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Eigen::SparseLU

template<typename _MatrixType, typename _OrderingType>
class Eigen::SparseLU< _MatrixType, _OrderingType >

Sparse supernodal LU factorization for general matrices.

This class implements the supernodal LU factorization for general matrices. It uses the main techniques from the sequential SuperLU package (http://crd-legacy.lbl.gov/~xiaoye/SuperLU/). It handles transparently real and complex arithmetic with single and double precision, depending on the scalar type of your input matrix. The code has been optimized to provide BLAS-3 operations during supernode-panel updates. It benefits directly from the built-in high-performant Eigen BLAS routines. Moreover, when the size of a supernode is very small, the BLAS calls are avoided to enable a better optimization from the compiler. For best performance, you should compile it with NDEBUG flag to avoid the numerous bounds checking on vectors.

An important parameter of this class is the ordering method. It is used to reorder the columns (and eventually the rows) of the matrix to reduce the number of new elements that are created during numerical factorization. The cheapest method available is COLAMD. See the OrderingMethods module for the list of built-in and external ordering methods.

Simple example with key steps

VectorXd x(n), b(n);
SparseMatrix<double> A;
SparseLU<SparseMatrix<double>, COLAMDOrdering<int> >   solver;
// fill A and b;
// Compute the ordering permutation vector from the structural pattern of A
solver.analyzePattern(A); 
// Compute the numerical factorization 
solver.factorize(A); 
//Use the factors to solve the linear system 
x = solver.solve(b);

Warning: The input matrix A should be in a compressed and column-major form. Otherwise an expensive copy will be made. You can call the inexpensive makeCompressed() to get a compressed matrix.

Note: Unlike the initial SuperLU implementation, there is no step to equilibrate the matrix. For badly scaled matrices, this step can be useful to reduce the pivoting during factorization. If this is the case for your matrices, you can try the basic scaling method at "unsupported/Eigen/src/IterativeSolvers/Scaling.h"

Template Parameters

_MatrixType	The type of the sparse matrix. It must be a column-major SparseMatrix<>
_OrderingType	The ordering method to use, either AMD, COLAMD or METIS. Default is COLMAD

This class follows the sparse solver concept .

See also: Sparse solver concept; OrderingMethods module


Scalar	absDeterminant ()

const SparseLUTransposeView< true, SparseLU< _MatrixType, _OrderingType > >	adjoint ()

void	analyzePattern (const MatrixType &matrix)

const PermutationType &	colsPermutation () const

void	compute (const MatrixType &matrix)

Scalar	determinant ()

void	factorize (const MatrixType &matrix)

ComputationInfo	info () const
	Reports whether previous computation was successful. More...

void	isSymmetric (bool sym)

std::string	lastErrorMessage () const

Scalar	logAbsDeterminant () const

SparseLUMatrixLReturnType< SCMatrix >	matrixL () const

SparseLUMatrixUReturnType< SCMatrix, MappedSparseMatrix< Scalar, ColMajor, StorageIndex > >	matrixU () const

const PermutationType &	rowsPermutation () const

void	setPivotThreshold (const RealScalar &thresh)

Scalar	signDeterminant ()

template<typename Rhs >
const Solve< SparseLU, Rhs >	solve (const MatrixBase< Rhs > &B) const

const SparseLUTransposeView< false, SparseLU< _MatrixType, _OrderingType > >	transpose ()

Public Member Functions inherited from Eigen::SparseSolverBase< SparseLU< _MatrixType, _OrderingType > >
const Solve< SparseLU< _MatrixType, _OrderingType >, Rhs >	solve (const MatrixBase< Rhs > &b) const

const Solve< SparseLU< _MatrixType, _OrderingType >, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const

	SparseSolverBase ()

absDeterminant()

template<typename _MatrixType , typename _OrderingType >

Scalar Eigen::SparseLU< _MatrixType, _OrderingType >::absDeterminant

(

)

inline

Returns: the absolute value of the determinant of the matrix of which *this is the QR decomposition.

Warning: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.

See also: logAbsDeterminant(), signDeterminant()

adjoint()

template<typename _MatrixType , typename _OrderingType >

const SparseLUTransposeView<true, SparseLU<_MatrixType,_OrderingType> > Eigen::SparseLU< _MatrixType, _OrderingType >::adjoint

(

)

inline

Returns: an expression of the adjoint of the factored matrix

A typical usage is to solve for the adjoint problem A' x = b:

solver.compute(A);
x = solver.adjoint().solve(b);

For real scalar types, this function is equivalent to transpose().

See also: transpose(), solve()

analyzePattern()

template<typename MatrixType , typename OrderingType >

void Eigen::SparseLU< MatrixType, OrderingType >::analyzePattern

(

const MatrixType &

mat

)

Compute the column permutation to minimize the fill-in

Apply this permutation to the input matrix -
Compute the column elimination tree on the permuted matrix
Postorder the elimination tree and the column permutation

colsPermutation()

template<typename _MatrixType , typename _OrderingType >

const PermutationType& Eigen::SparseLU< _MatrixType, _OrderingType >::colsPermutation

(

)

const

inline

Returns: a reference to the column matrix permutation \( P_c^T \) such that \(P_r A P_c^T = L U\)

See also: rowsPermutation()

compute()

template<typename _MatrixType , typename _OrderingType >

void Eigen::SparseLU< _MatrixType, _OrderingType >::compute

(

const MatrixType &

matrix

)

inline

Compute the symbolic and numeric factorization of the input sparse matrix. The input matrix should be in column-major storage.

determinant()

template<typename _MatrixType , typename _OrderingType >

Scalar Eigen::SparseLU< _MatrixType, _OrderingType >::determinant

(

)

inline

Returns: The determinant of the matrix.

