510 lines
18 KiB
C++
510 lines
18 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
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// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_PARTIALLU_H
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#define EIGEN_PARTIALLU_H
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namespace Eigen {
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/** \ingroup LU_Module
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*
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* \class PartialPivLU
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*
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* \brief LU decomposition of a matrix with partial pivoting, and related features
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*
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* \param MatrixType the type of the matrix of which we are computing the LU decomposition
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*
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* This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
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* is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
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* is a permutation matrix.
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*
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* Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
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* matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
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* does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
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* matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
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*
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* The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
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* by class FullPivLU.
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*
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* This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
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* such as rank computation. If you need these features, use class FullPivLU.
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*
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* This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
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* in the general case.
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* On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
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*
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* The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
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*
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* \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
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*/
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template<typename _MatrixType> class PartialPivLU
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{
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public:
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typedef _MatrixType MatrixType;
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enum {
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RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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Options = MatrixType::Options,
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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};
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typedef typename MatrixType::Scalar Scalar;
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typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
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typedef typename MatrixType::Index Index;
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typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
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typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
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/**
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* \brief Default Constructor.
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*
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via PartialPivLU::compute(const MatrixType&).
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*/
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PartialPivLU();
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/** \brief Default Constructor with memory preallocation
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*
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* Like the default constructor but with preallocation of the internal data
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* according to the specified problem \a size.
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* \sa PartialPivLU()
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*/
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PartialPivLU(Index size);
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/** Constructor.
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*
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* \param matrix the matrix of which to compute the LU decomposition.
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*
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* \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
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* If you need to deal with non-full rank, use class FullPivLU instead.
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*/
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PartialPivLU(const MatrixType& matrix);
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PartialPivLU& compute(const MatrixType& matrix);
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/** \returns the LU decomposition matrix: the upper-triangular part is U, the
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* unit-lower-triangular part is L (at least for square matrices; in the non-square
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* case, special care is needed, see the documentation of class FullPivLU).
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*
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* \sa matrixL(), matrixU()
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*/
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inline const MatrixType& matrixLU() const
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{
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eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
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return m_lu;
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}
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/** \returns the permutation matrix P.
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*/
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inline const PermutationType& permutationP() const
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{
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eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
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return m_p;
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}
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/** This method returns the solution x to the equation Ax=b, where A is the matrix of which
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* *this is the LU decomposition.
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*
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* \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
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* the only requirement in order for the equation to make sense is that
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* b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
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*
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* \returns the solution.
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*
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* Example: \include PartialPivLU_solve.cpp
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* Output: \verbinclude PartialPivLU_solve.out
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*
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* Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
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* theoretically exists and is unique regardless of b.
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*
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* \sa TriangularView::solve(), inverse(), computeInverse()
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*/
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template<typename Rhs>
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inline const internal::solve_retval<PartialPivLU, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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{
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eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
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return internal::solve_retval<PartialPivLU, Rhs>(*this, b.derived());
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}
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/** \returns the inverse of the matrix of which *this is the LU decomposition.
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*
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* \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
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* invertibility, use class FullPivLU instead.
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*
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* \sa MatrixBase::inverse(), LU::inverse()
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*/
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inline const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse() const
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{
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eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
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return internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType>
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(*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
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}
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/** \returns the determinant of the matrix of which
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* *this is the LU decomposition. It has only linear complexity
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* (that is, O(n) where n is the dimension of the square matrix)
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* as the LU decomposition has already been computed.
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*
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* \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
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* optimized paths.
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*
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* \warning a determinant can be very big or small, so for matrices
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* of large enough dimension, there is a risk of overflow/underflow.
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*
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* \sa MatrixBase::determinant()
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*/
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typename internal::traits<MatrixType>::Scalar determinant() const;
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MatrixType reconstructedMatrix() const;
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inline Index rows() const { return m_lu.rows(); }
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inline Index cols() const { return m_lu.cols(); }
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protected:
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static void check_template_parameters()
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{
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EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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}
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MatrixType m_lu;
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PermutationType m_p;
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TranspositionType m_rowsTranspositions;
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Index m_det_p;
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bool m_isInitialized;
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};
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template<typename MatrixType>
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PartialPivLU<MatrixType>::PartialPivLU()
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: m_lu(),
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m_p(),
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m_rowsTranspositions(),
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m_det_p(0),
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m_isInitialized(false)
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{
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}
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template<typename MatrixType>
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PartialPivLU<MatrixType>::PartialPivLU(Index size)
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: m_lu(size, size),
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m_p(size),
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m_rowsTranspositions(size),
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m_det_p(0),
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m_isInitialized(false)
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{
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}
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template<typename MatrixType>
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PartialPivLU<MatrixType>::PartialPivLU(const MatrixType& matrix)
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: m_lu(matrix.rows(), matrix.rows()),
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m_p(matrix.rows()),
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m_rowsTranspositions(matrix.rows()),
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m_det_p(0),
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m_isInitialized(false)
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{
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compute(matrix);
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}
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namespace internal {
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/** \internal This is the blocked version of fullpivlu_unblocked() */
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template<typename Scalar, int StorageOrder, typename PivIndex>
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struct partial_lu_impl
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{
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// FIXME add a stride to Map, so that the following mapping becomes easier,
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// another option would be to create an expression being able to automatically
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// warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
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// a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
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// and Block.
