| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. Eigen itself is part of the KDE project. |
| // |
| // Copyright (C) 2006-2008 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 LU |
| * |
| * \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, so that the rank |
| * of A is the index of the first zero on the diagonal of U (with indices starting at 0) if any. |
| * |
| * This decomposition provides the generic approach to solving systems of linear equations, computing |
| * the rank, invertibility, inverse, kernel, and determinant. |
| * |
| * The data of the LU decomposition can be directly accessed through the methods matrixLU(), |
| * permutationP(), permutationQ(). Convenience methods matrixL(), matrixU() are also provided. |
| * |
| * As an exemple, here is how the original matrix can be retrieved, in the square case: |
| * \include class_LU_1.cpp |
| * Output: \verbinclude class_LU_1.out |
| * |
| * When the matrix is not square, matrixL() is no longer very useful: if one needs it, one has |
| * to construct the L matrix by hand, as shown in this example: |
| * \include class_LU_2.cpp |
| * Output: \verbinclude class_LU_2.out |
| * |
| * \sa MatrixBase::lu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse() |
| */ |
| template<typename MatrixType> class LU |
| { |
| public: |
| |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; |
| typedef Matrix<int, 1, MatrixType::ColsAtCompileTime> IntRowVectorType; |
| typedef Matrix<int, MatrixType::RowsAtCompileTime, 1> IntColVectorType; |
| typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType; |
| typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVectorType; |
| |
| enum { MaxSmallDimAtCompileTime = EIGEN_ENUM_MIN( |
| MatrixType::MaxColsAtCompileTime, |
| MatrixType::MaxRowsAtCompileTime) |
| }; |
| |
| typedef Matrix<typename MatrixType::Scalar, |
| MatrixType::ColsAtCompileTime, // the number of rows in the "kernel matrix" is the number of cols of the original matrix |
| // so that the product "matrix * kernel = zero" makes sense |
| Dynamic, // we don't know at compile-time the dimension of the kernel |
| MatrixType::Options, |
| MatrixType::MaxColsAtCompileTime, // see explanation for 2nd template parameter |
| MatrixType::MaxColsAtCompileTime // the kernel is a subspace of the domain space, whose dimension is the number |
| // of columns of the original matrix |
| > KernelResultType; |
| |
| typedef Matrix<typename MatrixType::Scalar, |
| MatrixType::RowsAtCompileTime, // the image is a subspace of the destination space, whose dimension is the number |
| // of rows of the original matrix |
| Dynamic, // we don't know at compile time the dimension of the image (the rank) |
| MatrixType::Options, |
| MatrixType::MaxRowsAtCompileTime, // the image matrix will consist of columns from the original matrix, |
| MatrixType::MaxColsAtCompileTime // so it has the same number of rows and at most as many columns. |
| > ImageResultType; |
| |
| /** Constructor. |
| * |
| * \param matrix the matrix of which to compute the LU decomposition. |
| */ |
| LU(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 LU). |
| * |
| * \sa matrixL(), matrixU() |
| */ |
| inline const MatrixType& matrixLU() const |
| { |
| return m_lu; |
| } |
| |
| /** \returns an expression of the unit-lower-triangular part of the LU matrix. In the square case, |
| * this is the L matrix. In the non-square, actually obtaining the L matrix takes some |
| * more care, see the documentation of class LU. |
| * |
| * \sa matrixLU(), matrixU() |
| */ |
| inline const Part<MatrixType, UnitLowerTriangular> matrixL() const |
| { |
| return m_lu; |
| } |
| |
| /** \returns an expression of the U matrix, i.e. the upper-triangular part of the LU matrix. |
| * |
| * \sa matrixLU(), matrixL() |
| */ |
| inline const Part<MatrixType, UpperTriangular> matrixU() const |
| { |
| return m_lu; |
| } |
| |
| /** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed, |
| * representing the P permutation i.e. the permutation of the rows. For its precise meaning, |
| * see the examples given in the documentation of class LU. |
| * |
| * \sa permutationQ() |
| */ |
| inline const IntColVectorType& permutationP() const |
| { |
| return m_p; |
| } |
| |
