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Diffstat (limited to 'extern/Eigen2/Eigen/src/LU/LU.h')
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diff --git a/extern/Eigen2/Eigen/src/LU/LU.h b/extern/Eigen2/Eigen/src/LU/LU.h new file mode 100644 index 00000000000..176e76a91a3 --- /dev/null +++ b/extern/Eigen2/Eigen/src/LU/LU.h @@ -0,0 +1,541 @@ +// 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. + * + * This LU decomposition is very stable and well tested with large matrices. Even exact rank computation + * works at sizes larger than 1000x1000. However there are use cases where the SVD decomposition is inherently + * more stable when dealing with numerically damaged input. For example, computing the kernel is more stable with + * SVD because the SVD can determine which singular values are negligible while LU has to work at the level of matrix + * coefficients that are less meaningful in this respect. + * + * 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_LU.cpp + * Output: \verbinclude class_LU.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 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; + RealScalar m_precision; +}; + +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(); + + // 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. + m_precision = machine_epsilon<Scalar>() * size; + + IntColVectorType rows_transpositions(matrix.rows()); + IntRowVectorType cols_transpositions(matrix.cols()); + int number_of_transpositions = 0; + + RealScalar biggest = RealScalar(0); + m_rank = size; + 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; + if(k==0) biggest = biggest_in_corner; + + // if the corner is negligible, then we have less than full rank, and we can finish early + if(ei_isMuchSmallerThan(biggest_in_corner, biggest, m_precision)) + { + m_rank = k; + for(int i = k; i < size; i++) + { + rows_transpositions.coeffRef(i) = i; + cols_transpositions.coeffRef(i) = i; + } + break; + } + + 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<rows-1) + m_lu.col(k).end(rows-k-1) /= m_lu.coeff(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; +} + +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: replace c by the solution x to Ux = c. Check if a solution really exists. + * Step 4: result = Qc; + */ + + const int rows = m_lu.rows(), cols = m_lu.cols(); + ei_assert(b.rows() == rows); + const int smalldim = std::min(rows, 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 + m_lu.corner(Eigen::TopLeft,smalldim,smalldim).template marked<UnitLowerTriangular>() + .solveTriangularInPlace( + c.corner(Eigen::TopLeft, smalldim, c.cols())); + if(rows>cols) + { + c.corner(Eigen::BottomLeft, rows-cols, c.cols()) + -= m_lu.corner(Eigen::BottomLeft, rows-cols, cols) * c.corner(Eigen::TopLeft, cols, c.cols()); + } + + // Step 3 + if(!isSurjective()) + { + // is c is in the image of U ? + RealScalar biggest_in_c = m_rank>0 ? c.corner(TopLeft, m_rank, c.cols()).cwise().abs().maxCoeff() : 0; + 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, m_precision)) + return false; + } + m_lu.corner(TopLeft, m_rank, m_rank) + .template marked<UpperTriangular>() + .solveTriangularInPlace(c.corner(TopLeft, m_rank, c.cols())); + + // Step 4 + result->resize(m_lu.cols(), b.cols()); + for(int i = 0; i < m_rank; ++i) result->row(m_q.coeff(i)) = c.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 |