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diff --git a/src/3rdparty/eigen/Eigen/src/LU/FullPivLU.h b/src/3rdparty/eigen/Eigen/src/LU/FullPivLU.h new file mode 100644 index 000000000..ba1749fa6 --- /dev/null +++ b/src/3rdparty/eigen/Eigen/src/LU/FullPivLU.h @@ -0,0 +1,877 @@ +// 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> +// +// 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_LU_H +#define EIGEN_LU_H + +namespace Eigen { + +namespace internal { +template<typename _MatrixType> struct traits<FullPivLU<_MatrixType> > + : traits<_MatrixType> +{ + typedef MatrixXpr XprKind; + typedef SolverStorage StorageKind; + typedef int StorageIndex; + enum { Flags = 0 }; +}; + +} // end namespace internal + +/** \ingroup LU_Module + * + * \class FullPivLU + * + * \brief LU decomposition of a matrix with complete pivoting, and related features + * + * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition + * + * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is + * decomposed as \f$ A = P^{-1} L U Q^{-1} \f$ where L is unit-lower-triangular, U is + * upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU + * decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any + * zeros are at the end. + * + * This decomposition provides the generic approach to solving systems of linear equations, computing + * the rank, invertibility, inverse, kernel, and determinant. + * + * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD + * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, + * working with the SVD allows to select the smallest singular values of the matrix, something that + * the LU decomposition doesn't see. + * + * The data of the LU decomposition can be directly accessed through the methods matrixLU(), + * permutationP(), permutationQ(). + * + * As an example, here is how the original matrix can be retrieved: + * \include class_FullPivLU.cpp + * Output: \verbinclude class_FullPivLU.out + * + * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism. + * + * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse() + */ +template<typename _MatrixType> class FullPivLU + : public SolverBase<FullPivLU<_MatrixType> > +{ + public: + typedef _MatrixType MatrixType; + typedef SolverBase<FullPivLU> Base; + friend class SolverBase<FullPivLU>; + + EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU) + enum { + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime + }; + typedef typename internal::plain_row_type<MatrixType, StorageIndex>::type IntRowVectorType; + typedef typename internal::plain_col_type<MatrixType, StorageIndex>::type IntColVectorType; + typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType; + typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType; + typedef typename MatrixType::PlainObject PlainObject; + + /** + * \brief Default Constructor. + * + * The default constructor is useful in cases in which the user intends to + * perform decompositions via LU::compute(const MatrixType&). + */ + FullPivLU(); + + /** \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 FullPivLU() + */ + FullPivLU(Index rows, Index cols); + + /** Constructor. + * + * \param matrix the matrix of which to compute the LU decomposition. + * It is required to be nonzero. + */ + template<typename InputType> + explicit FullPivLU(const EigenBase<InputType>& matrix); + + /** \brief Constructs a LU factorization from a given matrix + * + * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref. + * + * \sa FullPivLU(const EigenBase&) + */ + template<typename InputType> + explicit FullPivLU(EigenBase<InputType>& matrix); + + /** Computes the LU decomposition of the given matrix. + * + * \param matrix the matrix of which to compute the LU decomposition. + * It is required to be nonzero. + * + * \returns a reference to *this + */ + template<typename InputType> + FullPivLU& compute(const EigenBase<InputType>& matrix) { + m_lu = matrix.derived(); + computeInPlace(); + return *this; + } + + /** \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 && "LU is not initialized."); + return m_lu; + } + + /** \returns the number of nonzero pivots in the LU decomposition. + * Here nonzero is meant in the exact sense, not in a fuzzy sense. + * So that notion isn't really intrinsically interesting, but it is + * still useful when implementing algorithms. + * + * \sa rank() + */ + inline Index nonzeroPivots() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return m_nonzero_pivots; + } + + /** \returns the absolute value of the biggest pivot, i.e. the biggest + * diagonal coefficient of U. + */ + RealScalar maxPivot() const { return m_maxpivot; } + + /** \returns the permutation matrix P + * + * \sa permutationQ() + */ + EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return m_p; + } + + /** \returns the permutation matrix Q + * + * \sa permutationP() + */ + inline const PermutationQType& permutationQ() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return m_q; + } + + /** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix + * will form a basis of the kernel. + * + * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + * + * Example: \include FullPivLU_kernel.cpp + * Output: \verbinclude FullPivLU_kernel.out + * + * \sa image() + */ + inline