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Implement complete orthogonal decomposition in Eigen.

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This commit is contained in:
Rasmus Munk Larsen 2016-02-06 16:32:00 -08:00
parent 093f2b3c01
commit d904c8ac8f
6 changed files with 633 additions and 13 deletions
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1
Eigen/QR
View File

@ -34,6 +34,7 @@
#include "src/QR/HouseholderQR.h"
#include "src/QR/FullPivHouseholderQR.h"
#include "src/QR/ColPivHouseholderQR.h"
#include "src/QR/CompleteOrthogonalDecomposition.h"
#ifdef EIGEN_USE_LAPACKE
#include "src/QR/HouseholderQR_MKL.h"
#include "src/QR/ColPivHouseholderQR_MKL.h"

25
Eigen/src/Core/MatrixBase.h
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@ -66,7 +66,7 @@ template<typename Derived> class MatrixBase
using Base::MaxSizeAtCompileTime;
using Base::IsVectorAtCompileTime;
using Base::Flags;
using Base::derived;
using Base::const_cast_derived;
using Base::rows;
@ -175,7 +175,7 @@ template<typename Derived> class MatrixBase
#endif
template<typename OtherDerived>
EIGEN_DEVICE_FUNC
EIGEN_DEVICE_FUNC
const Product<Derived,OtherDerived,LazyProduct>
lazyProduct(const MatrixBase<OtherDerived> &other) const;
@ -214,7 +214,7 @@ template<typename Derived> class MatrixBase
typedef Diagonal<Derived> DiagonalReturnType;
EIGEN_DEVICE_FUNC
DiagonalReturnType diagonal();
typedef typename internal::add_const<Diagonal<const Derived> >::type ConstDiagonalReturnType;
EIGEN_DEVICE_FUNC
ConstDiagonalReturnType diagonal() const;
@ -222,14 +222,14 @@ template<typename Derived> class MatrixBase
template<int Index> struct DiagonalIndexReturnType { typedef Diagonal<Derived,Index> Type; };
template<int Index> struct ConstDiagonalIndexReturnType { typedef const Diagonal<const Derived,Index> Type; };
template<int Index>
template<int Index>
EIGEN_DEVICE_FUNC
typename DiagonalIndexReturnType<Index>::Type diagonal();
template<int Index>
EIGEN_DEVICE_FUNC
typename ConstDiagonalIndexReturnType<Index>::Type diagonal() const;
typedef Diagonal<Derived,DynamicIndex> DiagonalDynamicIndexReturnType;
typedef typename internal::add_const<Diagonal<const Derived,DynamicIndex> >::type ConstDiagonalDynamicIndexReturnType;
@ -251,7 +251,7 @@ template<typename Derived> class MatrixBase
template<unsigned int UpLo> struct SelfAdjointViewReturnType { typedef SelfAdjointView<Derived, UpLo> Type; };
template<unsigned int UpLo> struct ConstSelfAdjointViewReturnType { typedef const SelfAdjointView<const Derived, UpLo> Type; };
template<unsigned int UpLo>
template<unsigned int UpLo>
EIGEN_DEVICE_FUNC
typename SelfAdjointViewReturnType<UpLo>::Type selfadjointView();
template<unsigned int UpLo>
@ -340,7 +340,7 @@ template<typename Derived> class MatrixBase
EIGEN_DEVICE_FUNC
inline const Inverse<Derived> inverse() const;
template<typename ResultType>
inline void computeInverseAndDetWithCheck(
ResultType& inverse,
@ -366,6 +366,7 @@ template<typename Derived> class MatrixBase
inline const HouseholderQR<PlainObject> householderQr() const;
inline const ColPivHouseholderQR<PlainObject> colPivHouseholderQr() const;
inline const FullPivHouseholderQR<PlainObject> fullPivHouseholderQr() const;
inline const CompleteOrthogonalDecomposition<PlainObject> completeOrthogonalDecomposition() const;
/////////// Eigenvalues module ///////////
@ -394,23 +395,23 @@ template<typename Derived> class MatrixBase
inline PlainObject
#endif
cross(const MatrixBase<OtherDerived>& other) const;
template<typename OtherDerived>
EIGEN_DEVICE_FUNC
inline PlainObject cross3(const MatrixBase<OtherDerived>& other) const;
EIGEN_DEVICE_FUNC
inline PlainObject unitOrthogonal(void) const;
inline Matrix<Scalar,3,1> eulerAngles(Index a0, Index a1, Index a2) const;
inline ScalarMultipleReturnType operator*(const UniformScaling<Scalar>& s) const;
// put this as separate enum value to work around possible GCC 4.3 bug (?)
enum { HomogeneousReturnTypeDirection = ColsAtCompileTime==1&&RowsAtCompileTime==1 ? ((internal::traits<Derived>::Flags&RowMajorBit)==RowMajorBit ? Horizontal : Vertical)
: ColsAtCompileTime==1 ? Vertical : Horizontal };
typedef Homogeneous<Derived, HomogeneousReturnTypeDirection> HomogeneousReturnType;
inline HomogeneousReturnType homogeneous() const;
enum {
SizeMinusOne = SizeAtCompileTime==Dynamic ? Dynamic : SizeAtCompileTime-1
};

