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 $A = P^{-1} L U Q^{-1}$ 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:

typedef Matrix<double, 5, 3> Matrix5x3;
typedef Matrix<double, 5, 5> Matrix5x5;
Matrix5x3 m = Matrix5x3::Random();
cout << "Here is the matrix m:" << endl << m << endl;
Eigen::FullPivLU<Matrix5x3> lu(m);
cout << "Here is, up to permutations, its LU decomposition matrix:"
     << endl << lu.matrixLU() << endl;
cout << "Here is the L part:" << endl;
Matrix5x5 l = Matrix5x5::Identity();
l.block<5,3>(0,0).triangularView<StrictlyLower>() = lu.matrixLU();
cout << l << endl;
cout << "Here is the U part:" << endl;
Matrix5x3 u = lu.matrixLU().triangularView<Upper>();
cout << u << endl;
cout << "Let us now reconstruct the original matrix m:" << endl;
cout << lu.permutationP().inverse() * l * u * lu.permutationQ().inverse() << endl;

Output:

Here is the matrix m:
   0.68  -0.605 -0.0452
 -0.211   -0.33   0.258
  0.566   0.536   -0.27
  0.597  -0.444  0.0268
  0.823   0.108   0.904
Here is, up to permutations, its LU decomposition matrix:
 0.904  0.823  0.108
-0.299  0.812  0.569
 -0.05  0.888   -1.1
0.0296  0.705  0.768
 0.285 -0.549 0.0436
Here is the L part:
     1      0      0      0      0
-0.299      1      0      0      0
 -0.05  0.888      1      0      0
0.0296  0.705  0.768      1      0
 0.285 -0.549 0.0436      0      1
Here is the U part:
0.904 0.823 0.108
    0 0.812 0.569
    0     0  -1.1
    0     0     0
    0     0     0
Let us now reconstruct the original matrix m:
   0.68  -0.605 -0.0452
 -0.211   -0.33   0.258
  0.566   0.536   -0.27
  0.597  -0.444  0.0268
  0.823   0.108   0.904

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

Public Member Functions
FullPivLU &	compute (const MatrixType &matrix)

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

Index	dimensionOfKernel () const

	FullPivLU ()
	Default Constructor. More...

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

	FullPivLU (const MatrixType &matrix)

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

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

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

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

const MatrixType &	matrixLU () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

const PermutationPType &	permutationP () const

const PermutationQType &	permutationQ () const

Index	rank () const

MatrixType	reconstructedMatrix () const

FullPivLU &	setThreshold (const RealScalar &threshold)

FullPivLU &	setThreshold (Default_t)

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

RealScalar	threshold () const

Constructor & Destructor Documentation

FullPivLU ( )

Default Constructor.

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

FullPivLU ( const MatrixType & matrix )

Constructor.

Parameters

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

References FullPivLU< MatrixType >::compute().

Member Function Documentation

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

Referenced by FullPivLU< MatrixType >::FullPivLU().

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

Index 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&).

References FullPivLU< MatrixType >::rank().

const internal::image_retval<FullPivLU> 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:

Matrix3d m;
m << 1,1,0,
     1,3,2,
     0,1,1;
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Notice that the middle column is the sum of the two others, so the "
     << "columns are linearly dependent." << endl;
cout << "Here is a matrix whose columns have the same span but are linearly independent:"
     << endl << m.fullPivLu().image(m) << endl;

Output:

Here is the matrix m:
1 1 0
1 3 2
0 1 1
Notice that the middle column is the sum of the two others, so the columns are linearly dependent.
Here is a matrix whose columns have the same span but are linearly independent:
1 1
3 1
1 0

See Also: kernel()

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

bool 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&).

References FullPivLU< MatrixType >::rank().

Referenced by FullPivLU< MatrixType >::isInvertible().

bool 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&).

References FullPivLU< MatrixType >::isInjective().

bool 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&).

References FullPivLU< MatrixType >::rank().

const internal::kernel_retval<FullPivLU> 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:

MatrixXf m = MatrixXf::Random(3,5);
cout << "Here is the matrix m:" << endl << m << endl;
MatrixXf ker = m.fullPivLu().kernel();
cout << "Here is a matrix whose columns form a basis of the kernel of m:"
     << endl << ker << endl;
cout << "By definition of the kernel, m*ker is zero:"
     << endl << m*ker << endl;

Output:

Here is the matrix m:
   0.68   0.597   -0.33   0.108   -0.27
 -0.211   0.823   0.536 -0.0452  0.0268
  0.566  -0.605  -0.444   0.258   0.904
Here is a matrix whose columns form a basis of the kernel of m:
 -0.219   0.763
0.00335  -0.447
      0       1
      1       0
 -0.145  -0.285
By definition of the kernel, m*ker is zero:
 7.45e-09  1.49e-08
-1.86e-09 -4.05e-08
        0 -2.98e-08

See Also: image()

const 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()

RealScalar maxPivot ( ) const

inline

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

Index 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()

const PermutationPType& permutationP ( ) const

inline

Returns: the permutation matrix P

See Also: permutationQ()

const PermutationQType& permutationQ ( ) const

inline

Returns: the permutation matrix Q

See Also: permutationP()

Index 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&).

References FullPivLU< MatrixType >::threshold().

Referenced by FullPivLU< MatrixType >::dimensionOfKernel(), FullPivLU< MatrixType >::isInjective(), and FullPivLU< MatrixType >::isSurjective().

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 purposes.

FullPivLU& 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)

References FullPivLU< MatrixType >::threshold().

FullPivLU& 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&).

const internal::solve_retval<FullPivLU, Rhs> 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.

This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:

bool a_solution_exists = (A*result).isApprox(b, precision);

This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get inf or nan values.

If there exists more than one solution, this method will arbitrarily choose one. If you need a complete analysis of the space of solutions, take the one solution obtained by this method and add to it elements of the kernel, as determined by kernel().

Example:

Matrix<float,2,3> m = Matrix<float,2,3>::Random();
Matrix2f y = Matrix2f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix<float,3,2> x = m.fullPivLu().solve(y);
if((m*x).isApprox(y))
{
  cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;
}
else
  cout << "The equation mx=y does not have any solution." << endl;

Output:

Here is the matrix m:
  0.68  0.566  0.823
-0.211  0.597 -0.605
Here is the matrix y:
 -0.33 -0.444
 0.536  0.108
Here is a solution x to the equation mx=y:
     0      0
 0.291 -0.216
  -0.6 -0.391

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

RealScalar threshold ( ) const

inline

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

See the documentation of setThreshold(const RealScalar&).

Referenced by FullPivLU< MatrixType >::rank(), and FullPivLU< MatrixType >::setThreshold().

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

Detailed Description

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

Public Member Functions

Constructor & Destructor Documentation

Member Function Documentation

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