 Eigen  3.3.7 Eigen::SparseLU< _MatrixType, _OrderingType > Class Template Reference

## Detailed Description

### 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 arithmetics 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 .

Sparse solver concept
OrderingMethods module Inheritance diagram for Eigen::SparseLU< _MatrixType, _OrderingType >:

## Public Member Functions

Scalar absDeterminant ()

void analyzePattern (const MatrixType &matrix)

const PermutationTypecolsPermutation () 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 PermutationTyperowsPermutation () const

void setPivotThreshold (const RealScalar &thresh)

Scalar signDeterminant ()

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

## ◆ 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 such that 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.
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 succesful, `NumericalIssue` if the LU factorization reports a problem, zero diagonal for instance `InvalidInput` if the input matrix is invalid
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.
absDeterminant(), signDeterminant()

## ◆ matrixL()

template<typename _MatrixType , typename _OrderingType >
 SparseLUMatrixLReturnType 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 > 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 such that 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
absDeterminant(), logAbsDeterminant()

## ◆ solve()

template<typename _MatrixType , typename _OrderingType >
template<typename Rhs >
 const Solve Eigen::SparseLU< _MatrixType, _OrderingType >::solve ( const MatrixBase< Rhs > & B ) const
inline
Returns
the solution X of using the current decomposition of A.
Warning
the destination matrix X in X = this->solve(B) must be colmun-major.