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CompleteOrthogonalDecomposition.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2016 Rasmus Munk Larsen <rmlarsen@google.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_COMPLETEORTHOGONALDECOMPOSITION_H
#define EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H
namespace Eigen {
namespace internal {
template <typename _MatrixType>
struct traits<CompleteOrthogonalDecomposition<_MatrixType> >
: traits<_MatrixType> {
typedef MatrixXpr XprKind;
typedef SolverStorage StorageKind;
typedef int StorageIndex;
enum { Flags = 0 };
};
} // end namespace internal
/** \ingroup QR_Module
*
* \class CompleteOrthogonalDecomposition
*
* \brief Complete orthogonal decomposition (COD) of a matrix.
*
* \param MatrixType the type of the matrix of which we are computing the COD.
*
* This class performs a rank-revealing complete orthogonal decomposition of a
* matrix \b A into matrices \b P, \b Q, \b T, and \b Z such that
* \f[
* \mathbf{A} \, \mathbf{P} = \mathbf{Q} \,
* \begin{bmatrix} \mathbf{T} & \mathbf{0} \\
* \mathbf{0} & \mathbf{0} \end{bmatrix} \, \mathbf{Z}
* \f]
* by using Householder transformations. Here, \b P is a permutation matrix,
* \b Q and \b Z are unitary matrices and \b T an upper triangular matrix of
* size rank-by-rank. \b A may be rank deficient.
*
* This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
*
* \sa MatrixBase::completeOrthogonalDecomposition()
*/
template <typename _MatrixType> class CompleteOrthogonalDecomposition
: public SolverBase<CompleteOrthogonalDecomposition<_MatrixType> >
{
public:
typedef _MatrixType MatrixType;
typedef SolverBase<CompleteOrthogonalDecomposition> Base;
template<typename Derived>
friend struct internal::solve_assertion;
EIGEN_GENERIC_PUBLIC_INTERFACE(CompleteOrthogonalDecomposition)
enum {
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
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;
typedef typename MatrixType::PlainObject PlainObject;
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
* \c CompleteOrthogonalDecomposition::compute(const* MatrixType&).
*/
CompleteOrthogonalDecomposition() : m_cpqr(), m_zCoeffs(), m_temp() {}
/** \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 CompleteOrthogonalDecomposition()
*/
CompleteOrthogonalDecomposition(Index rows, Index cols)
: m_cpqr(rows, cols), m_zCoeffs((std::min)(rows, cols)), m_temp(cols) {}
/** \brief Constructs a complete orthogonal decomposition from a given
* matrix.
*
* This constructor computes the complete orthogonal decomposition of the
* matrix \a matrix by calling the method compute(). The default
* threshold for rank determination will be used. It is a short cut for:
*
* \code
* CompleteOrthogonalDecomposition<MatrixType> cod(matrix.rows(),
* matrix.cols());
* cod.setThreshold(Default);
* cod.compute(matrix);
* \endcode
*
* \sa compute()
*/
template <typename InputType>
explicit CompleteOrthogonalDecomposition(const EigenBase<InputType>& matrix)
: m_cpqr(matrix.rows(), matrix.cols()),
m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
m_temp(matrix.cols())
{
compute(matrix.derived());
}
/** \brief Constructs a complete orthogonal decomposition from a given matrix
*
* This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
*
* \sa CompleteOrthogonalDecomposition(const EigenBase&)
*/
template<typename InputType>
explicit CompleteOrthogonalDecomposition(EigenBase<InputType>& matrix)
: m_cpqr(matrix.derived()),
m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
m_temp(matrix.cols())
{
computeInPlace();
}
#ifdef EIGEN_PARSED_BY_DOXYGEN
/** This method computes the minimum-norm solution X to a least squares
* problem \f[\mathrm{minimize} \|A X - B\|, \f] where \b A is the matrix of
* which \c *this is the complete orthogonal decomposition.
*
* \param b the right-hand sides of the problem to solve.
*
* \returns a solution.
*
*/
template <typename Rhs>
inline const Solve<CompleteOrthogonalDecomposition, Rhs> solve(
const MatrixBase<Rhs>& b) const;
#endif
HouseholderSequenceType householderQ(void) const;
HouseholderSequenceType matrixQ(void) const { return m_cpqr.householderQ(); }
/** \returns the matrix \b Z.
