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Diffstat (limited to 'eigen/Eigen/src/LU/PartialPivLU.h')
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diff --git a/eigen/Eigen/src/LU/PartialPivLU.h b/eigen/Eigen/src/LU/PartialPivLU.h new file mode 100644 index 0000000..7d1db94 --- /dev/null +++ b/eigen/Eigen/src/LU/PartialPivLU.h @@ -0,0 +1,509 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com> +// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr> +// +// This Source Code Form is subject to the terms of the Mozilla +// Public License v. 2.0. If a copy of the MPL was not distributed +// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. + +#ifndef EIGEN_PARTIALLU_H +#define EIGEN_PARTIALLU_H + +namespace Eigen { + +/** \ingroup LU_Module + * + * \class PartialPivLU + * + * \brief LU decomposition of a matrix with partial pivoting, and related features + * + * \param MatrixType the type of the matrix of which we are computing the LU decomposition + * + * This class represents a LU decomposition of a \b square \b 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 \b 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 \b 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(). + * + * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU + */ +template<typename _MatrixType> class PartialPivLU +{ + public: + + typedef _MatrixType MatrixType; + enum { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + Options = MatrixType::Options, + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime + }; + typedef typename MatrixType::Scalar Scalar; + typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; + typedef typename internal::traits<MatrixType>::StorageKind StorageKind; + typedef typename MatrixType::Index Index; + typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType; + typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType; + + + /** + * \brief Default Constructor. + * + * The default constructor is useful in cases in which the user intends to + * perform decompositions via PartialPivLU::compute(const MatrixType&). + */ + PartialPivLU(); + + /** \brief Default Constructor with memory preallocation + * + * Like the default constructor but with preallocation of the internal data + * according to the specified problem \a size. + * \sa PartialPivLU() + */ + PartialPivLU(Index size); + + /** Constructor. + * + * \param 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. + */ + PartialPivLU(const MatrixType& matrix); + + PartialPivLU& compute(const MatrixType& matrix); + + /** \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). + * + * \sa matrixL(), matrixU() + */ + inline const MatrixType& matrixLU() const + { + eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); + return m_lu; + } + + /** \returns the permutation matrix P. + */ + inline const PermutationType& permutationP() const + { + eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); + return m_p; + } + + /** This method returns the solution x to the equation Ax=b, where A is the matrix of which + * *this is the LU decomposition. + * + * \param 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: \include PartialPivLU_solve.cpp + * Output: \verbinclude PartialPivLU_solve.out + * + * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution + * theoretically exists and is unique regardless of b. + * + * \sa TriangularView::solve(), inverse(), computeInverse() + */ + template<typename Rhs> + inline const internal::solve_retval<PartialPivLU, Rhs> + solve(const MatrixBase<Rhs>& b) const + { + eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); + return internal::solve_retval<PartialPivLU, Rhs>(*this, b.derived()); + } + + /** \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. + * + * \sa MatrixBase::inverse(), LU::inverse() + */ + inline const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse() const + { + eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); + return internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> + (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols())); + } + + /** \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. + * + * \sa MatrixBase::determinant() + */ + typename internal::traits<MatrixType>::Scalar determinant() const; + + MatrixType reconstructedMatrix() const; + + inline Index rows() const { return m_lu.rows(); } + inline Index cols() const { return m_lu.cols(); } + + protected: + + static void check_template_parameters() + { + EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); + } + + MatrixType m_lu; + PermutationType m_p; + TranspositionType m_rowsTranspositions; + Index m_det_p; + bool m_isInitialized; +}; + +template<typename MatrixType> +PartialPivLU<MatrixType>::PartialPivLU() + : m_lu(), + m_p(), + m_rowsTranspositions(), + m_det_p(0), + m_isInitialized(false) +{ +} + +template<typename MatrixType> +PartialPivLU<MatrixType>::PartialPivLU(Index size) + : m_lu(size, size), + m_p(size), + m_rowsTranspositions(size), + m_det_p(0), + m_isInitialized(false) +{ +} + +template<typename MatrixType> +PartialPivLU<MatrixType>::PartialPivLU(const MatrixType& matrix) + : m_lu(matrix.rows(), matrix.rows()), + m_p(matrix.rows()), + m_rowsTranspositions(matrix.rows()), + m_det_p(0), + m_isInitialized(false) +{ + compute(matrix); +} + +namespace internal { + +/** \internal This is the blocked version of fullpivlu_unblocked() */ +template<typename Scalar, int StorageOrder, typename PivIndex> +struct partial_lu_impl +{ + // FIXME add a stride to Map, so that the following mapping becomes easier, + // another option would be to create an expression being able to automatically + // warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly + // a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix, + // and Block. + typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU; + typedef Block<MapLU, Dynamic, Dynamic> MatrixType; + typedef Block<MatrixType,Dynamic,Dynamic> BlockType; + typedef typename MatrixType::RealScalar RealScalar; + typedef typename MatrixType::Index Index; + + /** \internal performs the LU decomposition in-place of the matrix \a lu + * using an unblocked algorithm. + * + * In addition, this function returns the row transpositions in the + * vector \a row_transpositions which must have a size equal to the number + * of columns of the matrix \a lu, and an integer \a nb_transpositions + * which returns the actual number of transpositions. + * + * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise. + */ + static Index unblocked_lu(MatrixType& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions) + { + const Index rows = lu.rows(); + const Index cols = lu.cols(); + const Index size = (std::min)(rows,cols); + nb_transpositions = 0; + Index first_zero_pivot = -1; + for(Index k = 0; k < size; ++k) + { + Index rrows = rows-k-1; + Index rcols = cols-k-1; + + Index row_of_biggest_in_col; + RealScalar biggest_in_corner + = lu.col(k).tail(rows-k).cwiseAbs().maxCoeff(&row_of_biggest_in_col); + row_of_biggest_in_col += k; + + row_transpositions[k] = PivIndex(row_of_biggest_in_col); + + if(biggest_in_corner != RealScalar(0)) + { + if(k != row_of_biggest_in_col) + { + lu.row(k).swap(lu.row(row_of_biggest_in_col)); + ++nb_transpositions; + } + + // FIXME shall we introduce a safe quotient expression in cas 1/lu.coeff(k,k) + // overflow but not the actual quotient? + lu.col(k).tail(rrows) /= lu.coeff(k,k); + } + else if(first_zero_pivot==-1) + { + // the pivot is exactly zero, we record the index of the first pivot which is exactly 0, + // and continue the factorization such we still have A = PLU + first_zero_pivot = k; + } + + if(k<rows-1) + lu.bottomRightCorner(rrows,rcols).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rcols); + } + return first_zero_pivot; + } + + /** \internal performs the LU decomposition in-place of the matrix represented + * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a + * recursive, blocked algorithm. + * + * In addition, this function returns the row transpositions in the + * vector \a row_transpositions which must have a size equal to the number + * of columns of the matrix \a lu, and an integer \a nb_transpositions + * which returns the actual number of transpositions. + * + * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise. + * + * \note This very low level interface using pointers, etc. is to: + * 1 - reduce the number of instanciations to the strict minimum + * 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > > + */ + static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256) + { + MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols); + MatrixType lu(lu1,0,0,rows,cols); + + const Index size = (std::min)(rows,cols); + + // if the matrix is too small, no blocking: + if(size<=16) + { + return unblocked_lu(lu, row_transpositions, nb_transpositions); + } + + // automatically adjust the number of subdivisions to the size + // of the matrix so that there is enough sub blocks: + Index blockSize; + { + blockSize = size/8; + blockSize = (blockSize/16)*16; + blockSize = (std::min)((std::max)(blockSize,Index(8)), maxBlockSize); + } + + nb_transpositions = 0; + Index first_zero_pivot = -1; + for(Index k = 0; k < size; k+=blockSize) + { + Index bs = (std::min)(size-k,blockSize); // actual size of the block + Index trows = rows - k - bs; // trailing rows + Index tsize = size - k - bs; // trailing size + + // partition the matrix: + // A00 | A01 | A02 + // lu = A_0 | A_1 | A_2 = A10 | A11 | A12 + // A20 | A21 | A22 + BlockType A_0(lu,0,0,rows,k); + BlockType