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typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
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typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
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typedef Block<MatrixType,Dynamic,Dynamic> BlockType;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef typename MatrixType::Index Index;
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/** \internal performs the LU decomposition in-place of the matrix \a lu
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* using an unblocked algorithm.
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*
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* In addition, this function returns the row transpositions in the
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* vector \a row_transpositions which must have a size equal to the number
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* of columns of the matrix \a lu, and an integer \a nb_transpositions
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* which returns the actual number of transpositions.
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*
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* \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
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*/
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static Index unblocked_lu(MatrixType& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
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{
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const Index rows = lu.rows();
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const Index cols = lu.cols();
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const Index size = (std::min)(rows,cols);
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nb_transpositions = 0;
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Index first_zero_pivot = -1;
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for(Index k = 0; k < size; ++k)
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{
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Index rrows = rows-k-1;
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Index rcols = cols-k-1;
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Index row_of_biggest_in_col;
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RealScalar biggest_in_corner
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= lu.col(k).tail(rows-k).cwiseAbs().maxCoeff(&row_of_biggest_in_col);
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row_of_biggest_in_col += k;
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row_transpositions[k] = PivIndex(row_of_biggest_in_col);
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if(biggest_in_corner != RealScalar(0))
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{
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if(k != row_of_biggest_in_col)
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{
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lu.row(k).swap(lu.row(row_of_biggest_in_col));
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++nb_transpositions;
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}
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// FIXME shall we introduce a safe quotient expression in cas 1/lu.coeff(k,k)
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// overflow but not the actual quotient?
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lu.col(k).tail(rrows) /= lu.coeff(k,k);
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}
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else if(first_zero_pivot==-1)
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{
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// the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
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// and continue the factorization such we still have A = PLU
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first_zero_pivot = k;
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}
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if(k<rows-1)
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lu.bottomRightCorner(rrows,rcols).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rcols);
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}
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return first_zero_pivot;
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}
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/** \internal performs the LU decomposition in-place of the matrix represented
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* by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
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* recursive, blocked algorithm.
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*
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* In addition, this function returns the row transpositions in the
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* vector \a row_transpositions which must have a size equal to the number
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* of columns of the matrix \a lu, and an integer \a nb_transpositions
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* which returns the actual number of transpositions.
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*
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* \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
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*
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* \note This very low level interface using pointers, etc. is to:
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* 1 - reduce the number of instanciations to the strict minimum
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* 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
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*/
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static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
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{
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MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
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MatrixType lu(lu1,0,0,rows,cols);
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const Index size = (std::min)(rows,cols);
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// if the matrix is too small, no blocking:
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if(size<=16)
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{
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return unblocked_lu(lu, row_transpositions, nb_transpositions);
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}
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// automatically adjust the number of subdivisions to the size
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// of the matrix so that there is enough sub blocks:
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Index blockSize;
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{
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blockSize = size/8;
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blockSize = (blockSize/16)*16;
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blockSize = (std::min)((std::max)(blockSize,Index(8)), maxBlockSize);
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}
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nb_transpositions = 0;
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Index first_zero_pivot = -1;
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for(Index k = 0; k < size; k+=blockSize)
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{
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Index bs = (std::min)(size-k,blockSize); // actual size of the block
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Index trows = rows - k - bs; // trailing rows
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Index tsize = size - k - bs; // trailing size
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// partition the matrix:
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// A00 | A01 | A02
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// lu = A_0 | A_1 | A_2 = A10 | A11 | A12
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// A20 | A21 | A22
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BlockType A_0(lu,0,0,rows,k);
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BlockType A_2(lu,0,k+bs,rows,tsize);
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BlockType A11(lu,k,k,bs,bs);
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BlockType A12(lu,k,k+bs,bs,tsize);
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BlockType A21(lu,k+bs,k,trows,bs);
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BlockType A22(lu,k+bs,k+bs,trows,tsize);
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PivIndex nb_transpositions_in_panel;
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// recursively call the blocked LU algorithm on [A11^T A21^T]^T
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// with a very small blocking size:
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Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
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row_transpositions+k, nb_transpositions_in_panel, 16);
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if(ret>=0 && first_zero_pivot==-1)
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first_zero_pivot = k+ret;
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nb_transpositions += nb_transpositions_in_panel;
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// update permutations and apply them to A_0
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for(Index i=k; i<k+bs; ++i)
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{
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Index piv = (row_transpositions[i] += k);
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A_0.row(i).swap(A_0.row(piv));
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}
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if(trows)
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{
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// apply permutations to A_2
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for(Index i=k;i<k+bs; ++i)
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A_2.row(i).swap(A_2.row(row_transpositions[i]));
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// A12 = A11^-1 A12
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A11.template triangularView<UnitLower>().solveInPlace(A12);
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A22.noalias() -= A21 * A12;
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}
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}
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return first_zero_pivot;
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}
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};
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/** \internal performs the LU decomposition with partial pivoting in-place.
