summaryrefslogtreecommitdiff
path: root/runtimes/nn/depend/external/eigen/Eigen/src/LU/FullPivLU.h
diff options
context:
space:
mode:
Diffstat (limited to 'runtimes/nn/depend/external/eigen/Eigen/src/LU/FullPivLU.h')
-rw-r--r--runtimes/nn/depend/external/eigen/Eigen/src/LU/FullPivLU.h891
1 files changed, 0 insertions, 891 deletions
diff --git a/runtimes/nn/depend/external/eigen/Eigen/src/LU/FullPivLU.h b/runtimes/nn/depend/external/eigen/Eigen/src/LU/FullPivLU.h
deleted file mode 100644
index 03b6af706..000000000
--- a/runtimes/nn/depend/external/eigen/Eigen/src/LU/FullPivLU.h
+++ /dev/null
@@ -1,891 +0,0 @@
-// This file is part of Eigen, a lightweight C++ template library
-// for linear algebra.
-//
-// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
-//
-// This Source Code Form is subject to the terms of the Mozilla
-// Public License v. 2.0. If a copy of the MPL was not distributed
-// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
-
-#ifndef EIGEN_LU_H
-#define EIGEN_LU_H
-
-namespace Eigen {
-
-namespace internal {
-template<typename _MatrixType> struct traits<FullPivLU<_MatrixType> >
- : traits<_MatrixType>
-{
- typedef MatrixXpr XprKind;
- typedef SolverStorage StorageKind;
- enum { Flags = 0 };
-};
-
-} // end namespace internal
-
-/** \ingroup LU_Module
- *
- * \class FullPivLU
- *
- * \brief LU decomposition of a matrix with complete pivoting, and related features
- *
- * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition
- *
- * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is
- * decomposed as \f$ A = P^{-1} L U Q^{-1} \f$ where L is unit-lower-triangular, U is
- * upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU
- * decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any
- * zeros are at the end.
- *
- * This decomposition provides the generic approach to solving systems of linear equations, computing
- * the rank, invertibility, inverse, kernel, and determinant.
- *
- * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD
- * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix,
- * working with the SVD allows to select the smallest singular values of the matrix, something that
- * the LU decomposition doesn't see.
- *
- * The data of the LU decomposition can be directly accessed through the methods matrixLU(),
- * permutationP(), permutationQ().
- *
- * As an exemple, here is how the original matrix can be retrieved:
- * \include class_FullPivLU.cpp
- * Output: \verbinclude class_FullPivLU.out
- *
- * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
- *
- * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()
- */
-template<typename _MatrixType> class FullPivLU
- : public SolverBase<FullPivLU<_MatrixType> >
-{
- public:
- typedef _MatrixType MatrixType;
- typedef SolverBase<FullPivLU> Base;
-
- EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU)
- // FIXME StorageIndex defined in EIGEN_GENERIC_PUBLIC_INTERFACE should be int
- enum {
- MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
- MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
- };
- typedef typename internal::plain_row_type<MatrixType, StorageIndex>::type IntRowVectorType;
- typedef typename internal::plain_col_type<MatrixType, StorageIndex>::type IntColVectorType;
- typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
- typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;
- typedef typename MatrixType::PlainObject PlainObject;
-
- /**
- * \brief Default Constructor.
- *
- * The default constructor is useful in cases in which the user intends to
- * perform decompositions via LU::compute(const MatrixType&).
- */
- FullPivLU();
-
- /** \brief Default Constructor with memory preallocation
- *
- * Like the default constructor but with preallocation of the internal data
- * according to the specified problem \a size.
- * \sa FullPivLU()
- */
- FullPivLU(Index rows, Index cols);
-
- /** Constructor.
- *
- * \param matrix the matrix of which to compute the LU decomposition.
- * It is required to be nonzero.
- */
- template<typename InputType>
- explicit FullPivLU(const EigenBase<InputType>& matrix);
-
- /** \brief Constructs a LU factorization from a given matrix
- *
- * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
- *
- * \sa FullPivLU(const EigenBase&)
- */
- template<typename InputType>
- explicit FullPivLU(EigenBase<InputType>& matrix);
-
- /** Computes the LU decomposition of the given matrix.
