| // 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> |
| // |
| // Eigen is free software; you can redistribute it and/or |
| // modify it under the terms of the GNU Lesser General Public |
| // License as published by the Free Software Foundation; either |
| // version 3 of the License, or (at your option) any later version. |
| // |
| // Alternatively, you can redistribute it and/or |
| // modify it under the terms of the GNU General Public License as |
| // published by the Free Software Foundation; either version 2 of |
| // the License, or (at your option) any later version. |
| // |
| // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY |
| // // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS |
| // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the |
| // GNU General Public License for more details. |
| // |
| // You should have received a copy of the GNU Lesser General Public |
| // License and a copy of the GNU General Public License along with |
| // Eigen. If not, see <http://www.gnu.org/licenses/>. |
| |
| #ifndef EIGEN_LU_H |
| #define EIGEN_LU_H |
| |
| /** \ingroup LU_Module |
| * |
| * \class FullPivLU |
| * |
| * \brief LU decomposition of a matrix with complete 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 any matrix, with complete pivoting: the matrix A |
| * is decomposed as A = PLUQ 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 exemple, here is how the original matrix can be retrieved: |
| * \include class_FullPivLU.cpp |
| * Output: \verbinclude class_FullPivLU.out |
| * |
| * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse() |
| */ |
| template<typename _MatrixType> class FullPivLU |
| { |
| 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 ei_plain_row_type<MatrixType, int>::type IntRowVectorType; |
| typedef typename ei_plain_col_type<MatrixType, int>::type IntColVectorType; |
| typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType; |
| typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType; |
| |
| /** |
| * \brief Default Constructor. |
| * |
| * The default constructor is useful in cases in which the user intends to |
| * perform decompositions via LU::compute(const MatrixType&). |
| */ |
| FullPivLU(); |
| |
| /** Constructor. |
| * |
| * \param matrix the matrix of which to compute the LU decomposition. |
| * It is required to be nonzero. |
| */ |
| FullPivLU(const MatrixType& 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 |
| */ |
| FullPivLU& 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 |
| { |
| ei_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 int nonzeroPivots() const |
| { |
| ei_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() |
| */ |
| inline const PermutationPType& permutationP() const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| return m_p; |
| } |
| |
| /** \returns the permutation matrix Q |
| * |
| * \sa permutationP() |
| */ |
| inline const PermutationQType& permutationQ() const |
| { |
| ei_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 ei_kernel_retval<FullPivLU> kernel() const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| return ei_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 kernel. |
| * |
| * \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 ei_image_retval<FullPivLU> |
| image(const MatrixType& originalMatrix) const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| return ei_image_retval<FullPivLU>(*this, originalMatrix); |
| } |
| |
| /** \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 ei_solve_retval<FullPivLU, Rhs> |
| solve(const MatrixBase<Rhs>& b) const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| return ei_solve_retval<FullPivLU, Rhs>(*this, b.derived()); |
| } |
| |
| /** \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 ei_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; |
| } |
| |
| /** Returns the threshold that will be used by certain methods such as rank(). |
| * |
| * See the documentation of setThreshold(const RealScalar&). |
| */ |
| RealScalar threshold() const |
| { |
| ei_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() * 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 int rank() const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| RealScalar premultiplied_threshold = ei_abs(m_maxpivot) * threshold(); |
| int result = 0; |
| for(int i = 0; i < m_nonzero_pivots; ++i) |
| result += (ei_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 int dimensionOfKernel() const |
| { |
| ei_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 |
| { |
| ei_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 |
| { |
| ei_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 |
| { |
| ei_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 ei_solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> inverse() const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| ei_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!"); |
| return ei_solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> |
| (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols())); |
| } |
| |
| MatrixType reconstructedMatrix() const; |
| |
| inline int rows() const { return m_lu.rows(); } |
| inline int cols() const { return m_lu.cols(); } |
| |
| protected: |
| MatrixType m_lu; |
| PermutationPType m_p; |
| PermutationQType m_q; |
| int m_det_pq, m_nonzero_pivots; |
| RealScalar m_maxpivot, m_prescribedThreshold; |
| bool m_isInitialized, m_usePrescribedThreshold; |
| }; |
| |
| template<typename MatrixType> |
| FullPivLU<MatrixType>::FullPivLU() |
| : m_isInitialized(false), m_usePrescribedThreshold(false) |
| { |
| } |
| |
| template<typename MatrixType> |
| FullPivLU<MatrixType>::FullPivLU(const MatrixType& matrix) |
| : m_isInitialized(false), m_usePrescribedThreshold(false) |
| { |
| compute(matrix); |
| } |
| |
| template<typename MatrixType> |
| FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix) |
| { |
| m_isInitialized = true; |
| m_lu = matrix; |
| |
| const int size = matrix.diagonalSize(); |
| const int rows = matrix.rows(); |
| const int cols = matrix.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. |
| IntColVectorType rows_transpositions(matrix.rows()); |
| IntRowVectorType cols_transpositions(matrix.cols()); |
| int number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. rows_transpositions[i]!=i |
| |
| m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case) |
| m_maxpivot = RealScalar(0); |
| RealScalar cutoff(0); |
| |
| for(int 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) |
