// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2006-2009 Benoit Jacob // // 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 { /** \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 \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 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 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::Real RealScalar; typedef typename internal::traits::StorageKind StorageKind; typedef typename MatrixType::Index Index; typedef typename internal::plain_row_type::type IntRowVectorType; typedef typename internal::plain_col_type::type IntColVectorType; typedef PermutationMatrix PermutationQType; typedef PermutationMatrix 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(); /** \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. */ 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 { 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() */ 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 kernel() const { eigen_assert(m_isInitialized && "LU is not initialized."); return internal::kernel_retval(*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 internal::image_retval image(const MatrixType& originalMatrix) const { eigen_assert(m_isInitialized && "LU is not initialized."); return internal::image_retval(*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 inline const internal::solve_retval solve(const MatrixBase& b) const { eigen_assert(m_isInitialized && "LU is not initialized."); return internal::solve_retval(*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 internal::traits::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::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 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 internal::solve_retval 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 internal::solve_retval (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols())); } 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; PermutationPType m_p; PermutationQType m_q; IntColVectorType m_rowsTranspositions; IntRowVectorType m_colsTranspositions; Index m_det_pq, m_nonzero_pivots; RealScalar m_maxpivot, m_prescribedThreshold; bool m_isInitialized, m_usePrescribedThreshold; }; template FullPivLU::FullPivLU() : m_isInitialized(false), m_usePrescribedThreshold(false) { } template FullPivLU::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 FullPivLU::FullPivLU(const MatrixType& 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); } template FullPivLU& FullPivLU::compute(const MatrixType& matrix) { check_template_parameters(); // the permutations are stored as int indices, so just to be sure: eigen_assert(matrix.rows()<=NumTraits::highest() && matrix.cols()<=NumTraits::highest()); m_isInitialized = true; m_lu = matrix; const Index size = matrix.diagonalSize(); const Index rows = matrix.rows(); const Index 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. m_rowsTranspositions.resize(matrix.rows()); m_colsTranspositions.resize(matrix.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; RealScalar biggest_in_corner; biggest_in_corner = m_lu.bottomRightCorner(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. if(biggest_in_corner==RealScalar(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) = i; m_colsTranspositions.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). m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner; m_colsTranspositions.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= 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; return *this; } template typename internal::traits::Scalar FullPivLU::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 MatrixType FullPivLU::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().toDenseMatrix() * m_lu.topRows(smalldim) .template triangularView().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 struct kernel_retval > : kernel_retval_base > { EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>) enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime) }; template 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 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 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().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().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 struct image_retval > : image_retval_base > { EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>) enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime) }; template 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 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() *****************************************************/ template struct solve_retval, Rhs> : solve_retval_base, Rhs> { EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs) template 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 Index rows = dec().rows(), cols = dec().cols(), nonzero_pivots = dec().nonzeroPivots(); eigen_assert(rhs().rows() == rows); const Index 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() .topLeftCorner(smalldim,smalldim) .template triangularView() .solveInPlace(c.topRows(smalldim)); if(rows>cols) { c.bottomRows(rows-cols) -= dec().matrixLU().bottomRows(rows-cols) * c.topRows(cols); } // Step 3 dec().matrixLU() .topLeftCorner(nonzero_pivots, nonzero_pivots) .template triangularView() .solveInPlace(c.topRows(nonzero_pivots)); // Step 4 for(Index i = 0; i < nonzero_pivots; ++i) dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i); for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero(); } }; } // end namespace internal /******* MatrixBase methods *****************************************************************/ /** \lu_module * * \return the full-pivoting LU decomposition of \c *this. * * \sa class FullPivLU */ template inline const FullPivLU::PlainObject> MatrixBase::fullPivLu() const { return FullPivLU(eval()); } } // end namespace Eigen #endif // EIGEN_LU_H