- *
- * \param matrix the matrix of which to compute the LU decomposition.
- * It is required to be nonzero.
- *
- * \returns a reference to *this
- */
- template<typename InputType>
- FullPivLU& compute(const EigenBase<InputType>& matrix) {
- m_lu = matrix.derived();
- computeInPlace();
- return *this;
- }
-
- /** \returns the LU decomposition matrix: the upper-triangular part is U, the
- * unit-lower-triangular part is L (at least for square matrices; in the non-square
- * case, special care is needed, see the documentation of class FullPivLU).
- *
- * \sa matrixL(), matrixU()
- */
- inline const MatrixType& matrixLU() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return m_lu;
- }
-
- /** \returns the number of nonzero pivots in the LU decomposition.
- * Here nonzero is meant in the exact sense, not in a fuzzy sense.
- * So that notion isn't really intrinsically interesting, but it is
- * still useful when implementing algorithms.
- *
- * \sa rank()
- */
- inline Index nonzeroPivots() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return m_nonzero_pivots;
- }
-
- /** \returns the absolute value of the biggest pivot, i.e. the biggest
- * diagonal coefficient of U.
- */
- RealScalar maxPivot() const { return m_maxpivot; }
-
- /** \returns the permutation matrix P
- *
- * \sa permutationQ()
- */
- EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return m_p;
- }
-
- /** \returns the permutation matrix Q
- *
- * \sa permutationP()
- */
- inline const PermutationQType& permutationQ() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return m_q;
- }
-
- /** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
- * will form a basis of the kernel.
- *
- * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- *
- * Example: \include FullPivLU_kernel.cpp
- * Output: \verbinclude FullPivLU_kernel.out
- *
- * \sa image()
- */
- inline const internal::kernel_retval<FullPivLU> kernel() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return internal::kernel_retval<FullPivLU>(*this);
- }
-
- /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
- * will form a basis of the image (column-space).
- *
- * \param originalMatrix the original matrix, of which *this is the LU decomposition.
- * The reason why it is needed to pass it here, is that this allows
- * a large optimization, as otherwise this method would need to reconstruct it
- * from the LU decomposition.
- *
- * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- *
- * Example: \include FullPivLU_image.cpp
- * Output: \verbinclude FullPivLU_image.out
- *
- * \sa kernel()
- */
- inline const internal::image_retval<FullPivLU>
- image(const MatrixType& originalMatrix) const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return internal::image_retval<FullPivLU>(*this, originalMatrix);
- }
-
- /** \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()
- */
- // FIXME this is a copy-paste of the base-class member to add the isInitialized assertion.
- template<typename Rhs>
- inline const Solve<FullPivLU, Rhs>
- solve(const MatrixBase<Rhs>& b) const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return Solve<FullPivLU, Rhs>(*this, b.derived());
- }
-
- /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
- the LU decomposition.
- */
- inline RealScalar rcond() const
- {
- eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
- return internal::rcond_estimate_helper(m_l1_norm, *this);
- }
-
- /** \returns the determinant of the matrix of which
- * *this is the LU decomposition. It has only linear complexity
- * (that is, O(n) where n is the dimension of the square matrix)
- * as the LU decomposition has already been computed.
- *
- * \note This is only for square matrices.
- *
- * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
- * optimized paths.
- *
- * \warning a determinant can be very big or small, so for matrices
- * of large enough dimension, there is a risk of overflow/underflow.
- *
- * \sa MatrixBase::determinant()
- */
- typename internal::traits<MatrixType>::Scalar determinant() const;
-
- /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
- * who need to determine when pivots are to be considered nonzero. This is not used for the
- * LU decomposition itself.
- *
- * When it needs to get the threshold value, Eigen calls threshold(). By default, this
- * uses a formula to automatically determine a reasonable threshold.
- * Once you have called the present method setThreshold(const RealScalar&),
- * your value is used instead.
- *
- * \param threshold The new value to use as the threshold.
- *
- * A pivot will be considered nonzero if its absolute value is strictly greater than
- * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
- * where maxpivot is the biggest pivot.