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*/
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template<typename MatrixType, typename TranspositionType>
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void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::Index& nb_transpositions)
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{
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eigen_assert(lu.cols() == row_transpositions.size());
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eigen_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);
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partial_lu_impl
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<typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor, typename TranspositionType::Index>
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::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
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}
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} // end namespace internal
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template<typename MatrixType>
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PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const MatrixType& matrix)
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{
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check_template_parameters();
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// the row permutation is stored as int indices, so just to be sure:
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eigen_assert(matrix.rows()<NumTraits<int>::highest());
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m_lu = matrix;
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eigen_assert(matrix.rows() == matrix.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
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const Index size = matrix.rows();
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m_rowsTranspositions.resize(size);
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typename TranspositionType::Index nb_transpositions;
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internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
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m_det_p = (nb_transpositions%2) ? -1 : 1;
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m_p = m_rowsTranspositions;
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m_isInitialized = true;
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return *this;
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}
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template<typename MatrixType>
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typename internal::traits<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
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{
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eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
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return Scalar(m_det_p) * m_lu.diagonal().prod();
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}
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/** \returns the matrix represented by the decomposition,
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* i.e., it returns the product: P^{-1} L U.
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* This function is provided for debug purpose. */
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template<typename MatrixType>
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MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
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{
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eigen_assert(m_isInitialized && "LU is not initialized.");
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// LU
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MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
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* m_lu.template triangularView<Upper>();
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// P^{-1}(LU)
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res = m_p.inverse() * res;
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return res;
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}
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/***** Implementation of solve() *****************************************************/
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namespace internal {
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template<typename _MatrixType, typename Rhs>
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struct solve_retval<PartialPivLU<_MatrixType>, Rhs>
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: solve_retval_base<PartialPivLU<_MatrixType>, Rhs>
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{
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EIGEN_MAKE_SOLVE_HELPERS(PartialPivLU<_MatrixType>,Rhs)
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template<typename Dest> void evalTo(Dest& dst) const
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{
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/* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
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* So we proceed as follows:
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* Step 1: compute c = Pb.
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* Step 2: replace c by the solution x to Lx = c.
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* Step 3: replace c by the solution x to Ux = c.
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*/
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eigen_assert(rhs().rows() == dec().matrixLU().rows());
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// Step 1
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dst = dec().permutationP() * rhs();
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// Step 2
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dec().matrixLU().template triangularView<UnitLower>().solveInPlace(dst);
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// Step 3
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dec().matrixLU().template triangularView<Upper>().solveInPlace(dst);
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}
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};
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} // end namespace internal
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/******** MatrixBase methods *******/
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/** \lu_module
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*
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* \return the partial-pivoting LU decomposition of \c *this.
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*
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* \sa class PartialPivLU
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*/
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template<typename Derived>
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inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
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MatrixBase<Derived>::partialPivLu() const
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{
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return PartialPivLU<PlainObject>(eval());
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}
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#if EIGEN2_SUPPORT_STAGE > STAGE20_RESOLVE_API_CONFLICTS
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/** \lu_module
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|
*
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|
* Synonym of partialPivLu().
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|
*
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|
* \return the partial-pivoting LU decomposition of \c *this.
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|
*
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|
* \sa class PartialPivLU
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|
*/
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|
template<typename Derived>
|
|
inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
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|
MatrixBase<Derived>::lu() const
|
|
{
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|
return PartialPivLU<PlainObject>(eval());
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|
}
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#endif
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|
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} // end namespace Eigen
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|
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#endif // EIGEN_PARTIALLU_H
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