Yoji KURODA
Eigne Matrix Class Library
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PartialPivLU< _MatrixType > Class Template Reference
[LU module]
LU decomposition of a matrix with partial pivoting, and related features. More...
#include <PartialPivLU.h>
Public Member Functions | |
| PartialPivLU () | |
| Default Constructor. | |
| PartialPivLU (Index size) | |
| Default Constructor with memory preallocation. | |
| PartialPivLU (const MatrixType &matrix) | |
| Constructor. | |
| const MatrixType & | matrixLU () const |
| const PermutationType & | permutationP () const |
| template<typename Rhs > | |
| const internal::solve_retval < PartialPivLU, Rhs > | solve (const MatrixBase< Rhs > &b) const |
| This method returns the solution x to the equation Ax=b, where A is the matrix of which *this is the LU decomposition. | |
| const internal::solve_retval < PartialPivLU, typename MatrixType::IdentityReturnType > | inverse () const |
| internal::traits< MatrixType > ::Scalar | determinant () const |
| MatrixType | reconstructedMatrix () 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
Definition at line 47 of file PartialPivLU.h.
Constructor & Destructor Documentation
| PartialPivLU | ( | ) |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via PartialPivLU::compute(const MatrixType&).
Definition at line 188 of file PartialPivLU.h.
| 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.
- See also:
- PartialPivLU()
Definition at line 198 of file PartialPivLU.h.
| PartialPivLU | ( | const MatrixType & | matrix ) |
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.
Definition at line 208 of file PartialPivLU.h.
Member Function Documentation
| internal::traits< MatrixType >::Scalar determinant | ( | ) | const |
- 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()
Definition at line 418 of file PartialPivLU.h.
| const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse | ( | ) | const |
- 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()
Definition at line 146 of file PartialPivLU.h.
| const MatrixType& matrixLU | ( | ) | const |
- 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()
Definition at line 100 of file PartialPivLU.h.
| const PermutationType& permutationP | ( | ) | const |
- Returns:
- the permutation matrix P.
Definition at line 108 of file PartialPivLU.h.
| MatrixType reconstructedMatrix | ( | ) | const |
- Returns:
- the matrix represented by the decomposition, i.e., it returns the product: P^{-1} L U. This function is provided for debug purpose.
Definition at line 428 of file PartialPivLU.h.
| const internal::solve_retval<PartialPivLU, Rhs> solve | ( | const MatrixBase< Rhs > & | b ) | const |
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:
Output:
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()
Definition at line 133 of file PartialPivLU.h.
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