- *
- * If you want to come back to the default behavior, call setThreshold(Default_t)
- */
- FullPivLU& setThreshold(const RealScalar& threshold)
- {
- m_usePrescribedThreshold = true;
- m_prescribedThreshold = threshold;
- return *this;
- }
-
- /** Allows to come back to the default behavior, letting Eigen use its default formula for
- * determining the threshold.
- *
- * You should pass the special object Eigen::Default as parameter here.
- * \code lu.setThreshold(Eigen::Default); \endcode
- *
- * See the documentation of setThreshold(const RealScalar&).
- */
- FullPivLU& setThreshold(Default_t)
- {
- m_usePrescribedThreshold = false;
- return *this;
- }
-
- /** Returns the threshold that will be used by certain methods such as rank().
- *
- * See the documentation of setThreshold(const RealScalar&).
- */
- RealScalar threshold() const
- {
- eigen_assert(m_isInitialized || m_usePrescribedThreshold);
- return m_usePrescribedThreshold ? m_prescribedThreshold
- // this formula comes from experimenting (see "LU precision tuning" thread on the list)
- // and turns out to be identical to Higham's formula used already in LDLt.
- : NumTraits<Scalar>::epsilon() * m_lu.diagonalSize();
- }
-
- /** \returns the rank of the matrix of which *this is the LU decomposition.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline Index rank() const
- {
- using std::abs;
- eigen_assert(m_isInitialized && "LU is not initialized.");
- RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
- Index result = 0;
- for(Index i = 0; i < m_nonzero_pivots; ++i)
- result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold);
- return result;
- }
-
- /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline Index dimensionOfKernel() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return cols() - rank();
- }
-
- /** \returns true if the matrix of which *this is the LU decomposition represents an injective
- * linear map, i.e. has trivial kernel; false otherwise.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline bool isInjective() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return rank() == cols();
- }
-
- /** \returns true if the matrix of which *this is the LU decomposition represents a surjective
- * linear map; false otherwise.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline bool isSurjective() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return rank() == rows();
- }
-
- /** \returns true if the matrix of which *this is the LU decomposition is invertible.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline bool isInvertible() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return isInjective() && (m_lu.rows() == m_lu.cols());
- }
-
- /** \returns the inverse of the matrix of which *this is the LU decomposition.
- *
- * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
- * Use isInvertible() to first determine whether this matrix is invertible.
- *
- * \sa MatrixBase::inverse()
- */
- inline const Inverse<FullPivLU> inverse() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
- return Inverse<FullPivLU>(*this);
- }
-
- MatrixType reconstructedMatrix() const;
-
- EIGEN_DEVICE_FUNC inline Index rows() const { return m_lu.rows(); }
- EIGEN_DEVICE_FUNC inline Index cols() const { return m_lu.cols(); }
-
- #ifndef EIGEN_PARSED_BY_DOXYGEN
- template<typename RhsType, typename DstType>
- EIGEN_DEVICE_FUNC
- void _solve_impl(const RhsType &rhs, DstType &dst) const;
-
- template<bool Conjugate, typename RhsType, typename DstType>
- EIGEN_DEVICE_FUNC
- void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
- #endif
-
- protected:
-
- static void check_template_parameters()
- {
- EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
- }
-
- void computeInPlace();
-
- MatrixType m_lu;
- PermutationPType m_p;
- PermutationQType m_q;
- IntColVectorType m_rowsTranspositions;
- IntRowVectorType m_colsTranspositions;
- Index m_nonzero_pivots;
- RealScalar m_l1_norm;
- RealScalar m_maxpivot, m_prescribedThreshold;
- signed char m_det_pq;
- bool m_isInitialized, m_usePrescribedThreshold;
-};
-
-template<typename MatrixType>
-FullPivLU<MatrixType>::FullPivLU()
- : m_isInitialized(false), m_usePrescribedThreshold(false)
-{
-}
-
-template<typename MatrixType>
-FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols)
- : m_lu(rows, cols),
- m_p(rows),
- m_q(cols),
