// 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/.
/** \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 example, 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; friendclass SolverBase<FullPivLU>;
/** * \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()
*/ inlineconst 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 inlineconst PermutationPType& permutationP() const
{
eigen_assert(m_isInitialized && "LU is not initialized."); return m_p;
}
/** \returns the permutation matrix Q * * \sa permutationP()
*/ inlineconst 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()
*/ inlineconst 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()
*/ inlineconst internal::image_retval<FullPivLU>
image(const MatrixType& originalMatrix) const
{
eigen_assert(m_isInitialized && "LU is not initialized."); return internal::image_retval<FullPivLU>(*this, originalMatrix);
}
#ifdef EIGEN_PARSED_BY_DOXYGEN /** \return a solution x to the equation Ax=b, where A is the matrix of which * *this is the LU decomposition. * * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix, * the only requirement in order for the equation to make sense is that * b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition. * * \returns a solution. * * \note_about_checking_solutions * * \note_about_arbitrary_choice_of_solution * \note_about_using_kernel_to_study_multiple_solutions * * Example: \include FullPivLU_solve.cpp * Output: \verbinclude FullPivLU_solve.out * * \sa TriangularView::solve(), kernel(), inverse()
*/ template<typename Rhs> inlineconst Solve<FullPivLU, Rhs>
solve(const MatrixBase<Rhs>& b) const; #endif
/** \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() * RealScalar(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&).
*/ inlinebool 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&).
*/ inlinebool 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&).
*/ inlinebool 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()
*/ inlineconst 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);
}
// 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());
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; typedeftypename 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) = internal::convert_index<StorageIndex>(i);
m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
} break;
}
// 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() **************************************************/
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 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(); const Index smalldim = (std::min)(rows, cols);
// 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(); const Index smalldim = (std::min)(rows, cols);
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