PartialPivLU< _MatrixType > Class Template Reference
[LU module]

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

List of all members.

Public Member Functions

internal::traits< MatrixType >
::Scalar 
determinant () const
const internal::solve_retval
< PartialPivLU, typename
MatrixType::IdentityReturnType > 
inverse () const
const MatrixType & matrixLU () const
 PartialPivLU (const MatrixType &matrix)
 PartialPivLU (Index size)
 Default Constructor with memory preallocation.
 PartialPivLU ()
 Default Constructor.
const PermutationTypepermutationP () const
MatrixType reconstructedMatrix () const
template<typename Rhs >
const internal::solve_retval
< PartialPivLU, Rhs > 
solve (const MatrixBase< Rhs > &b) const

Detailed Description

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

LU decomposition of a matrix with partial 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 a square invertible matrix, with partial pivoting: the matrix A is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P is a permutation matrix.

Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.

The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided by class FullPivLU.

This is not a rank-revealing LU decomposition. Many features are intentionally absent from this class, such as rank computation. If you need these features, use class FullPivLU.

This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses in the general case. On the other hand, it is not suitable to determine whether a given matrix is invertible.

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

See also:
MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU

Constructor & Destructor Documentation

PartialPivLU (  )  [inline]

Default Constructor.

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

PartialPivLU ( Index  size  )  [inline]

Default Constructor with memory preallocation.

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

See also:
PartialPivLU()
PartialPivLU ( const MatrixType &  matrix  )  [inline]

Constructor.

Parameters:
matrix the matrix of which to compute the LU decomposition.
Warning:
The matrix should have full rank (e.g. if it's square, it should be invertible). If you need to deal with non-full rank, use class FullPivLU instead.

Member Function Documentation

internal::traits< MatrixType >::Scalar determinant (  )  const [inline]
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:
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()
const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse (  )  const [inline]
Returns:
the inverse of the matrix of which *this is the LU decomposition.
Warning:
The matrix being decomposed here is assumed to be invertible. If you need to check for invertibility, use class FullPivLU instead.
See also:
MatrixBase::inverse(), LU::inverse()
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()
const PermutationType& permutationP (  )  const [inline]
Returns:
the permutation matrix P.
MatrixType reconstructedMatrix (  )  const [inline]
Returns:
the matrix represented by the decomposition, i.e., it returns the product: P^{-1} L U. This function is provided for debug purpose.
const internal::solve_retval<PartialPivLU, Rhs> solve ( const MatrixBase< Rhs > &  b  )  const [inline]

This method returns the 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:
the solution.

Example:

MatrixXd A = MatrixXd::Random(3,3);
MatrixXd B = MatrixXd::Random(3,2);
cout << "Here is the invertible matrix A:" << endl << A << endl;
cout << "Here is the matrix B:" << endl << B << endl;
MatrixXd X = A.lu().solve(B);
cout << "Here is the (unique) solution X to the equation AX=B:" << endl << X << endl;
cout << "Relative error: " << (A*X-B).norm() / B.norm() << endl;

Output:

Here is the invertible matrix A:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Here is the matrix B:
 0.108  -0.27
-0.0452 0.0268
 0.258  0.904
Here is the (unique) solution X to the equation AX=B:
0.609  2.68
-0.231 -1.57
 0.51  3.51
Relative error: 2.56e-16

Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.

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

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

Generated on 15 Aug 2012 for Eigen by  doxygen 1.6.1