- m_rowsTranspositions(rows),
- m_colsTranspositions(cols),
- m_isInitialized(false),
- m_usePrescribedThreshold(false)
-{
-}
-
-template<typename MatrixType>
-template<typename InputType>
-FullPivLU<MatrixType>::FullPivLU(const EigenBase<InputType>& matrix)
- : m_lu(matrix.rows(), matrix.cols()),
- m_p(matrix.rows()),
- m_q(matrix.cols()),
- m_rowsTranspositions(matrix.rows()),
- m_colsTranspositions(matrix.cols()),
- m_isInitialized(false),
- m_usePrescribedThreshold(false)
-{
- compute(matrix.derived());
-}
-
-template<typename MatrixType>
-template<typename InputType>
-FullPivLU<MatrixType>::FullPivLU(EigenBase<InputType>& matrix)
- : m_lu(matrix.derived()),
- m_p(matrix.rows()),
- m_q(matrix.cols()),
- m_rowsTranspositions(matrix.rows()),
- m_colsTranspositions(matrix.cols()),
- m_isInitialized(false),
- m_usePrescribedThreshold(false)
-{
- computeInPlace();
-}
-
-template<typename MatrixType>
-void FullPivLU<MatrixType>::computeInPlace()
-{
- check_template_parameters();
-
- // the permutations are stored as int indices, so just to be sure:
- eigen_assert(m_lu.rows()<=NumTraits<int>::highest() && m_lu.cols()<=NumTraits<int>::highest());
-
- m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
-
- const Index size = m_lu.diagonalSize();
- const Index rows = m_lu.rows();
- const Index cols = m_lu.cols();
-
- // will store the transpositions, before we accumulate them at the end.
- // can't accumulate on-the-fly because that will be done in reverse order for the rows.
- m_rowsTranspositions.resize(m_lu.rows());
- m_colsTranspositions.resize(m_lu.cols());
- Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i
-
- m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
- m_maxpivot = RealScalar(0);
-
- for(Index k = 0; k < size; ++k)
- {
- // First, we need to find the pivot.
-
- // biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
- Index row_of_biggest_in_corner, col_of_biggest_in_corner;
- typedef internal::scalar_score_coeff_op<Scalar> Scoring;
- typedef typename Scoring::result_type Score;
- Score biggest_in_corner;
- biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
- .unaryExpr(Scoring())
- .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
- row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
- col_of_biggest_in_corner += k; // need to add k to them.
-
- if(biggest_in_corner==Score(0))
- {
- // before exiting, make sure to initialize the still uninitialized transpositions
- // in a sane state without destroying what we already have.
- m_nonzero_pivots = k;
- for(Index i = k; i < size; ++i)
- {
- m_rowsTranspositions.coeffRef(i) = i;
- m_colsTranspositions.coeffRef(i) = i;
- }
- break;
- }
-
- RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner);
- if(abs_pivot > m_maxpivot) m_maxpivot = abs_pivot;
-
- // Now that we've found the pivot, we need to apply the row/col swaps to
- // bring it to the location (k,k).
-
- m_rowsTranspositions.coeffRef(k) = 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<rows-1)
- m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
- if(k<size-1)
- m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1);
- }
-
- // the main loop is over, we still have to accumulate the transpositions to find the
- // permutations P and Q
-
- m_p.setIdentity(rows);
- for(Index k = size-1; k >= 0; --k)
- m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
-
- m_q.setIdentity(cols);
- for(Index k = 0; k < size; ++k)
- m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
-
- m_det_pq = (number_of_transpositions%2) ? -1 : 1;
-
- m_isInitialized = true;
-}
-
-template<typename MatrixType>
-typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const
-{
- eigen_assert(m_isInitialized && "LU is not initialized.");
- eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
- return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
-}
-
-/** \returns the matrix represented by the decomposition,
- * i.e., it returns the product: \f$ P^{-1} L U Q^{-1} \f$.
- * This function is provided for debug purposes. */
-template<typename MatrixType>
-MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const
-{
- eigen_assert(m_isInitialized && "LU is not initialized.");
- const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
- // LU
- MatrixType res(m_lu.rows(),m_lu.cols());
- // FIXME the .toDenseMatrix() should not be needed...