*/
MatrixType matrixZ() const {
MatrixType Z = MatrixType::Identity(m_cpqr.cols(), m_cpqr.cols());
applyZOnTheLeftInPlace<false>(Z);
return Z;
}
/** \returns a reference to the matrix where the complete orthogonal
* decomposition is stored
*/
const MatrixType& matrixQTZ() const { return m_cpqr.matrixQR(); }
/** \returns a reference to the matrix where the complete orthogonal
* decomposition is stored.
* \warning The strict lower part and \code cols() - rank() \endcode right
* columns of this matrix contains internal values.
* Only the upper triangular part should be referenced. To get it, use
* \code matrixT().template triangularView<Upper>() \endcode
* For rank-deficient matrices, use
* \code
* matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
* \endcode
*/
const MatrixType& matrixT() const { return m_cpqr.matrixQR(); }
template <typename InputType>
CompleteOrthogonalDecomposition& compute(const EigenBase<InputType>& matrix) {
// Compute the column pivoted QR factorization A P = Q R.
m_cpqr.compute(matrix);
computeInPlace();
return *this;
}
/** \returns a const reference to the column permutation matrix */
const PermutationType& colsPermutation() const {
return m_cpqr.colsPermutation();
}
/** \returns the absolute value of the determinant of the matrix of which
* *this is the complete orthogonal decomposition. It has only linear
* complexity (that is, O(n) where n is the dimension of the square matrix)
* as the complete orthogonal 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 complete orthogonal decomposition. It has
* only linear complexity (that is, O(n) where n is the dimension of the
* square matrix) as the complete orthogonal 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 complete orthogonal
* 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 { return m_cpqr.rank(); }
/** \returns the dimension of the kernel of the matrix of which *this is the
* complete orthogonal 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 { return m_cpqr.dimensionOfKernel(); }
/** \returns true if the matrix of which *this is the 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 { return m_cpqr.isInjective(); }
/** \returns true if the matrix of which *this is the 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 { return m_cpqr.isSurjective(); }
/** \returns true if the matrix of which *this is the complete orthogonal
* 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 { return m_cpqr.isInvertible(); }
/** \returns the pseudo-inverse of the matrix of which *this is the complete
* orthogonal decomposition.
* \warning: Do not compute \c this->pseudoInverse()*rhs to solve a linear systems.
* It is more efficient and numerically stable to call \c this->solve(rhs).
*/
inline const Inverse<CompleteOrthogonalDecomposition> pseudoInverse() const
{
eigen_assert(m_cpqr.m_isInitialized && "CompleteOrthogonalDecomposition is not initialized.");
return Inverse<CompleteOrthogonalDecomposition>(*this);
}
inline Index rows() const { return m_cpqr.rows(); }
inline Index cols() const { return m_cpqr.cols(); }
/** \returns a const reference to the vector of Householder coefficients used
* to represent the factor \c Q.
*
* For advanced uses only.
*/
inline const HCoeffsType& hCoeffs() const { return m_cpqr.hCoeffs(); }
/** \returns a const reference to the vector of Householder coefficients
* used to represent the factor \c Z.
*
* For advanced uses only.
*/
const HCoeffsType& zCoeffs() const { return m_zCoeffs; }
/** 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.
* Most be called before calling compute().
*
* 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)
*/
CompleteOrthogonalDecomposition& setThreshold(const RealScalar& threshold) {
m_cpqr.setThreshold(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&).
*/
CompleteOrthogonalDecomposition& setThreshold(Default_t) {
m_cpqr.setThreshold(Default);
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 { return m_cpqr.threshold(); }
/** \returns the number of nonzero pivots in the complete orthogonal
* 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 { return m_cpqr.nonzeroPivots(); }
/** \returns the absolute value of the biggest pivot, i.e. the biggest
* diagonal coefficient of R.
*/
inline RealScalar maxPivot() const { return m_cpqr.maxPivot(); }
/** \brief Reports whether the complete orthogonal decomposition was
* successful.
*
* \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_cpqr.m_isInitialized && "Decomposition is not initialized.");
return Success;
}
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename RhsType, typename DstType>
void _solve_impl(const RhsType& rhs, DstType& dst) const;
template<bool Conjugate, typename RhsType, typename DstType>
void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
#endif
protected:
static void check_template_parameters() {
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
}
template<bool Transpose_, typename Rhs>
void _check_solve_assertion(const Rhs& b) const {
EIGEN_ONLY_USED_FOR_DEBUG(b);
eigen_assert(m_cpqr.m_isInitialized && "CompleteOrthogonalDecomposition is not initialized.");
eigen_assert((Transpose_?derived().cols():derived().rows())==b.rows() && "CompleteOrthogonalDecomposition::solve(): invalid number of rows of the right hand side matrix b");
}
void computeInPlace();
/** Overwrites \b rhs with \f$ \mathbf{Z} * \mathbf{rhs} \f$ or
* \f$ \mathbf{\overline Z} * \mathbf{rhs} \f$ if \c Conjugate
* is set to \c true.