| /** \returns a vector of integers, whose size is the number of columns of the matrix being |
| * decomposed, representing the Q permutation i.e. the permutation of the columns. |
| * For its precise meaning, see the examples given in the documentation of class LU. |
| * |
| * \sa permutationP() |
| */ |
| inline const IntRowVectorType& permutationQ() const |
| { |
| return m_q; |
| } |
| |
| /** Computes a basis of the kernel of the matrix, also called the null-space of the matrix. |
| * |
| * \note This method is only allowed on non-invertible matrices, as determined by |
| * isInvertible(). Calling it on an invertible matrix will make an assertion fail. |
| * |
| * \param result a pointer to the matrix in which to store the kernel. The columns of this |
| * matrix will be set to form a basis of the kernel (it will be resized |
| * if necessary). |
| * |
| * Example: \include LU_computeKernel.cpp |
| * Output: \verbinclude LU_computeKernel.out |
| * |
| * \sa kernel(), computeImage(), image() |
| */ |
| template<typename KernelMatrixType> |
| void computeKernel(KernelMatrixType *result) const; |
| |
| /** Computes a basis of the image of the matrix, also called the column-space or range of he matrix. |
| * |
| * \note Calling this method on the zero matrix will make an assertion fail. |
| * |
| * \param result a pointer to the matrix in which to store the image. The columns of this |
| * matrix will be set to form a basis of the image (it will be resized |
| * if necessary). |
| * |
| * Example: \include LU_computeImage.cpp |
| * Output: \verbinclude LU_computeImage.out |
| * |
| * \sa image(), computeKernel(), kernel() |
| */ |
| template<typename ImageMatrixType> |
| void computeImage(ImageMatrixType *result) const; |
| |
| /** \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: this method is only allowed on non-invertible matrices, as determined by |
| * isInvertible(). Calling it on an invertible matrix will make an assertion fail. |
| * |
| * \note: this method returns a matrix by value, which induces some inefficiency. |
| * If you prefer to avoid this overhead, use computeKernel() instead. |
| * |
| * Example: \include LU_kernel.cpp |
| * Output: \verbinclude LU_kernel.out |
| * |
| * \sa computeKernel(), image() |
| */ |
| const KernelResultType kernel() const; |
| |
| /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix |
| * will form a basis of the kernel. |
| * |
| * \note: Calling this method on the zero matrix will make an assertion fail. |
| * |
| * \note: this method returns a matrix by value, which induces some inefficiency. |
| * If you prefer to avoid this overhead, use computeImage() instead. |
| * |
| * Example: \include LU_image.cpp |
| * Output: \verbinclude LU_image.out |
| * |
| * \sa computeImage(), kernel() |
| */ |
| const ImageResultType image() const; |
| |
| /** This method finds a solution x to the equation Ax=b, where A is the matrix of which |
| * *this is the LU decomposition, if any exists. |
| * |
| * \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. |
| * \param result a pointer to the vector or matrix in which to store the solution, if any exists. |
| * Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols(). |
| * If no solution exists, *result is left with undefined coefficients. |
| * |
| * \returns true if any solution exists, false if no solution exists. |
| * |
| * \note If there exist more than one solution, this method will arbitrarily choose one. |
| * If you need a complete analysis of the space of solutions, take the one solution obtained |
| * by this method and add to it elements of the kernel, as determined by kernel(). |
| * |
| * Example: \include LU_solve.cpp |
| * Output: \verbinclude LU_solve.out |
| * |
| * \sa MatrixBase::solveTriangular(), kernel(), computeKernel(), inverse(), computeInverse() |
| */ |
| template<typename OtherDerived, typename ResultType> |
| bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const; |
| |
| /** \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; |
| |
| /** \returns the rank of the matrix of which *this is the LU decomposition. |
| * |
| * \note This is computed at the time of the construction of the LU decomposition. This |
| * method does not perform any further computation. |