- res = m_lu.leftCols(smalldim)
- .template triangularView<UnitLower>().toDenseMatrix()
- * m_lu.topRows(smalldim)
- .template triangularView<Upper>().toDenseMatrix();
-
- // P^{-1}(LU)
- res = m_p.inverse() * res;
-
- // (P^{-1}LU)Q^{-1}
- res = res * m_q.inverse();
-
- return res;
-}
-
-/********* Implementation of kernel() **************************************************/
-
-namespace internal {
-template<typename _MatrixType>
-struct kernel_retval<FullPivLU<_MatrixType> >
- : kernel_retval_base<FullPivLU<_MatrixType> >
-{
- EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)
-
- enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
- MatrixType::MaxColsAtCompileTime,
- MatrixType::MaxRowsAtCompileTime)
- };
-
- template<typename Dest> void evalTo(Dest& dst) const
- {
- using std::abs;
- const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
- if(dimker == 0)
- {
- // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
- // avoid crashing/asserting as that depends on floating point calculations. Let's
- // just return a single column vector filled with zeros.
- dst.setZero();
- return;
- }
-
- /* Let us use the following lemma:
- *
- * Lemma: If the matrix A has the LU decomposition PAQ = LU,
- * then Ker A = Q(Ker U).
- *
- * Proof: trivial: just keep in mind that P, Q, L are invertible.
- */
-
- /* Thus, all we need to do is to compute Ker U, and then apply Q.
- *
- * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
- * Thus, the diagonal of U ends with exactly
- * dimKer zero's. Let us use that to construct dimKer linearly
- * independent vectors in Ker U.
- */
-
- Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
- RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
- Index p = 0;
- for(Index i = 0; i < dec().nonzeroPivots(); ++i)
- if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
- pivots.coeffRef(p++) = i;
- eigen_internal_assert(p == rank());
-
- // we construct a temporaty trapezoid matrix m, by taking the U matrix and
- // permuting the rows and cols to bring the nonnegligible pivots to the top of
- // the main diagonal. We need that to be able to apply our triangular solvers.
- // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified
- Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
- MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
- m(dec().matrixLU().block(0, 0, rank(), cols));
- for(Index i = 0; i < rank(); ++i)
- {
- if(i) m.row(i).head(i).setZero();
- m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
- }
- m.block(0, 0, rank(), rank());
- m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
- for(Index i = 0; i < rank(); ++i)
- m.col(i).swap(m.col(pivots.coeff(i)));
-
- // ok, we have our trapezoid matrix, we can apply the triangular solver.
- // notice that the math behind this suggests that we should apply this to the
- // negative of the RHS, but for performance we just put the negative sign elsewhere, see below.
- m.topLeftCorner(rank(), rank())
- .template triangularView<Upper>().solveInPlace(
- m.topRightCorner(rank(), dimker)
- );
-
- // now we must undo the column permutation that we had applied!
- for(Index i = rank()-1; i >= 0; --i)
- m.col(i).swap(m.col(pivots.coeff(i)));
-
- // see the negative sign in the next line, that's what we were talking about above.
- for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
- for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
- for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
- }
-};
-
-/***** Implementation of image() *****************************************************/
-
-template<typename _MatrixType>
-struct image_retval<FullPivLU<_MatrixType> >
- : image_retval_base<FullPivLU<_MatrixType> >
-{
- EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)
-
- enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
- MatrixType::MaxColsAtCompileTime,
- MatrixType::MaxRowsAtCompileTime)
- };
-
- template<typename Dest> void evalTo(Dest& dst) const
- {
- using std::abs;
- if(rank() == 0)
- {
- // The Image is just {0}, so it doesn't have a basis properly speaking, but let's
- // avoid crashing/asserting as that depends on floating point calculations. Let's
- // just return a single column vector filled with zeros.
- dst.setZero();
- return;
- }
-
- Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
- RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
- Index p = 0;
- for(Index i = 0; i < dec().nonzeroPivots(); ++i)
- if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
- pivots.coeffRef(p++) = i;
- eigen_internal_assert(p == rank());
-
- for(Index i = 0; i < rank(); ++i)
- dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
- }
-};
-
-/***** Implementation of solve() *****************************************************/
-
-} // end namespace internal
-
-#ifndef EIGEN_PARSED_BY_DOXYGEN
-template<typename _MatrixType>
-template<typename RhsType, typename DstType>
-void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
-{
- /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
- * So we proceed as follows:
- * Step 1: compute c = P * rhs.
- * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
- * Step 3: replace c by the solution x to Ux = c. May or may not exist.
- * Step 4: result = Q * c;
- */
-
- const Index rows = this->rows(),
- cols = this->cols(),
- nonzero_pivots = this->rank();
- eigen_assert(rhs.rows() == rows);
- const Index smalldim = (std::min)(rows, cols);
-
- if(nonzero_pivots == 0)
- {
- dst.setZero();
- return;
- }
-
- typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
-
- // Step 1
- c = permutationP() * rhs;
-
- // Step 2
- m_lu.topLeftCorner(smalldim,smalldim)
- .template triangularView<UnitLower>()
- .solveInPlace(c.topRows(smalldim));
- if(rows>cols)
- c.bottomRows(rows-cols) -= m_lu.bottomRows(rows-cols) * c.topRows(cols);
-
- // Step 3
- m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
- .template triangularView<Upper>()
- .solveInPlace(c.topRows(nonzero_pivots));
-
- // Step 4
- for(Index i = 0; i < nonzero_pivots; ++i)
- dst.row(permutationQ().indices().coeff(i)) = c.row(i);
- for(Index i = nonzero_pivots; i < m_lu.cols(); ++i)
- dst.row(permutationQ().indices().coeff(i)).setZero();
-}
-
-template<typename _MatrixType>
-template<bool Conjugate, typename RhsType, typename DstType>
-void FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
-{
- /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1},
- * and since permutations are real and unitary, we can write this
- * as A^T = Q U^T L^T P,
- * So we proceed as follows:
- * Step 1: compute c = Q^T rhs.
- * Step 2: replace c by the solution x to U^T x = c. May or may not exist.
- * Step 3: replace c by the solution x to L^T x = c.
- * Step 4: result = P^T c.
- * If Conjugate is true, replace "^T" by "^*" above.
- */
-
- const Index rows = this->rows(), cols = this->cols(),
- nonzero_pivots = this->rank();
- eigen_assert(rhs.rows() == cols);
- const Index smalldim = (std::min)(rows, cols);
-
- if(nonzero_pivots == 0)
- {
- dst.setZero();
- return;
- }
-
- typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
-
- // Step 1
- c = permutationQ().inverse() * rhs;
-
- if (Conjugate) {
- // Step 2
- m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
- .template triangularView<Upper>()
- .adjoint()
- .solveInPlace(c.topRows(nonzero_pivots));
- // Step 3
- m_lu.topLeftCorner(smalldim, smalldim)
- .template triangularView<UnitLower>()
- .adjoint()
- .solveInPlace(c.topRows(smalldim));
- } else {
- // Step 2
- m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
- .template triangularView<Upper>()
- .transpose()
- .solveInPlace(c.topRows(nonzero_pivots));
- // Step 3
- m_lu.topLeftCorner(smalldim, smalldim)
- .template triangularView<UnitLower>()
- .transpose()
- .solveInPlace(c.topRows(smalldim));
- }
-
- // Step 4
- PermutationPType invp = permutationP().inverse().eval();
- for(Index i = 0; i < smalldim; ++i)
- dst.row(invp.indices().coeff(i)) = c.row(i);
- for(Index i = smalldim; i < rows; ++i)
- dst.row(invp.indices().coeff(i)).setZero();
-}
-
-#endif
-
-namespace internal {
-
-
-/***** Implementation of inverse() *****************************************************/
-template<typename DstXprType, typename MatrixType>
-struct Assignment<DstXprType, Inverse<FullPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename FullPivLU<MatrixType>::Scalar>, Dense2Dense>
-{
- typedef FullPivLU<MatrixType> LuType;
- typedef Inverse<LuType> SrcXprType;
- static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename MatrixType::Scalar> &)
- {
- dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
- }
-};
-} // end namespace internal
-
-/******* MatrixBase methods *****************************************************************/
-
-/** \lu_module
- *
- * \return the full-pivoting LU decomposition of \c *this.
- *
- * \sa class FullPivLU
- */
-template<typename Derived>
-inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
-MatrixBase<Derived>::fullPivLu() const
-{
- return FullPivLU<PlainObject>(eval());
-}
-
-} // end namespace Eigen
-
-#endif // EIGEN_LU_H