Eigen libary for mbed
Diff: src/QR/ColPivHouseholderQR.h
- Revision:
- 0:13a5d365ba16
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/QR/ColPivHouseholderQR.h Thu Oct 13 04:07:23 2016 +0000
@@ -0,0 +1,580 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
+// Copyright (C) 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_COLPIVOTINGHOUSEHOLDERQR_H
+#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
+
+namespace Eigen {
+
+/** \ingroup QR_Module
+ *
+ * \class ColPivHouseholderQR
+ *
+ * \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
+ *
+ * \param MatrixType the type of the matrix of which we are computing the QR decomposition
+ *
+ * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
+ * such that
+ * \f[
+ * \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
+ * \f]
+ * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
+ * upper triangular matrix.
+ *
+ * This decomposition performs column pivoting in order to be rank-revealing and improve
+ * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
+ *
+ * \sa MatrixBase::colPivHouseholderQr()
+ */
+template<typename _MatrixType> class ColPivHouseholderQR
+{
+ public:
+
+ typedef _MatrixType MatrixType;
+ enum {
+ RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+ ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+ Options = MatrixType::Options,
+ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+ MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
+ };
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename MatrixType::RealScalar RealScalar;
+ typedef typename MatrixType::Index Index;
+ typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
+ typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
+ typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
+ typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
+ typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
+ typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
+ typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
+
+ private:
+
+ typedef typename PermutationType::Index PermIndexType;
+
+ public:
+
+ /**
+ * \brief Default Constructor.
+ *
+ * The default constructor is useful in cases in which the user intends to
+ * perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
+ */
+ ColPivHouseholderQR()
+ : m_qr(),
+ m_hCoeffs(),
+ m_colsPermutation(),
+ m_colsTranspositions(),
+ m_temp(),
+ m_colSqNorms(),
+ m_isInitialized(false),
+ m_usePrescribedThreshold(false) {}
+
+ /** \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 ColPivHouseholderQR()
+ */
+ ColPivHouseholderQR(Index rows, Index cols)
+ : m_qr(rows, cols),
+ m_hCoeffs((std::min)(rows,cols)),
+ m_colsPermutation(PermIndexType(cols)),
+ m_colsTranspositions(cols),
+ m_temp(cols),
+ m_colSqNorms(cols),
+ m_isInitialized(false),
+ m_usePrescribedThreshold(false) {}
+
+ /** \brief Constructs a QR factorization from a given matrix
+ *
+ * This constructor computes the QR factorization of the matrix \a matrix by calling
+ * the method compute(). It is a short cut for:
+ *
+ * \code
+ * ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
+ * qr.compute(matrix);
+ * \endcode
+ *
+ * \sa compute()
+ */
+ ColPivHouseholderQR(const MatrixType& matrix)
+ : m_qr(matrix.rows(), matrix.cols()),
+ m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
+ m_colsPermutation(PermIndexType(matrix.cols())),
+ m_colsTranspositions(matrix.cols()),
+ m_temp(matrix.cols()),
+ m_colSqNorms(matrix.cols()),
+ m_isInitialized(false),
+ m_usePrescribedThreshold(false)
+ {
+ compute(matrix);
+ }
+
+ /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
+ * *this is the QR decomposition, if any exists.
+ *
+ * \param b the right-hand-side of the equation to solve.
+ *
+ * \returns a solution.
+ *
+ * \note The case where b is a matrix is not yet implemented. Also, this
+ * code is space inefficient.
+ *
+ * \note_about_checking_solutions
+ *
+ * \note_about_arbitrary_choice_of_solution
+ *
+ * Example: \include ColPivHouseholderQR_solve.cpp
+ * Output: \verbinclude ColPivHouseholderQR_solve.out
+ */
+ template<typename Rhs>
+ inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
+ solve(const MatrixBase<Rhs>& b) const
+ {
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
+ }
+
+ HouseholderSequenceType householderQ(void) const;
+ HouseholderSequenceType matrixQ(void) const
+ {
+ return householderQ();
+ }
+
+ /** \returns a reference to the matrix where the Householder QR decomposition is stored
+ */
+ const MatrixType& matrixQR() const
+ {
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ return m_qr;
+ }
+
+ /** \returns a reference to the matrix where the result Householder QR is stored
+ * \warning The strict lower part of this matrix contains internal values.