3
Eigen/src/Core/util/ForwardDeclarations.h
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@ -95,7 +95,7 @@ template<typename Decomposition, typename Rhstype> class Solve;
template<typename XprType> class Inverse;
template<typename Lhs, typename Rhs, int Option = DefaultProduct> class Product;
template<typename Derived> class DiagonalBase;
template<typename _DiagonalVectorType> class DiagonalWrapper;
template<typename _Scalar, int SizeAtCompileTime, int MaxSizeAtCompileTime=SizeAtCompileTime> class DiagonalMatrix;
@ -248,6 +248,7 @@ template<typename MatrixType> struct inverse_impl;
template<typename MatrixType> class HouseholderQR;
template<typename MatrixType> class ColPivHouseholderQR;
template<typename MatrixType> class FullPivHouseholderQR;
template<typename MatrixType> class CompleteOrthogonalDecomposition;
template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner> class JacobiSVD;
template<typename MatrixType> class BDCSVD;
template<typename MatrixType, int UpLo = Lower> class LLT;

2
Eigen/src/QR/ColPivHouseholderQR.h
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@ -404,6 +404,8 @@ template<typename _MatrixType> class ColPivHouseholderQR
protected:
friend class CompleteOrthogonalDecomposition<MatrixType>;
static void check_template_parameters()
{
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);