| int row_of_biggest_in_corner, col_of_biggest_in_corner; |
| RealScalar biggest_in_corner; |
| biggest_in_corner = m_lu.corner(Eigen::BottomRight, rows-k, cols-k) |
| .cwiseAbs() |
| .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. |
| |
| // when k==0, biggest_in_corner is the biggest coeff absolute value in the original matrix |
| if(k == 0) cutoff = biggest_in_corner * NumTraits<Scalar>::epsilon(); |
| |
| // if the pivot (hence the corner) is "zero", terminate to avoid generating nan/inf values. |
| // Notice that using an exact comparison (biggest_in_corner==0) here, as Golub-van Loan do in |
| // their pseudo-code, results in numerical instability! The cutoff here has been validated |
| // by running the unit test 'lu' with many repetitions. |
| if(biggest_in_corner < cutoff) |
| { |
| // 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(int i = k; i < size; ++i) |
| { |
| rows_transpositions.coeffRef(i) = i; |
| cols_transpositions.coeffRef(i) = i; |
| } |
| break; |
| } |
| |
| if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner; |
| |
| // Now that we've found the pivot, we need to apply the row/col swaps to |
| // bring it to the location (k,k). |
| |
| rows_transpositions.coeffRef(k) = row_of_biggest_in_corner; |
| cols_transpositions.coeffRef(k) = 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(int k = size-1; k >= 0; --k) |
| m_p.applyTranspositionOnTheRight(k, rows_transpositions.coeff(k)); |
| |
| m_q.setIdentity(cols); |
| for(int k = 0; k < size; ++k) |
| m_q.applyTranspositionOnTheRight(k, cols_transpositions.coeff(k)); |
| |
| m_det_pq = (number_of_transpositions%2) ? -1 : 1; |
| return *this; |
| } |
| |
| template<typename MatrixType> |
| typename ei_traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| ei_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: P^{-1} L U Q^{-1}. |
| * This function is provided for debug purpose. */ |
| template<typename MatrixType> |
| MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const |
| { |
| ei_assert(m_isInitialized && "LU is not initialized."); |
| const int 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.corner(TopLeft,m_lu.rows(),smalldim) |
| .template triangularView<UnitLower>().toDenseMatrix() |
| * m_lu.corner(TopLeft,smalldim,m_lu.cols()) |
| .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() **************************************************/ |
| |
| template<typename _MatrixType> |
| struct ei_kernel_retval<FullPivLU<_MatrixType> > |
| : ei_kernel_retval_base<FullPivLU<_MatrixType> > |
| { |
| EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>) |
| |
| enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN( |
| MatrixType::MaxColsAtCompileTime, |
| MatrixType::MaxRowsAtCompileTime) |
| }; |
| |
| template<typename Dest> void evalTo(Dest& dst) const |
| { |
| const int 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<int, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); |
| RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); |
| int p = 0; |
| for(int i = 0; i < dec().nonzeroPivots(); ++i) |
| if(ei_abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) |
| pivots.coeffRef(p++) = i; |
| ei_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(int 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(int 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.corner(TopLeft, rank(), rank()) |
| .template triangularView<Upper>().solveInPlace( |
| m.corner(TopRight, rank(), dimker) |
| ); |
| |
| // now we must undo the column permutation that we had applied! |
| for(int 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(int i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker); |
| for(int i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero(); |
| for(int k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1); |
| } |
| }; |
| |
| /***** Implementation of image() *****************************************************/ |
| |
| template<typename _MatrixType> |
| struct ei_image_retval<FullPivLU<_MatrixType> > |
| : ei_image_retval_base<FullPivLU<_MatrixType> > |
| { |
| EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>) |
| |
| enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN( |
| MatrixType::MaxColsAtCompileTime, |
| MatrixType::MaxRowsAtCompileTime) |
| }; |
| |
| template<typename Dest> void evalTo(Dest& dst) const |
| { |
| 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<int, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); |
| RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); |
| int p = 0; |
| for(int i = 0; i < dec().nonzeroPivots(); ++i) |
| if(ei_abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) |
| pivots.coeffRef(p++) = i; |
| ei_internal_assert(p == rank()); |
| |
| for(int i = 0; i < rank(); ++i) |
| dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i))); |
| } |
| }; |
| |
| /***** Implementation of solve() *****************************************************/ |
| |
| template<typename _MatrixType, typename Rhs> |
| struct ei_solve_retval<FullPivLU<_MatrixType>, Rhs> |
| : ei_solve_retval_base<FullPivLU<_MatrixType>, Rhs> |
| { |
| EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs) |
| |
| template<typename Dest> void evalTo(Dest& 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 int rows = dec().rows(), cols = dec().cols(), |
| nonzero_pivots = dec().nonzeroPivots(); |
| ei_assert(rhs().rows() == rows); |
| const int smalldim = std::min(rows, cols); |
| |
| if(nonzero_pivots == 0) |
| { |
| dst.setZero(); |
| return; |
| } |
| |
| typename Rhs::PlainObject c(rhs().rows(), rhs().cols()); |
| |
| // Step 1 |
| c = dec().permutationP() * rhs(); |
| |
| // Step 2 |
| dec().matrixLU() |
| .corner(Eigen::TopLeft,smalldim,smalldim) |
| .template triangularView<UnitLower>() |
| .solveInPlace(c.corner(Eigen::TopLeft, smalldim, c.cols())); |
| if(rows>cols) |
| { |
| c.corner(Eigen::BottomLeft, rows-cols, c.cols()) |
| -= dec().matrixLU().corner(Eigen::BottomLeft, rows-cols, cols) |
| * c.corner(Eigen::TopLeft, cols, c.cols()); |
| } |
| |
| // Step 3 |
| dec().matrixLU() |
| .corner(TopLeft, nonzero_pivots, nonzero_pivots) |
| .template triangularView<Upper>() |
| .solveInPlace(c.corner(TopLeft, nonzero_pivots, c.cols())); |
| |
| // Step 4 |
| for(int i = 0; i < nonzero_pivots; ++i) |
| dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i); |
| for(int i = nonzero_pivots; i < dec().matrixLU().cols(); ++i) |
| dst.row(dec().permutationQ().indices().coeff(i)).setZero(); |
| } |
| }; |
| |
| /******* 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()); |
| } |
| |
| #endif // EIGEN_LU_H |