| */ |
| inline int rank() const |
| { |
| return m_rank; |
| } |
| |
| /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition. |
| * |
| * \note Since the rank is computed at the time of the construction of the LU decomposition, this |
| * method almost does not perform any further computation. |
| */ |
| inline int dimensionOfKernel() const |
| { |
| return m_lu.cols() - m_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 Since the rank is computed at the time of the construction of the LU decomposition, this |
| * method almost does not perform any further computation. |
| */ |
| inline bool isInjective() const |
| { |
| return m_rank == m_lu.cols(); |
| } |
| |
| /** \returns true if the matrix of which *this is the LU decomposition represents a surjective |
| * linear map; false otherwise. |
| * |
| * \note Since the rank is computed at the time of the construction of the LU decomposition, this |
| * method almost does not perform any further computation. |
| */ |
| inline bool isSurjective() const |
| { |
| return m_rank == m_lu.rows(); |
| } |
| |
| /** \returns true if the matrix of which *this is the LU decomposition is invertible. |
| * |
| * \note Since the rank is computed at the time of the construction of the LU decomposition, this |
| * method almost does not perform any further computation. |
| */ |
| inline bool isInvertible() const |
| { |
| return isInjective() && isSurjective(); |
| } |
| |
| /** Computes the inverse of the matrix of which *this is the LU decomposition. |
| * |
| * \param result a pointer to the matrix into which to store the inverse. Resized if needed. |
| * |
| * \note If this matrix is not invertible, *result is left with undefined coefficients. |
| * Use isInvertible() to first determine whether this matrix is invertible. |
| * |
| * \sa MatrixBase::computeInverse(), inverse() |
| */ |
| inline void computeInverse(MatrixType *result) const |
| { |
| solve(MatrixType::Identity(m_lu.rows(), m_lu.cols()), result); |
| } |
| |
| /** \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 computeInverse(), MatrixBase::inverse() |
| */ |
| inline MatrixType inverse() const |
| { |
| MatrixType result; |
| computeInverse(&result); |
| return result; |
| } |
| |
| protected: |
| const MatrixType& m_originalMatrix; |
| MatrixType m_lu; |
| IntColVectorType m_p; |
| IntRowVectorType m_q; |
| int m_det_pq; |
| int m_rank; |
| }; |
| |
| template<typename MatrixType> |
| LU<MatrixType>::LU(const MatrixType& matrix) |
| : m_originalMatrix(matrix), |
| m_lu(matrix), |
| m_p(matrix.rows()), |
| m_q(matrix.cols()) |
| { |
| const int size = matrix.diagonal().size(); |
| const int rows = matrix.rows(); |
| const int cols = matrix.cols(); |
| |
| IntColVectorType rows_transpositions(matrix.rows()); |
| IntRowVectorType cols_transpositions(matrix.cols()); |
| int number_of_transpositions = 0; |
| |
| RealScalar biggest = RealScalar(0); |
| for(int k = 0; k < size; ++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) |
| .cwise().abs() |
| .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner); |
| row_of_biggest_in_corner += k; |
| col_of_biggest_in_corner += 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; |
| } |
| |
| if(k==0) biggest = biggest_in_corner; |
| const Scalar lu_k_k = m_lu.coeff(k,k); |
| if(ei_isMuchSmallerThan(lu_k_k, biggest)) continue; |
| if(k<rows-1) |
| m_lu.col(k).end(rows-k-1) /= lu_k_k; |
| if(k<size-1) |
| for(int col = k + 1; col < cols; ++col) |
| m_lu.col(col).end(rows-k-1) -= m_lu.col(k).end(rows-k-1) * m_lu.coeff(k,col); |
| } |
| |
| for(int k = 0; k < matrix.rows(); ++k) m_p.coeffRef(k) = k; |
| for(int k = size-1; k >= 0; --k) |
| std::swap(m_p.coeffRef(k), m_p.coeffRef(rows_transpositions.coeff(k))); |
| |
| for(int k = 0; k < matrix.cols(); ++k) m_q.coeffRef(k) = k; |
| for(int k = 0; k < size; ++k) |
| std::swap(m_q.coeffRef(k), m_q.coeffRef(cols_transpositions.coeff(k))); |
| |
| m_det_pq = (number_of_transpositions%2) ? -1 : 1; |
| |
| for(m_rank = 0; m_rank < size; ++m_rank) |
| if(ei_isMuchSmallerThan(m_lu.diagonal().coeff(m_rank), m_lu.diagonal().coeff(0))) |
| break; |
| } |
| |
| template<typename MatrixType> |
| typename ei_traits<MatrixType>::Scalar LU<MatrixType>::determinant() const |
