LU decomposition of a matrix with partial pivoting, and related features. More...
#include <PartialPivLU.h>
Public Types | |
enum | { RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime } |
typedef MatrixType::Index | Index |
typedef _MatrixType | MatrixType |
typedef PermutationMatrix < RowsAtCompileTime, MaxRowsAtCompileTime > | PermutationType |
typedef NumTraits< typename MatrixType::Scalar >::Real | RealScalar |
typedef MatrixType::Scalar | Scalar |
typedef internal::traits < MatrixType >::StorageKind | StorageKind |
typedef Transpositions < RowsAtCompileTime, MaxRowsAtCompileTime > | TranspositionType |
Public Member Functions | |
Index | cols () const |
PartialPivLU & | compute (const MatrixType &matrix) |
internal::traits< MatrixType > ::Scalar | determinant () const |
const internal::solve_retval < PartialPivLU, typename MatrixType::IdentityReturnType > | inverse () const |
const MatrixType & | matrixLU () const |
PartialPivLU () | |
Default Constructor. | |
PartialPivLU (Index size) | |
Default Constructor with memory preallocation. | |
PartialPivLU (const MatrixType &matrix) | |
const PermutationType & | permutationP () const |
MatrixType | reconstructedMatrix () const |
Index | rows () const |
template<typename Rhs > | |
const internal::solve_retval < PartialPivLU, Rhs > | solve (const MatrixBase< Rhs > &b) const |
Protected Attributes | |
Index | m_det_p |
bool | m_isInitialized |
MatrixType | m_lu |
PermutationType | m_p |
TranspositionType | m_rowsTranspositions |
LU decomposition of a matrix with partial 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 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().
typedef MatrixType::Index Index |
typedef _MatrixType MatrixType |
typedef NumTraits<typename MatrixType::Scalar>::Real RealScalar |
typedef MatrixType::Scalar Scalar |
typedef internal::traits<MatrixType>::StorageKind StorageKind |
anonymous enum |
PartialPivLU | ( | ) |
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 | ) |
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
PartialPivLU | ( | const MatrixType & | matrix | ) |
Constructor.
matrix | the matrix of which to compute the LU decomposition. |
References PartialPivLU< _MatrixType >::compute().
References PartialPivLU< _MatrixType >::m_lu.
PartialPivLU< MatrixType > & compute | ( | const MatrixType & | matrix | ) |
References Eigen::internal::partial_lu_inplace(), and PartialPivLU< _MatrixType >::rows().
Referenced by PartialPivLU< _MatrixType >::PartialPivLU().
internal::traits< MatrixType >::Scalar determinant | ( | ) | const |
const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse | ( | ) | const [inline] |
References PartialPivLU< _MatrixType >::m_isInitialized, and PartialPivLU< _MatrixType >::m_lu.
const MatrixType& matrixLU | ( | ) | const [inline] |
References PartialPivLU< _MatrixType >::m_isInitialized, and PartialPivLU< _MatrixType >::m_lu.
const PermutationType& permutationP | ( | ) | const [inline] |
References PartialPivLU< _MatrixType >::m_isInitialized, and PartialPivLU< _MatrixType >::m_p.
MatrixType reconstructedMatrix | ( | ) | const |
References PartialPivLU< _MatrixType >::m_lu.
Referenced by PartialPivLU< _MatrixType >::compute().
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.
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. |
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.
References PartialPivLU< _MatrixType >::m_isInitialized.
bool m_isInitialized [protected] |
MatrixType m_lu [protected] |
PermutationType m_p [protected] |
Referenced by PartialPivLU< _MatrixType >::permutationP().
TranspositionType m_rowsTranspositions [protected] |