*/
template <bool Conjugate, typename Rhs>
void applyZOnTheLeftInPlace(Rhs& rhs) const;
/** Overwrites \b rhs with \f$ \mathbf{Z}^* * \mathbf{rhs} \f$.
*/
template <typename Rhs>
void applyZAdjointOnTheLeftInPlace(Rhs& rhs) const;
ColPivHouseholderQR<MatrixType> m_cpqr;
HCoeffsType m_zCoeffs;
RowVectorType m_temp;
};
template <typename MatrixType>
typename MatrixType::RealScalar
CompleteOrthogonalDecomposition<MatrixType>::absDeterminant() const {
return m_cpqr.absDeterminant();
}
template <typename MatrixType>
typename MatrixType::RealScalar
CompleteOrthogonalDecomposition<MatrixType>::logAbsDeterminant() const {
return m_cpqr.logAbsDeterminant();
}
/** Performs the complete orthogonal decomposition 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 CompleteOrthogonalDecomposition,
* CompleteOrthogonalDecomposition(const MatrixType&)
*/
template <typename MatrixType>
void CompleteOrthogonalDecomposition<MatrixType>::computeInPlace()
{
check_template_parameters();
// the column permutation is stored as int indices, so just to be sure:
eigen_assert(m_cpqr.cols() <= NumTraits<int>::highest());
const Index rank = m_cpqr.rank();
const Index cols = m_cpqr.cols();
const Index rows = m_cpqr.rows();
m_zCoeffs.resize((std::min)(rows, cols));
m_temp.resize(cols);
if (rank < cols) {
// We have reduced the (permuted) matrix to the form
// [R11 R12]
// [ 0 R22]
// where R11 is r-by-r (r = rank) upper triangular, R12 is
// r-by-(n-r), and R22 is empty or the norm of R22 is negligible.
// We now compute the complete orthogonal decomposition by applying
// Householder transformations from the right to the upper trapezoidal
// matrix X = [R11 R12] to zero out R12 and obtain the factorization
// [R11 R12] = [T11 0] * Z, where T11 is r-by-r upper triangular and
// Z = Z(0) * Z(1) ... Z(r-1) is an n-by-n orthogonal matrix.
// We store the data representing Z in R12 and m_zCoeffs.
for (Index k = rank - 1; k >= 0; --k) {
if (k != rank - 1) {
// Given the API for Householder reflectors, it is more convenient if
// we swap the leading parts of columns k and r-1 (zero-based) to form
// the matrix X_k = [X(0:k, k), X(0:k, r:n)]
m_cpqr.m_qr.col(k).head(k + 1).swap(
m_cpqr.m_qr.col(rank - 1).head(k + 1));
}
// Construct Householder reflector Z(k) to zero out the last row of X_k,
// i.e. choose Z(k) such that
// [X(k, k), X(k, r:n)] * Z(k) = [beta, 0, .., 0].
RealScalar beta;
m_cpqr.m_qr.row(k)
.tail(cols - rank + 1)
.makeHouseholderInPlace(m_zCoeffs(k), beta);
m_cpqr.m_qr(k, rank - 1) = beta;
if (k > 0) {
// Apply Z(k) to the first k rows of X_k
m_cpqr.m_qr.topRightCorner(k, cols - rank + 1)
.applyHouseholderOnTheRight(
m_cpqr.m_qr.row(k).tail(cols - rank).adjoint(), m_zCoeffs(k),
&m_temp(0));
}
if (k != rank - 1) {
// Swap X(0:k,k) back to its proper location.
