rowsTranspositions(rows),
     m_colsTranspositions(cols),
     m_isInitialized(false),
     m_usePrescribedThreshold(false)
 {
 }

template<typename MatrixType >

Eigen::FullPivLU< MatrixType >::FullPivLU ( const MatrixType & matrix )

Constructor.

Parameters

matrix the matrix of which to compute the LU decomposition. It is required to be nonzero.

Definition at line 405 of file FullPivLU.h.

   : m_lu(matrix.rows(), matrix.cols()),
     m_p(matrix.rows()),
     m_q(matrix.cols()),
     m_rowsTranspositions(matrix.rows()),
     m_colsTranspositions(matrix.cols()),
     m_isInitialized(false),
     m_usePrescribedThreshold(false)
 {
   compute(matrix);
 }

Member Function Documentation

template<typename MatrixType >

FullPivLU< MatrixType > & Eigen::FullPivLU< MatrixType >::compute ( const MatrixType & matrix )

Computes the LU decomposition of the given matrix.

Parameters

matrix the matrix of which to compute the LU decomposition. It is required to be nonzero.

Returns: a reference to *this

Definition at line 418 of file FullPivLU.h.

 {
   // the permutations are stored as int indices, so just to be sure:
   eigen_assert(matrix.rows()<=NumTraits<int>::highest() && matrix.cols()<=NumTraits<int>::highest());
   
   m_isInitialized = true;
   m_lu = matrix;
 
   const Index size = matrix.diagonalSize();
   const Index rows = matrix.rows();
   const Index cols = matrix.cols();
 
   // will store the transpositions, before we accumulate them at the end.
   // can't accumulate on-the-fly because that will be done in reverse order for the rows.
   m_rowsTranspositions.resize(matrix.rows());
   m_colsTranspositions.resize(matrix.cols());
   Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i
 
   m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
   m_maxpivot = RealScalar(0);
 
   for(Index k = 0; k < size; ++k)
   {
     // First, we need to find the pivot.
 
     // biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
     Index row_of_biggest_in_corner, col_of_biggest_in_corner;
     RealScalar biggest_in_corner;
     biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
                         .cwiseAbs()
                         .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
     row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
     col_of_biggest_in_corner += k; // need to add k to them.
 
     if(biggest_in_corner==RealScalar(0))
     {
       // before exiting, make sure to initialize the still uninitialized transpositions
       // in a sane state without destroying what we already have.
       m_nonzero_pivots = k;
       for(Index i = k; i < size; ++i)
       {
         m_rowsTranspositions.coeffRef(i) = i;
         m_colsTranspositions.coeffRef(i) = i;
       }
       break;
     }
 
     if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner;
 
     // Now that we've found the pivot, we need to apply the row/col swaps to
     // bring it to the location (k,k).
 
     m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner;
     m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner;
     if(k != row_of_biggest_in_corner) {
       m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
       ++number_of_transpositions;
     }
     if(k != col_of_biggest_in_corner) {
       m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
       ++number_of_transpositions;
     }
 
     // Now that the pivot is at the right location, we update the remaining
     // bottom-right corner by Gaussian elimination.
 
     if(k<rows-1)
       m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
     if(k<size-1)
       m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1);
   }
 
   // the main loop is over, we still have to accumulate the transpositions to find the
   // permutations P and Q
 
   m_p.setIdentity(rows);
   for(Index k = size-1; k >= 0; --k)
     m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
 
   m_q.setIdentity(cols);
   for(Index k = 0; k < size; ++k)
     m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
 
   m_det_pq = (number_of_transpositions%2) ? -1 : 1;
   return *this;
 }

template<typename MatrixType >

internal::traits< MatrixType >::Scalar Eigen::FullPivLU< 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: This is only for square matrices.; 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 506 of file FullPivLU.h.

 {
   eigen_assert(m_isInitialized && "LU is not initialized.");
   eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
   return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
 }

template<typename _MatrixType>

Index Eigen::FullPivLU< _MatrixType >::dimensionOfKernel ( ) const

inline

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

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 311 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return cols() - rank();
     }

template<typename _MatrixType>

const internal::image_retval<FullPivLU> Eigen::FullPivLU< _MatrixType >::image ( const MatrixType & originalMatrix ) const

inline

Returns: the image of the matrix, also called its column-space. The columns of the returned matrix will form a basis of the kernel.

Parameters

originalMatrix the original matrix, of which *this is the LU decomposition. The reason why it is needed to pass it here, is that this allows a large optimization, as otherwise this method would need to reconstruct it from the LU decomposition.

Note: If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.; This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Example:

Output:

See also: kernel()

Definition at line 187 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return internal::image_retval<FullPivLU>(*this, originalMatrix);
     }

template<typename _MatrixType>

const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> Eigen::FullPivLU< _MatrixType >::inverse ( void ) const

inline

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

Note: If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.