+ * Only the upper triangular part should be referenced. To get it, use
+ * \code matrixR().template triangularView<Upper>() \endcode
+ * For rank-deficient matrices, use
+ * \code
+ * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
+ * \endcode
+ */
+ const MatrixType& matrixR() const
+ {
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ return m_qr;
+ }
+
+ ColPivHouseholderQR& compute(const MatrixType& matrix);
+
+ /** \returns a const reference to the column permutation matrix */
+ const PermutationType& colsPermutation() const
+ {
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ return m_colsPermutation;
+ }
+
+ /** \returns the absolute value of the determinant of the matrix of which
+ * *this is the QR decomposition. It has only linear complexity
+ * (that is, O(n) where n is the dimension of the square matrix)
+ * as the QR decomposition has already been computed.
+ *
+ * \note This is only for square matrices.
+ *
+ * \warning a determinant can be very big or small, so for matrices
+ * of large enough dimension, there is a risk of overflow/underflow.
+ * One way to work around that is to use logAbsDeterminant() instead.
+ *
+ * \sa logAbsDeterminant(), MatrixBase::determinant()
+ */
+ typename MatrixType::RealScalar absDeterminant() const;
+
+ /** \returns the natural log of the absolute value of the determinant of the matrix of which
+ * *this is the QR decomposition. It has only linear complexity
+ * (that is, O(n) where n is the dimension of the square matrix)
+ * as the QR decomposition has already been computed.
+ *
+ * \note This is only for square matrices.
+ *
+ * \note This method is useful to work around the risk of overflow/underflow that's inherent
+ * to determinant computation.
+ *
+ * \sa absDeterminant(), MatrixBase::determinant()
+ */
+ typename MatrixType::RealScalar logAbsDeterminant() const;
+
+ /** \returns the rank of the matrix of which *this is the QR 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 && "ColPivHouseholderQR 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_qr.coeff(i,i)) > premultiplied_threshold);
+ return result;
+ }
+
+ /** \returns the dimension of the kernel of the matrix of which *this is the QR 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 && "ColPivHouseholderQR is not initialized.");
+ return cols() - rank();
+ }
+
+ /** \returns true if the matrix of which *this is the QR 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 && "ColPivHouseholderQR is not initialized.");
+ return rank() == cols();
+ }
+
+ /** \returns true if the matrix of which *this is the QR 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 && "ColPivHouseholderQR is not initialized.");
+ return rank() == rows();
+ }
+
+ /** \returns true if the matrix of which *this is the QR 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 && "ColPivHouseholderQR is not initialized.");
+ return isInjective() && isSurjective();
+ }
+
+ /** \returns the inverse of the matrix of which *this is the QR decomposition.
+ *
+ * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
+ * Use isInvertible() to first determine whether this matrix is invertible.
+ */
+ inline const
+ internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
+ inverse() const
+ {
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ return internal::solve_retval<ColPivHouseholderQR,typename MatrixType::IdentityReturnType>
+ (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
+ }
+
+ inline Index rows() const { return m_qr.rows(); }
+ inline Index cols() const { return m_qr.cols(); }
+
+ /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
+ *
+ * For advanced uses only.
+ */
+ const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
+
+ /** 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
+ * QR 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)
+ */
+ ColPivHouseholderQR& 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 qr.setThreshold(Eigen::Default); \endcode
+ *
+ * See the documentation of setThreshold(const RealScalar&).
+ */
+ ColPivHouseholderQR& 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_qr.diagonalSize());
+ }
+
+ /** \returns the number of nonzero pivots in the QR 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 && "ColPivHouseholderQR is not initialized.");
+ return m_nonzero_pivots;
+ }
+
+ /** \returns the absolute value of the biggest pivot, i.e. the biggest
+ * diagonal coefficient of R.
+ */
+ RealScalar maxPivot() const { return m_maxpivot; }
+
+ /** \brief Reports whether the QR factorization was succesful.
+ *
+ * \note This function always returns \c Success. It is provided for compatibility
+ * with other factorization routines.
+ * \returns \c Success
+ */
+ ComputationInfo info() const
+ {
+ eigen_assert(m_isInitialized && "Decomposition is not initialized.");
+ return Success;
+ }
+
+ protected:
+
+ static void check_template_parameters()
+ {
+ EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
+ }
+
+ MatrixType m_qr;
+ HCoeffsType m_hCoeffs;
+ PermutationType m_colsPermutation;
+ IntRowVectorType m_colsTranspositions;
+ RowVectorType m_temp;
+ RealRowVectorType m_colSqNorms;
+ bool m_isInitialized, m_usePrescribedThreshold;
+ RealScalar m_prescribedThreshold, m_maxpivot;
+ Index m_nonzero_pivots;
+ Index m_det_pq;
+};
+
+template<typename MatrixType>
+typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
+{
+ using std::abs;
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
+ return abs(m_qr.diagonal().prod());
+}
+
+template<typename MatrixType>
+typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
+{
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
+ return m_qr.diagonal().cwiseAbs().array().log().sum();
+}
+
+/** Performs the QR factorization of the given matrix \a matrix. The result of
+ * the factorization is stored into \c *this, and a reference to \c *this
+ * is returned.