| { |
| return Scalar(m_det_pq) * m_lu.diagonal().redux(ei_scalar_product_op<Scalar>()); |
| } |
| |
| template<typename MatrixType> |
| template<typename KernelMatrixType> |
| void LU<MatrixType>::computeKernel(KernelMatrixType *result) const |
| { |
| ei_assert(!isInvertible()); |
| const int dimker = dimensionOfKernel(), cols = m_lu.cols(); |
| result->resize(cols, dimker); |
| |
| /* 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 |
| * m_dimKer zero's. Let us use that to construct m_dimKer linearly |
| * independent vectors in Ker U. |
| */ |
| |
| Matrix<Scalar, Dynamic, Dynamic, MatrixType::Options, |
| MatrixType::MaxColsAtCompileTime, MatrixType::MaxColsAtCompileTime> |
| y(-m_lu.corner(TopRight, m_rank, dimker)); |
| |
| m_lu.corner(TopLeft, m_rank, m_rank) |
| .template marked<UpperTriangular>() |
| .solveTriangularInPlace(y); |
| |
| for(int i = 0; i < m_rank; ++i) |
| result->row(m_q.coeff(i)) = y.row(i); |
| for(int i = m_rank; i < cols; ++i) result->row(m_q.coeff(i)).setZero(); |
| for(int k = 0; k < dimker; ++k) result->coeffRef(m_q.coeff(m_rank+k), k) = Scalar(1); |
| } |
| |
| template<typename MatrixType> |
| const typename LU<MatrixType>::KernelResultType |
| LU<MatrixType>::kernel() const |
| { |
| KernelResultType result(m_lu.cols(), dimensionOfKernel()); |
| computeKernel(&result); |
| return result; |
| } |
| |
| template<typename MatrixType> |
| template<typename ImageMatrixType> |
| void LU<MatrixType>::computeImage(ImageMatrixType *result) const |
| { |
| ei_assert(m_rank > 0); |
| result->resize(m_originalMatrix.rows(), m_rank); |
| for(int i = 0; i < m_rank; ++i) |
| result->col(i) = m_originalMatrix.col(m_q.coeff(i)); |
| } |
| |
| template<typename MatrixType> |
| const typename LU<MatrixType>::ImageResultType |
| LU<MatrixType>::image() const |
| { |
| ImageResultType result(m_originalMatrix.rows(), m_rank); |
| computeImage(&result); |
| return result; |
| } |
| |
| template<typename MatrixType> |
| template<typename OtherDerived, typename ResultType> |
| bool LU<MatrixType>::solve( |
| const MatrixBase<OtherDerived>& b, |
| ResultType *result |
| ) 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 = Pb. |
| * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible. |
| * Step 3: compute d such that Ud = c. Check if such d really exists. |
| * Step 4: result = Qd; |
| */ |
| |
| const int rows = m_lu.rows(); |
| ei_assert(b.rows() == rows); |
| const int smalldim = std::min(rows, m_lu.cols()); |
| |
| typename OtherDerived::PlainMatrixType c(b.rows(), b.cols()); |
| |
| // Step 1 |
| for(int i = 0; i < rows; ++i) c.row(m_p.coeff(i)) = b.row(i); |
| |
| // Step 2 |
| Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime, |
| MatrixType::Options, |
| MatrixType::MaxRowsAtCompileTime, |
| MatrixType::MaxRowsAtCompileTime> l(rows, rows); |
| l.setZero(); |
| l.corner(Eigen::TopLeft,rows,smalldim) |
| = m_lu.corner(Eigen::TopLeft,rows,smalldim); |
| l.template marked<UnitLowerTriangular>().solveTriangularInPlace(c); |
| |
| // Step 3 |
| if(!isSurjective()) |
| { |
| // is c is in the image of U ? |
| RealScalar biggest_in_c = c.corner(TopLeft, m_rank, c.cols()).cwise().abs().maxCoeff(); |
| for(int col = 0; col < c.cols(); ++col) |
| for(int row = m_rank; row < c.rows(); ++row) |
| if(!ei_isMuchSmallerThan(c.coeff(row,col), biggest_in_c)) |
| return false; |
| } |
| Matrix<Scalar, Dynamic, OtherDerived::ColsAtCompileTime, |
| MatrixType::Options, |
| MatrixType::MaxRowsAtCompileTime, OtherDerived::MaxColsAtCompileTime> |
| d(c.corner(TopLeft, m_rank, c.cols())); |
| m_lu.corner(TopLeft, m_rank, m_rank) |
| .template marked<UpperTriangular>() |
| .solveTriangularInPlace(d); |
| |
| // Step 4 |
| result->resize(m_lu.cols(), b.cols()); |
| for(int i = 0; i < m_rank; ++i) result->row(m_q.coeff(i)) = d.row(i); |
| for(int i = m_rank; i < m_lu.cols(); ++i) result->row(m_q.coeff(i)).setZero(); |
| return true; |
| } |
| |
| /** \lu_module |
| * |
| * \return the LU decomposition of \c *this. |
| * |
| * \sa class LU |
| */ |
| template<typename Derived> |
| inline const LU<typename MatrixBase<Derived>::PlainMatrixType> |
| MatrixBase<Derived>::lu() const |
| { |
| return LU<PlainMatrixType>(eval()); |
| } |
| |
| #endif // EIGEN_LU_H |
