Eigen
3.2.8

LU decomposition of a matrix with complete pivoting, and related features.
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 where L is unitlowertriangular, U is uppertriangular, and P and Q are permutation matrices. This is a rankrevealing 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:
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
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 
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.
FullPivLU  (  const MatrixType &  matrix  ) 
Constructor.
matrix  the matrix of which to compute the LU decomposition. It is required to be nonzero. 
References FullPivLU< MatrixType >::compute().
FullPivLU< MatrixType > & compute  (  const MatrixType &  matrix  ) 
Computes the LU decomposition of the given matrix.
matrix  the matrix of which to compute the LU decomposition. It is required to be nonzero. 
Referenced by FullPivLU< MatrixType >::FullPivLU().
internal::traits< MatrixType >::Scalar determinant  (  )  const 

inline 
References FullPivLU< MatrixType >::rank().

inline 
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. 
Example:
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

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

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References FullPivLU< MatrixType >::isInjective().

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References FullPivLU< MatrixType >::rank().

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Example:
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: 7.45e09 1.49e08 1.86e09 4.05e08 0 2.98e08

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References FullPivLU< MatrixType >::threshold().
Referenced by FullPivLU< MatrixType >::dimensionOfKernel(), FullPivLU< MatrixType >::isInjective(), and FullPivLU< MatrixType >::isSurjective().
MatrixType reconstructedMatrix  (  )  const 

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.
threshold  The new value to use as the threshold. 
A pivot will be considered nonzero if its absolute value is strictly greater than where maxpivot is the biggest pivot.
If you want to come back to the default behavior, call setThreshold(Default_t)
References FullPivLU< MatrixType >::threshold().

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.
See the documentation of setThreshold(const RealScalar&).

inline 
b  the righthandside 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. 
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:
This method avoids dividing by zero, so that the nonexistence 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:
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

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().