See also: MatrixBase::inverse()

Definition at line 362 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
       return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
                (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
     }

template<typename _MatrixType>

bool Eigen::FullPivLU< _MatrixType >::isInjective ( ) const

inline

Returns: true if the matrix of which *this is the LU decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 324 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return rank() == cols();
     }

template<typename _MatrixType>

bool Eigen::FullPivLU< _MatrixType >::isInvertible ( ) const

inline

Returns: true if the matrix of which *this is the LU decomposition is invertible.

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 349 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return isInjective() && (m_lu.rows() == m_lu.cols());
     }

template<typename _MatrixType>

bool Eigen::FullPivLU< _MatrixType >::isSurjective ( ) const

inline

Returns: true if the matrix of which *this is the LU decomposition represents a surjective linear map; false otherwise.

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 337 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return rank() == rows();
     }

template<typename _MatrixType>

const internal::kernel_retval<FullPivLU> Eigen::FullPivLU< _MatrixType >::kernel ( ) const

inline

Returns: the kernel of the matrix, also called its null-space. The columns of the returned matrix will form a basis of the kernel.

Note: If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.; This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Example:

Output:

See also: image()

Definition at line 161 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return internal::kernel_retval<FullPivLU>(*this);
     }

template<typename _MatrixType>

const MatrixType& Eigen::FullPivLU< _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 103 of file FullPivLU.h.

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

template<typename _MatrixType>

RealScalar Eigen::FullPivLU< _MatrixType >::maxPivot ( ) const

inline

Returns: the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.

Definition at line 125 of file FullPivLU.h.

125 { return m_maxpivot; }

template<typename _MatrixType>

Index Eigen::FullPivLU< _MatrixType >::nonzeroPivots ( ) const

inline

Returns: the number of nonzero pivots in the LU decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.

See also: rank()

Definition at line 116 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return m_nonzero_pivots;
     }

template<typename _MatrixType>

const PermutationPType& Eigen::FullPivLU< _MatrixType >::permutationP ( ) const

inline

Returns: the permutation matrix P

See also: permutationQ()

Definition at line 131 of file FullPivLU.h.

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

template<typename _MatrixType>

const PermutationQType& Eigen::FullPivLU< _MatrixType >::permutationQ ( ) const

inline

Returns: the permutation matrix Q

See also: permutationP()

Definition at line 141 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return m_q;
     }

template<typename _MatrixType>

Index Eigen::FullPivLU< _MatrixType >::rank ( ) const

inline

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

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 294 of file FullPivLU.h.

     {
       using std::abs;
       eigen_assert(m_isInitialized && "LU is not initialized.");
       RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
       Index result = 0;
       for(Index i = 0; i < m_nonzero_pivots; ++i)
         result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold);
       return result;
     }

template<typename MatrixType >

MatrixType Eigen::FullPivLU< MatrixType >::reconstructedMatrix ( ) const

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

Definition at line 517 of file FullPivLU.h.

 {
   eigen_assert(m_isInitialized && "LU is not initialized.");
   const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
   // LU
   MatrixType res(m_lu.rows(),m_lu.cols());
   // FIXME the .toDenseMatrix() should not be needed...
   res = m_lu.leftCols(smalldim)
             .template triangularView<UnitLower>().toDenseMatrix()
       * m_lu.topRows(smalldim)
             .template triangularView<Upper>().toDenseMatrix();
 
   // P^{-1}(LU)
   res = m_p.inverse() * res;
 
   // (P^{-1}LU)Q^{-1}
   res = res * m_q.inverse();
 
   return res;
 }

template<typename _MatrixType>

FullPivLU& Eigen::FullPivLU< _MatrixType >::setThreshold ( const RealScalar & threshold )

inline

Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. This is not used for the LU decomposition itself.

When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.

Parameters

threshold The new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than $\vert pivot \vert \leqslant threshold \times \vert maxpivot \vert$ where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

Definition at line 254 of file FullPivLU.h.

     {
       m_usePrescribedThreshold = true;
       m_prescribedThreshold = threshold;
       return *this;
     }

template<typename _MatrixType>

FullPivLU& Eigen::FullPivLU< _MatrixType >::setThreshold ( Default_t )

inline

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

lu.setThreshold(Eigen::Default);

See the documentation of setThreshold(const RealScalar&).

Definition at line 269 of file FullPivLU.h.

     {
       m_usePrescribedThreshold = false;
       return *this;
     }

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<FullPivLU, Rhs> Eigen::FullPivLU< _MatrixType >::solve ( const MatrixBase< Rhs > & b ) const

inline

Returns: a 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: a solution.

Example:

Output:

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

Definition at line 214 of file FullPivLU.h.

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

template<typename _MatrixType>

RealScalar Eigen::FullPivLU< _MatrixType >::threshold ( ) const

inline

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

Definition at line 279 of file FullPivLU.h.

     {
       eigen_assert(m_isInitialized || m_usePrescribedThreshold);
       return m_usePrescribedThreshold ? m_prescribedThreshold
       // this formula comes from experimenting (see "LU precision tuning" thread on the list)
       // and turns out to be identical to Higham's formula used already in LDLt.
                                       : NumTraits<Scalar>::epsilon() * m_lu.diagonalSize();
     }

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/FullPivLU.h

Public Types

Public Member Functions

Protected Attributes

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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