const internal::kernel_retval<FullPivLU> kernel() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return internal::kernel_retval<FullPivLU>(*this); + } + + /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix + * will form a basis of the image (column-space). + * + * \param originalMatrix the original matrix, of which *this is the LU decomposition. + * The reason why it is needed to pass it here, is that this allows + * a large optimization, as otherwise this method would need to reconstruct it + * from the LU decomposition. + * + * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + * + * Example: \include FullPivLU_image.cpp + * Output: \verbinclude FullPivLU_image.out + * + * \sa kernel() + */ + inline const internal::image_retval<FullPivLU> + image(const MatrixType& originalMatrix) const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return internal::image_retval<FullPivLU>(*this, originalMatrix); + } + + #ifdef EIGEN_PARSED_BY_DOXYGEN + /** \return a 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 a solution. + * + * \note_about_checking_solutions + * + * \note_about_arbitrary_choice_of_solution + * \note_about_using_kernel_to_study_multiple_solutions + * + * Example: \include FullPivLU_solve.cpp + * Output: \verbinclude FullPivLU_solve.out + * + * \sa TriangularView::solve(), kernel(), inverse() + */ + template<typename Rhs> + inline const Solve<FullPivLU, Rhs> + solve(const MatrixBase<Rhs>& b) const; + #endif + + /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is + the LU decomposition. + */ + inline RealScalar rcond() const + { + eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); + return internal::rcond_estimate_helper(m_l1_norm, *this); + } + + /** \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 This is only for square matrices. + * + * \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; + + /** Allows to prescribe a threshold to be used by certain methods, such as rank(), + * who need to determine when pivots are to be considered nonzero. This is not used for the + * LU decomposition itself. + * + * When it needs to get the threshold value, Eigen calls threshold(). By default, this + * uses a formula to automatically determine a reasonable threshold. + * Once you have called the present method setThreshold(const RealScalar&), + * your value is used instead. + * + * \param threshold The new value to use as the threshold. + * + * A pivot will be considered nonzero if its absolute value is strictly greater than + * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$ + * where maxpivot is the biggest pivot. + * + * If you want to come back to the default behavior, call setThreshold(Default_t) + */ + FullPivLU& setThreshold(const RealScalar& threshold) + { + m_usePrescribedThreshold = true; + m_prescribedThreshold = threshold; + return *this; + } + + /** Allows to come back to the default behavior, letting Eigen use its default formula for + * determining the threshold. + * + * You should pass the special object Eigen::Default as parameter here. + * \code lu.setThreshold(Eigen::Default); \endcode + * + * See the documentation of setThreshold(const RealScalar&). + */ + FullPivLU& setThreshold(Default_t) + { + m_usePrescribedThreshold = false; + return *this; + } + + /** Returns the threshold that will be used by certain methods such as rank(). + * + * See the documentation of setThreshold(const RealScalar&). + */ + RealScalar threshold() const + { + eigen_assert(m_isInitialized || m_usePrescribedThreshold); + return m_usePrescribedThreshold ? m_prescribedThreshold + // this formula comes from experimenting (see "LU precision tuning" thread on the list) + // and turns out to be identical to Higham's formula used already in LDLt. + : NumTraits<Scalar>::epsilon() * RealScalar(m_lu.diagonalSize()); + } + + /** \returns the rank of the matrix of which *this is the LU decomposition. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + */ + inline Index rank() const + { + using std::abs; + eigen_assert(m_isInitialized && "LU is not initialized."); + RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold(); + Index result = 0; + for(Index i = 0; i < m_nonzero_pivots; ++i) + result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold); + return result; + } + + /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + */ + inline Index dimensionOfKernel() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return cols() - rank(); + } + + /** \returns true if the matrix of which *this is the LU decomposition represents an injective + * linear map, i.e. has trivial kernel; false otherwise. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + */ + inline bool isInjective() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return rank() == cols(); + } + + /** \returns true if the matrix of which *this is the LU decomposition represents a surjective + * linear map; false otherwise. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + */ + inline bool isSurjective() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return rank() == rows(); + } + + /** \returns true if the matrix of which *this is the LU decomposition is invertible. + * + * \note This method has to determine which pivots should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + */ + inline bool isInvertible() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + return isInjective() && (m_lu.rows() == m_lu.cols()); + } + + /** \returns the inverse of the matrix of which *this is the LU decomposition. + * + * \note If this matrix is not invertible, the returned matrix has undefined coefficients. + * Use isInvertible() to first determine whether this matrix is invertible. + * + * \sa MatrixBase::inverse() + */ + inline const Inverse<FullPivLU> inverse() const + { + eigen_assert(m_isInitialized && "LU is not initialized."); + eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!"); + return Inverse<FullPivLU>(*this); + } + + MatrixType reconstructedMatrix() const; + + EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR + inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); } + EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR + inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); } + + #ifndef EIGEN_PARSED_BY_DOXYGEN + template<typename RhsType, typename DstType> + void _solve_impl(const RhsType &rhs, DstType &dst) const; + + template<bool Conjugate, typename RhsType, typename DstType> + void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const; + #endif + + protected: + + static void check_template_parameters() + { + EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); + } + + void computeInPlace(); + + MatrixType m_lu; + PermutationPType m_p; + PermutationQType m_q; + IntColVectorType m_rowsTranspositions; + IntRowVectorType m_colsTranspositions; + Index m_nonzero_pivots; + RealScalar m_l1_norm; + RealScalar m_maxpivot, m_prescribedThreshold; + signed char m_det_pq; + bool m_isInitialized, m_usePrescribedThreshold; +}; + +template<typename MatrixType> +FullPivLU<MatrixType>::FullPivLU() + : m_isInitialized(false), m_usePrescribedThreshold(false) +{ +} + +template<typename MatrixType> +FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols) + : m_lu(rows, cols), + m_p(rows), + m_q(cols), + m_rowsTranspositions(rows), + m_colsTranspositions(cols), + m_isInitialized(false), + m_usePrescribedThreshold(false) +{ +} + +template<typename MatrixType> +template<typename InputType> +FullPivLU<MatrixType>::FullPivLU(const EigenBase<InputType>& matrix) + : m_lu(matrix.rows(), matrix.cols()), + m_p(matrix.rows()), + m_q(matrix.cols()), + m_rowsTranspositions(matrix.rows()), + m_colsTranspositions(matrix.cols()), + m_isInitialized(false), + m_usePrescribedThreshold(false) +{ + compute(matrix.derived()); +} + +template<typename MatrixType> +template<typename InputType> +FullPivLU<MatrixType>::FullPivLU(EigenBase<InputType>& matrix) + : m_lu(matrix.derived()), + m_p(matrix.rows()), + m_q(matrix.cols()), + m_rowsTranspositions(matrix.rows()), + m_colsTranspositions(matrix.cols()), + m_isInitialized(false), + m_usePrescribedThreshold(false) +{ + computeInPlace(); +} + +template<typename MatrixType> +void FullPivLU<MatrixType>::computeInPlace() +{ + check_template_parameters(); + + // the permutations are stored as int indices, so just to be sure: + eigen_assert(m_lu.rows()<=NumTraits<int>::highest() && m_lu.cols()<=NumTraits<int>::highest()); + + m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff(); + + const Index size = m_lu.diagonalSize(); + const Index rows = m_lu.rows(); + const Index cols = m_lu.cols(); + + // will store the transpositions, before we accumulate them at the end. + // can't accumulate on-the-fly because that will be done in reverse order for the rows. + m_rowsTranspositions.resize(m_lu.rows()); + m_colsTranspositions.resize(m_lu.cols()); + Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i + + m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case) + m_maxpivot = RealScalar(0); + + for(Index k = 0; k < size; ++k) + { + // First, we need to find the pivot. + + // biggest coefficient in the remaining bottom-right corner (starting at row k, col k) + Index row_of_biggest_in_corner, col_of_biggest_in_corner; + typedef internal::scalar_score_coeff_op<Scalar> Scoring; + typedef typename Scoring::result_type Score; + Score biggest_in_corner; + biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k) + .unaryExpr(Scoring()) + .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner); + row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner, + col_of_biggest_in_corner += k; // need to add k to them. + + if(biggest_in_corner==Score(0)) + { + // before exiting, make sure to initialize the still uninitialized transpositions + // in a sane state without destroying what we already have. + m_nonzero_pivots = k; + for(Index i = k; i < size; ++i) + { + m_rowsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i); + m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i); + } + break; + } + + RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner); + if(abs_pivot > m_maxpivot) m_maxpivot = abs_pivot; + + // Now that we've found the pivot, we need to apply the row/col swaps to + // bring it to the location (k,k). + + m_rowsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner); + m_colsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(col_of_biggest_in_corner); + if(k != row_of_biggest_in_corner) { + m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner)); + ++number_of_transpositions; + } + if(k != col_of_biggest_in_corner) { + m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner)); + ++number_of_transpositions; + } + + // Now that the pivot is at the right location, we update the remaining + // bottom-right corner by Gaussian elimination. + + if(k<rows-1) + m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k); + if(k<size-1) + m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1); + } + + // the main loop is over, we still have to accumulate the transpositions to find the + // permutations P and Q + + m_p.setIdentity(rows); + for(Index k = size-1; k >= 0; --k) + m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k)); + + m_q.setIdentity(cols); + for(Index k = 0; k < size; ++k) + m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k)); + + m_det_pq = (number_of_transpositions%2) ? -1 : 1; + + m_isInitialized = true; +} + +template<typename MatrixType> +typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const +{ + eigen_assert(m_isInitialized && "LU is not initialized."); + eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!"); + return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod()); +} + +/** \returns the matrix represented by the decomposition, + * i.e., it returns the product: \f$ P^{-1} L U Q^{-1} \f$. + * This function is provided for debug purposes. */ +template<typename MatrixType> +MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const +{ + eigen_assert(m_isInitialized && "LU is not initialized."); + const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols()); + // LU + MatrixType res(m_lu.rows(),m_lu.cols()); + // FIXME the .toDenseMatrix() should not be needed... + res = m_lu.leftCols(smalldim) + .template triangularView<UnitLower>().toDenseMatrix() + * m_lu.topRows(smalldim) + .template triangularView<Upper>().toDenseMatrix(); + + // P^{-1}(LU) + res = m_p.inverse() * res; + + // (P^{-1}LU)Q^{-1} + res = res * m_q.inverse(); + + return res; +} + +/********* Implementation of kernel() **************************************************/ + +namespace internal { +template<typename _MatrixType> +struct kernel_retval<FullPivLU<_MatrixType> > + : kernel_retval_base<FullPivLU<_MatrixType> > +{ + EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>) + + enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( + MatrixType::MaxColsAtCompileTime, + MatrixType::MaxRowsAtCompileTime) + }; + + template<typename Dest> void evalTo(Dest& dst) const + { + using std::abs; + const Index cols = dec().matrixLU().cols(), dimker = cols - rank(); + if(dimker == 0) + { + // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's + // avoid crashing/asserting as that depends on floating point calculations. Let's + // just return a single column vector filled with zeros. + dst.setZero(); + return; + } + + /* Let us use the following lemma: + * + * Lemma: If the matrix A has the LU decomposition PAQ = LU, + * then Ker A = Q(Ker U). + * + * Proof: trivial: just keep in mind that P, Q, L are invertible. + */ + + /* Thus, all we need to do is to compute Ker U, and then apply Q. + * + * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end. + * Thus, the diagonal of U ends with exactly + * dimKer zero's. Let us use that to construct dimKer linearly + * independent vectors in Ker U. + */ + + Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); + RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); + Index p = 0; + for(Index i = 0; i < dec().nonzeroPivots(); ++i) + if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) + pivots.coeffRef(p++) = i; + eigen_internal_assert(p == rank()); + + // we construct a temporaty trapezoid matrix m, by taking the U matrix and + // permuting the rows and cols to bring the nonnegligible pivots to the top of + // the main diagonal. We need that to be able to apply our triangular solvers. + // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified + Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options, + MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime> + m(dec().matrixLU().block(0, 0, rank(), cols)); + for(Index i = 0; i < rank(); ++i) + { + if(i) m.row(i).head(i).setZero(); + m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i); + } + m.block(0, 0, rank(), rank()); + m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero(); + for(Index i = 0; i < rank(); ++i) + m.col(i).swap(m.col(pivots.coeff(i))); + + // ok, we have our trapezoid matrix, we can apply the triangular solver. + // notice that the math behind this suggests that we should apply this to the + // negative of the RHS, but for performance we just put the negative sign elsewhere, see below. + m.topLeftCorner(rank(), rank()) + .template triangularView<Upper>().solveInPlace( + m.topRightCorner(rank(), dimker) + ); + + // now we must undo the column permutation that we had applied! + for(Index i = rank()-1; i >= 0; --i) + m.col(i).swap(m.col(pivots.coeff(i))); + + // see the negative sign in the next line, that's what we were talking about above. + for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker); + for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero(); + for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1); + } +}; + +/***** Implementation of