+ *
+ * \sa class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
+ */
+template<typename MatrixType>
+ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
+{
+ check_template_parameters();
+
+ using std::abs;
+ Index rows = matrix.rows();
+ Index cols = matrix.cols();
+ Index size = matrix.diagonalSize();
+
+ // the column permutation is stored as int indices, so just to be sure:
+ eigen_assert(cols<=NumTraits<int>::highest());
+
+ m_qr = matrix;
+ m_hCoeffs.resize(size);
+
+ m_temp.resize(cols);
+
+ m_colsTranspositions.resize(matrix.cols());
+ Index number_of_transpositions = 0;
+
+ m_colSqNorms.resize(cols);
+ for(Index k = 0; k < cols; ++k)
+ m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
+
+ RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);
+
+ 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 look up in our table m_colSqNorms which column has the biggest squared norm
+ Index biggest_col_index;
+ RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
+ biggest_col_index += k;
+
+ // since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
+ // the actual squared norm of the selected column.
+ // Note that not doing so does result in solve() sometimes returning inf/nan values
+ // when running the unit test with 1000 repetitions.
+ biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();
+
+ // we store that back into our table: it can't hurt to correct our table.
+ m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;
+
+ // Track the number of meaningful pivots but do not stop the decomposition to make
+ // sure that the initial matrix is properly reproduced. See bug 941.
+ if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
+ m_nonzero_pivots = k;
+
+ // apply the transposition to the columns
+ m_colsTranspositions.coeffRef(k) = biggest_col_index;
+ if(k != biggest_col_index) {
+ m_qr.col(k).swap(m_qr.col(biggest_col_index));
+ std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.coeffRef(biggest_col_index));
+ ++number_of_transpositions;
+ }
+
+ // generate the householder vector, store it below the diagonal
+ RealScalar beta;
+ m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
+
+ // apply the householder transformation to the diagonal coefficient
+ m_qr.coeffRef(k,k) = beta;
+
+ // remember the maximum absolute value of diagonal coefficients
+ if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
+
+ // apply the householder transformation
+ m_qr.bottomRightCorner(rows-k, cols-k-1)
+ .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
+
+ // update our table of squared norms of the columns
+ m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
+ }
+
+ m_colsPermutation.setIdentity(PermIndexType(cols));
+ for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
+ m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
+
+ m_det_pq = (number_of_transpositions%2) ? -1 : 1;
+ m_isInitialized = true;
+
+ return *this;
+}
+
+namespace internal {
+
+template<typename _MatrixType, typename Rhs>
+struct solve_retval<ColPivHouseholderQR<_MatrixType>, Rhs>
+ : solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
+{
+ EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)
+
+ template<typename Dest> void evalTo(Dest& dst) const
+ {
+ eigen_assert(rhs().rows() == dec().rows());
+
+ const Index cols = dec().cols(),
+ nonzero_pivots = dec().nonzeroPivots();
+
+ if(nonzero_pivots == 0)
+ {
+ dst.setZero();
+ return;
+ }
+
+ typename Rhs::PlainObject c(rhs());
+
+ // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
+ c.applyOnTheLeft(householderSequence(dec().matrixQR(), dec().hCoeffs())
+ .setLength(dec().nonzeroPivots())
+ .transpose()
+ );
+
+ dec().matrixR()
+ .topLeftCorner(nonzero_pivots, nonzero_pivots)
+ .template triangularView<Upper>()
+ .solveInPlace(c.topRows(nonzero_pivots));
+
+ for(Index i = 0; i < nonzero_pivots; ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
+ for(Index i = nonzero_pivots; i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
+ }
+};
+
+} // end namespace internal
+
+/** \returns the matrix Q as a sequence of householder transformations.
+ * You can extract the meaningful part only by using:
+ * \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
+template<typename MatrixType>
+typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
+ ::householderQ() const
+{
+ eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+ return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
+}
+
+/** \return the column-pivoting Householder QR decomposition of \c *this.
+ *
+ * \sa class ColPivHouseholderQR
+ */
+template<typename Derived>
+const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
+MatrixBase<Derived>::colPivHouseholderQr() const
+{
+ return ColPivHouseholderQR<PlainObject>(eval());
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
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