A_2(lu,0,k+bs,rows,tsize); + BlockType A11(lu,k,k,bs,bs); + BlockType A12(lu,k,k+bs,bs,tsize); + BlockType A21(lu,k+bs,k,trows,bs); + BlockType A22(lu,k+bs,k+bs,trows,tsize); + + PivIndex nb_transpositions_in_panel; + // recursively call the blocked LU algorithm on [A11^T A21^T]^T + // with a very small blocking size: + Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride, + row_transpositions+k, nb_transpositions_in_panel, 16); + if(ret>=0 && first_zero_pivot==-1) + first_zero_pivot = k+ret; + + nb_transpositions += nb_transpositions_in_panel; + // update permutations and apply them to A_0 + for(Index i=k; i<k+bs; ++i) + { + Index piv = (row_transpositions[i] += k); + A_0.row(i).swap(A_0.row(piv)); + } + + if(trows) + { + // apply permutations to A_2 + for(Index i=k;i<k+bs; ++i) + A_2.row(i).swap(A_2.row(row_transpositions[i])); + + // A12 = A11^-1 A12 + A11.template triangularView<UnitLower>().solveInPlace(A12); + + A22.noalias() -= A21 * A12; + } + } + return first_zero_pivot; + } +}; + +/** \internal performs the LU decomposition with partial pivoting in-place. + */ +template<typename MatrixType, typename TranspositionType> +void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::Index& nb_transpositions) +{ + eigen_assert(lu.cols() == row_transpositions.size()); + eigen_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1); + + partial_lu_impl + <typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor, typename TranspositionType::Index> + ::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions); +} + +} // end namespace internal + +template<typename MatrixType> +PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const MatrixType& matrix) +{ + check_template_parameters(); + + // the row permutation is stored as int indices, so just to be sure: + eigen_assert(matrix.rows()<NumTraits<int>::highest()); + + m_lu = matrix; + + eigen_assert(matrix.rows() == matrix.cols() && "PartialPivLU is only for square (and moreover invertible) matrices"); + const Index size = matrix.rows(); + + m_rowsTranspositions.resize(size); + + typename TranspositionType::Index nb_transpositions; + internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions); + m_det_p = (nb_transpositions%2) ? -1 : 1; + + m_p = m_rowsTranspositions; + + m_isInitialized = true; + return *this; +} + +template<typename MatrixType> +typename internal::traits<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const +{ + eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); + return Scalar(m_det_p) * m_lu.diagonal().prod(); +} + +/** \returns the matrix represented by the decomposition, + * i.e., it returns the product: P^{-1} L U. + * This function is provided for debug purpose. */ +template<typename MatrixType> +MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const +{ + eigen_assert(m_isInitialized && "LU is not initialized."); + // LU + MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix() + * m_lu.template triangularView<Upper>(); + + // P^{-1}(LU) + res = m_p.inverse() * res; + + return res; +} + +/***** Implementation of solve() *****************************************************/ + +namespace internal { + +template<typename _MatrixType, typename Rhs> +struct solve_retval<PartialPivLU<_MatrixType>, Rhs> + : solve_retval_base<PartialPivLU<_MatrixType>, Rhs> +{ + EIGEN_MAKE_SOLVE_HELPERS(PartialPivLU<_MatrixType>,Rhs) + + template<typename Dest> void evalTo(Dest& dst) const + { + /* The decomposition PA = LU can be rewritten as A = P^{-1} L U. + * So we proceed as follows: + * Step 1: compute c = Pb. + * Step 2: replace c by the solution x to Lx = c. + * Step 3: replace c by the solution x to Ux = c. + */ + + eigen_assert(rhs().rows() == dec().matrixLU().rows()); + + // Step 1 + dst = dec().permutationP() * rhs(); + + // Step 2 + dec().matrixLU().template triangularView<UnitLower>().solveInPlace(dst); + + // Step 3 + dec().matrixLU().template triangularView<Upper>().solveInPlace(dst); + } +}; + +} // end namespace internal + +/******** MatrixBase methods *******/ + +/** \lu_module + * + * \return the partial-pivoting LU decomposition of \c *this. + * + * \sa class PartialPivLU + */ +template<typename Derived> +inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject> +MatrixBase<Derived>::partialPivLu() const +{ + return PartialPivLU<PlainObject>(eval()); +} + +#if EIGEN2_SUPPORT_STAGE > STAGE20_RESOLVE_API_CONFLICTS +/** \lu_module + * + * Synonym of partialPivLu(). + * + * \return the partial-pivoting LU decomposition of \c *this. + * + * \sa class PartialPivLU + */ +template<typename Derived> +inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject> +MatrixBase<Derived>::lu() const +{ + return PartialPivLU<PlainObject>(eval()); +} +#endif + +} // end namespace Eigen + +#endif // EIGEN_PARTIALLU_H |