m_cpqr.m_qr.col(k).head(k + 1).swap(
m_cpqr.m_qr.col(rank - 1).head(k + 1));
}
}
}
}
template <typename MatrixType>
template <bool Conjugate, typename Rhs>
void CompleteOrthogonalDecomposition<MatrixType>::applyZOnTheLeftInPlace(
Rhs& rhs) const {
const Index cols = this->cols();
const Index nrhs = rhs.cols();
const Index rank = this->rank();
Matrix<typename Rhs::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
for (Index k = rank-1; k >= 0; --k) {
if (k != rank - 1) {
rhs.row(k).swap(rhs.row(rank - 1));
}
rhs.middleRows(rank - 1, cols - rank + 1)
.applyHouseholderOnTheLeft(
matrixQTZ().row(k).tail(cols - rank).transpose().template conjugateIf<!Conjugate>(), zCoeffs().template conjugateIf<Conjugate>()(k),
&temp(0));
if (k != rank - 1) {
rhs.row(k).swap(rhs.row(rank - 1));
}
}
}
template <typename MatrixType>
template <typename Rhs>
void CompleteOrthogonalDecomposition<MatrixType>::applyZAdjointOnTheLeftInPlace(
Rhs& rhs) const {
const Index cols = this->cols();
const Index nrhs = rhs.cols();
const Index rank = this->rank();
Matrix<typename Rhs::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
for (Index k = 0; k < rank; ++k) {
if (k != rank - 1) {
rhs.row(k).swap(rhs.row(rank - 1));
}
rhs.middleRows(rank - 1, cols - rank + 1)
.applyHouseholderOnTheLeft(
matrixQTZ().row(k).tail(cols - rank).adjoint(), zCoeffs()(k),
&temp(0));
if (k != rank - 1) {
rhs.row(k).swap(rhs.row(rank - 1));
}
}
}
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename _MatrixType>
template <typename RhsType, typename DstType>
void CompleteOrthogonalDecomposition<_MatrixType>::_solve_impl(
const RhsType& rhs, DstType& dst) const {
const Index rank = this->rank();
if (rank == 0) {
dst.setZero();
return;
}
// Compute c = Q^* * rhs
typename RhsType::PlainObject c(rhs);
c.applyOnTheLeft(matrixQ().setLength(rank).adjoint());
// Solve T z = c(1:rank, :)
dst.topRows(rank) = matrixT()
.topLeftCorner(rank, rank)
.template triangularView<Upper>()
.solve(c.topRows(rank));
const Index cols = this->cols();
if (rank < cols) {
// Compute y = Z^* * [ z ]
// [ 0 ]
dst.bottomRows(cols - rank).setZero();
applyZAdjointOnTheLeftInPlace(dst);
}
// Undo permutation to get x = P^{-1} * y.
dst = colsPermutation() * dst;
}
template<typename _MatrixType>
template<bool Conjugate, typename RhsType, typename DstType>
void CompleteOrthogonalDecomposition<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
{
const Index rank = this->rank();
if (rank == 0) {
dst.setZero();
return;
}
typename RhsType::PlainObject c(colsPermutation().transpose()*rhs);
if (rank < cols()) {
applyZOnTheLeftInPlace<!Conjugate>(c);
}
matrixT().topLeftCorner(rank, rank)
.template triangularView<Upper>()
.transpose().template conjugateIf<Conjugate>()
.solveInPlace(c.topRows(rank));
dst.topRows(rank) = c.topRows(rank);
dst.bottomRows(rows()-rank).setZero();
dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>() );
}
#endif
namespace internal {
template<typename MatrixType>
struct traits<Inverse<CompleteOrthogonalDecomposition<MatrixType> > >
: traits<typename Transpose<typename MatrixType::PlainObject>::PlainObject>
{
enum { Flags = 0 };
};
template<typename DstXprType, typename MatrixType>
struct Assignment<DstXprType, Inverse<CompleteOrthogonalDecomposition<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename CompleteOrthogonalDecomposition<MatrixType>::Scalar>, Dense2Dense>
{
typedef CompleteOrthogonalDecomposition<MatrixType> CodType;
typedef Inverse<CodType> SrcXprType;
static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename CodType::Scalar> &)
{
typedef Matrix<typename CodType::Scalar, CodType::RowsAtCompileTime, CodType::RowsAtCompileTime, 0, CodType::MaxRowsAtCompileTime, CodType::MaxRowsAtCompileTime> IdentityMatrixType;
dst = src.nestedExpression().solve(IdentityMatrixType::Identity(src.cols(), src.cols()));
}
};
} // end namespace internal
/** \returns the matrix Q as a sequence of householder transformations */
template <typename MatrixType>
typename CompleteOrthogonalDecomposition<MatrixType>::HouseholderSequenceType
CompleteOrthogonalDecomposition<MatrixType>::householderQ() const {
return m_cpqr.householderQ();
}
/** \return the complete orthogonal decomposition of \c *this.
*
* \sa class CompleteOrthogonalDecomposition
*/
template <typename Derived>
const CompleteOrthogonalDecomposition<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::completeOrthogonalDecomposition() const {
return CompleteOrthogonalDecomposition<PlainObject>(eval());
}
} // end namespace Eigen
#endif // EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H

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