image() *****************************************************/ + +template<typename _MatrixType> +struct image_retval<FullPivLU<_MatrixType> > + : image_retval_base<FullPivLU<_MatrixType> > +{ + EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>) + + enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( + MatrixType::MaxColsAtCompileTime, + MatrixType::MaxRowsAtCompileTime) + }; + + template<typename Dest> void evalTo(Dest& dst) const + { + using std::abs; + if(rank() == 0) + { + // The Image is just {0}, so it doesn't have a basis properly speaking, but let's + // avoid crashing/asserting as that depends on floating point calculations. Let's + // just return a single column vector filled with zeros. + dst.setZero(); + return; + } + + Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); + RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); + Index p = 0; + for(Index i = 0; i < dec().nonzeroPivots(); ++i) + if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) + pivots.coeffRef(p++) = i; + eigen_internal_assert(p == rank()); + + for(Index i = 0; i < rank(); ++i) + dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i))); + } +}; + +/***** Implementation of solve() *****************************************************/ + +} // end namespace internal + +#ifndef EIGEN_PARSED_BY_DOXYGEN +template<typename _MatrixType> +template<typename RhsType, typename DstType> +void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const +{ + /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}. + * So we proceed as follows: + * Step 1: compute c = P * rhs. + * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible. + * Step 3: replace c by the solution x to Ux = c. May or may not exist. + * Step 4: result = Q * c; + */ + + const Index rows = this->rows(), + cols = this->cols(), + nonzero_pivots = this->rank(); + const Index smalldim = (std::min)(rows, cols); + + if(nonzero_pivots == 0) + { + dst.setZero(); + return; + } + + typename RhsType::PlainObject c(rhs.rows(), rhs.cols()); + + // Step 1 + c = permutationP() * rhs; + + // Step 2 + m_lu.topLeftCorner(smalldim,smalldim) + .template triangularView<UnitLower>() + .solveInPlace(c.topRows(smalldim)); + if(rows>cols) + c.bottomRows(rows-cols) -= m_lu.bottomRows(rows-cols) * c.topRows(cols); + + // Step 3 + m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots) + .template triangularView<Upper>() + .solveInPlace(c.topRows(nonzero_pivots)); + + // Step 4 + for(Index i = 0; i < nonzero_pivots; ++i) + dst.row(permutationQ().indices().coeff(i)) = c.row(i); + for(Index i = nonzero_pivots; i < m_lu.cols(); ++i) + dst.row(permutationQ().indices().coeff(i)).setZero(); +} + +template<typename _MatrixType> +template<bool Conjugate, typename RhsType, typename DstType> +void FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const +{ + /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}, + * and since permutations are real and unitary, we can write this + * as A^T = Q U^T L^T P, + * So we proceed as follows: + * Step 1: compute c = Q^T rhs. + * Step 2: replace c by the solution x to U^T x = c. May or may not exist. + * Step 3: replace c by the solution x to L^T x = c. + * Step 4: result = P^T c. + * If Conjugate is true, replace "^T" by "^*" above. + */ + + const Index rows = this->rows(), cols = this->cols(), + nonzero_pivots = this->rank(); + const Index smalldim = (std::min)(rows, cols); + + if(nonzero_pivots == 0) + { + dst.setZero(); + return; + } + + typename RhsType::PlainObject c(rhs.rows(), rhs.cols()); + + // Step 1 + c = permutationQ().inverse() * rhs; + + // Step 2 + m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots) + .template triangularView<Upper>() + .transpose() + .template conjugateIf<Conjugate>() + .solveInPlace(c.topRows(nonzero_pivots)); + + // Step 3 + m_lu.topLeftCorner(smalldim, smalldim) + .template triangularView<UnitLower>() + .transpose() + .template conjugateIf<Conjugate>() + .solveInPlace(c.topRows(smalldim)); + + // Step 4 + PermutationPType invp = permutationP().inverse().eval(); + for(Index i = 0; i < smalldim; ++i) + dst.row(invp.indices().coeff(i)) = c.row(i); + for(Index i = smalldim; i < rows; ++i) + dst.row(invp.indices().coeff(i)).setZero(); +} + +#endif + +namespace internal { + + +/***** Implementation of inverse() *****************************************************/ +template<typename DstXprType, typename MatrixType> +struct Assignment<DstXprType, Inverse<FullPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename FullPivLU<MatrixType>::Scalar>, Dense2Dense> +{ + typedef FullPivLU<MatrixType> LuType; + typedef Inverse<LuType> SrcXprType; + static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename MatrixType::Scalar> &) + { + dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols())); + } +}; +} // end namespace internal + +/******* MatrixBase methods *****************************************************************/ + +/** \lu_module + * + * \return the full-pivoting LU decomposition of \c *this. + * + * \sa class FullPivLU + */ +template<typename Derived> +inline const FullPivLU<typename MatrixBase<Derived>::PlainObject> +MatrixBase<Derived>::fullPivLu() const +{ + return FullPivLU<PlainObject>(eval()); +} + +} // end namespace Eigen + +#endif // EIGEN_LU_H |