See also: absDeterminant(), logAbsDeterminant()

factorize()

template<typename MatrixType , typename OrderingType >

void Eigen::SparseLU< MatrixType, OrderingType >::factorize

(

const MatrixType &

matrix

)

Numerical factorization
Interleaved with the symbolic factorization On exit, info is

= 0: successful factorization

0: if info = i, and i is

<= A->ncol: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

> A->ncol: number of bytes allocated when memory allocation failure occurred, plus A->ncol. If lwork = -1, it is the estimated amount of space needed, plus A->ncol.

info()

template<typename _MatrixType , typename _OrderingType >

ComputationInfo Eigen::SparseLU< _MatrixType, _OrderingType >::info

(

)

const

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NumericalIssue if the LU factorization reports a problem, zero diagonal for instance InvalidInput if the input matrix is invalid

See also: iparm()

isSymmetric()

template<typename _MatrixType , typename _OrderingType >

void Eigen::SparseLU< _MatrixType, _OrderingType >::isSymmetric

(

bool

sym

)

inline

Indicate that the pattern of the input matrix is symmetric

lastErrorMessage()

template<typename _MatrixType , typename _OrderingType >

std::string Eigen::SparseLU< _MatrixType, _OrderingType >::lastErrorMessage

(

)

const

inline

Returns: A string describing the type of error

logAbsDeterminant()

template<typename _MatrixType , typename _OrderingType >

Scalar Eigen::SparseLU< _MatrixType, _OrderingType >::logAbsDeterminant

(

)

const

inline

Returns: the natural log of the absolute value of the determinant of the matrix of which **this is the QR decomposition

Note: This method is useful to work around the risk of overflow/underflow that's inherent to the determinant computation.

See also: absDeterminant(), signDeterminant()

matrixL()

template<typename _MatrixType , typename _OrderingType >

SparseLUMatrixLReturnType<SCMatrix> Eigen::SparseLU< _MatrixType, _OrderingType >::matrixL

(

)

const

inline

Returns

an expression of the matrix L, internally stored as supernodes The only operation available with this expression is the triangular solve

y = b; matrixL().solveInPlace(y);

matrixU()

template<typename _MatrixType , typename _OrderingType >

SparseLUMatrixUReturnType<SCMatrix,MappedSparseMatrix<Scalar,ColMajor,StorageIndex> > Eigen::SparseLU< _MatrixType, _OrderingType >::matrixU

(

)

const

inline

Returns

an expression of the matrix U, The only operation available with this expression is the triangular solve

y = b; matrixU().solveInPlace(y);

rowsPermutation()

template<typename _MatrixType , typename _OrderingType >

const PermutationType& Eigen::SparseLU< _MatrixType, _OrderingType >::rowsPermutation

(

)

const

inline

Returns: a reference to the row matrix permutation \( P_r \) such that \(P_r A P_c^T = L U\)

See also: colsPermutation()

setPivotThreshold()

template<typename _MatrixType , typename _OrderingType >

void Eigen::SparseLU< _MatrixType, _OrderingType >::setPivotThreshold

(

const RealScalar &

thresh

)

inline

Set the threshold used for a diagonal entry to be an acceptable pivot.

signDeterminant()

template<typename _MatrixType , typename _OrderingType >

Scalar Eigen::SparseLU< _MatrixType, _OrderingType >::signDeterminant

(

)

inline

Returns: A number representing the sign of the determinant

See also: absDeterminant(), logAbsDeterminant()

solve()

template<typename _MatrixType , typename _OrderingType >

template<typename Rhs >

const Solve<SparseLU, Rhs> Eigen::SparseLU< _MatrixType, _OrderingType >::solve

(

const MatrixBase< Rhs > &

)

const

inline

Returns: the solution X of \( A X = B \) using the current decomposition of A.

Warning: the destination matrix X in X = this->solve(B) must be colmun-major.

See also: compute()

transpose()

template<typename _MatrixType , typename _OrderingType >

const SparseLUTransposeView<false,SparseLU<_MatrixType,_OrderingType> > Eigen::SparseLU< _MatrixType, _OrderingType >::transpose

(

)

inline

Returns: an expression of the transposed of the factored matrix.

A typical usage is to solve for the transposed problem A^T x = b:

solver.compute(A);
x = solver.transpose().solve(b);

See also: adjoint(), solve()

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

SparseLU.h

Eigen::SparseLU

template<typename _MatrixType, typename _OrderingType> class Eigen::SparseLU< _MatrixType, _OrderingType >

absDeterminant()

adjoint()

analyzePattern()

colsPermutation()

compute()

determinant()

factorize()

info()

isSymmetric()

lastErrorMessage()

logAbsDeterminant()

matrixL()

matrixU()

rowsPermutation()

setPivotThreshold()

signDeterminant()

solve()

transpose()

template<typename _MatrixType, typename _OrderingType>
class Eigen::SparseLU< _MatrixType, _OrderingType >