538
Eigen/src/QR/CompleteOrthogonalDecomposition.h Normal file
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@ -0,0 +1,538 @@
// 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> {
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 ortogonal 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{matrix} \mathbf{T} &
* \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{matrix} \, \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.
*
* \sa MatrixBase::completeOrthogonalDecomposition()
*/
template <typename _MatrixType>
class CompleteOrthogonalDecomposition {
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::StorageIndex StorageIndex;
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
* \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());
}
/** 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 {
eigen_assert(m_cpqr.m_isInitialized &&
"CompleteOrthogonalDecomposition is not initialized.");
return Solve<CompleteOrthogonalDecomposition, Rhs>(*this, b.derived());
}
HouseholderSequenceType householderQ(void) const;
HouseholderSequenceType matrixQ(void) const { return m_cpqr.householderQ(); }
/** Overwrites \b rhs with \f$ \mathbf{Z}^* * \mathbf{rhs} \f$.
*/
template <typename Rhs>
void applyZAdjointOnTheLeftInPlace(Rhs& rhs) const;
/** \returns the matrix \b Z.
*/
MatrixType matrixZ() const {
MatrixType Z = MatrixType::Identity(m_cpqr.cols(), m_cpqr.cols());
applyZAdjointOnTheLeftInPlace(Z);
return Z.adjoint();
}
/** \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);
/** \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 inverse of the matrix of which *this is the complete
* orthogonal decomposition.
*
* \note If this matrix is not invertible, the returned matrix has undefined
* coefficients. Use isInvertible() to first determine whether this matrix is
* invertible.
*/
// TODO(rmlarsen): Add method for pseudo-inverse.
// inline const
// internal::solve_retval<CompleteOrthogonalDecomposition, typename
// MatrixType::IdentityReturnType>
// inverse() const
// {
// eigen_assert(m_isInitialized && "CompleteOrthogonalDecomposition is not
// initialized.");
// return internal::solve_retval<CompleteOrthogonalDecomposition,typename
// MatrixType::IdentityReturnType>
// (*this, MatrixType::Identity(m_cpqr.rows(), m_cpqr.cols()));
// }
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
* 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_cpqr.m_isInitialized && "Decomposition is not initialized.");
return Success;
}
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename RhsType, typename DstType>
EIGEN_DEVICE_FUNC void _solve_impl(const RhsType& rhs, DstType& dst) const;
#endif
protected:
static void check_template_parameters() {
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
}
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>
template <typename InputType>
CompleteOrthogonalDecomposition<MatrixType>& CompleteOrthogonalDecomposition<
MatrixType>::compute(const EigenBase<InputType>& matrix) {
check_template_parameters();
// the column permutation is stored as int indices, so just to be sure:
eigen_assert(matrix.cols() <= NumTraits<int>::highest());
// Compute the column pivoted QR factorization A P = Q R.
m_cpqr.compute(matrix);
const Index rank = m_cpqr.rank();
const Index cols = matrix.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).transpose(), 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));
}
}
}
return *this;
}
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 MatrixType::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 {
eigen_assert(rhs().rows() == this->rows());
const Index rank = this->rank();
if (rank == 0) {
dst.setZero();
return;
}
// Compute c = Q^* * rhs
// Note that the matrix Q = H_0^* H_1^*... so its inverse is
// Q^* = (H_0 H_1 ...)^T
typename RhsType::PlainObject c(rhs());
c.applyOnTheLeft(
householderSequence(matrixQTZ(), hCoeffs()).setLength(rank).transpose());
// 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;
}
#endif
/** \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();
}
#ifndef __CUDACC__
/** \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());
}
#endif // __CUDACC__
} // end namespace Eigen
#endif // EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H

77
test/qr_colpivoting.cpp
View File

@ -10,6 +10,83 @@
#include "main.h"
#include <Eigen/QR>
#include <Eigen/SVD>
template <typename MatrixType>
void cod() {
typedef typename MatrixType::Index Index;
Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
typedef typename MatrixType::Scalar Scalar;
typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
MatrixType::RowsAtCompileTime>
MatrixQType;
MatrixType matrix;
createRandomPIMatrixOfRank(rank, rows, cols, matrix);
CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
VERIFY(rank == cod.rank());
VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
VERIFY(!cod.isInjective());
VERIFY(!cod.isInvertible());
VERIFY(!cod.isSurjective());
MatrixQType q = cod.householderQ();
VERIFY_IS_UNITARY(q);
MatrixType z = cod.matrixZ();
VERIFY_IS_UNITARY(z);
MatrixType t;
t.setZero(rows, cols);
t.topLeftCorner(rank, rank) =
cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
MatrixType c = q * t * z * cod.colsPermutation().inverse();
VERIFY_IS_APPROX(matrix, c);
MatrixType exact_solution = MatrixType::Random(cols, cols2);
MatrixType rhs = matrix * exact_solution;
MatrixType cod_solution = cod.solve(rhs);
VERIFY_IS_APPROX(rhs, matrix * cod_solution);
// Verify that we get the same minimum-norm solution as the SVD.
JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
MatrixType svd_solution = svd.solve(rhs);
VERIFY_IS_APPROX(cod_solution, svd_solution);
}
template <typename MatrixType, int Cols2>
void cod_fixedsize() {
enum {
Rows = MatrixType::RowsAtCompileTime,
Cols = MatrixType::ColsAtCompileTime
};
typedef typename MatrixType::Scalar Scalar;
int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
Matrix<Scalar, Rows, Cols> matrix;
createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > cod(matrix);
VERIFY(rank == cod.rank());
VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
VERIFY(cod.isInjective() == (rank == Rows));
VERIFY(cod.isSurjective() == (rank == Cols));
VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
Matrix<Scalar, Cols, Cols2> exact_solution;
exact_solution.setRandom(Cols, Cols2);
Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
VERIFY_IS_APPROX(rhs, matrix * cod_solution);
// Verify that we get the same minimum-norm solution as the SVD.
JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
VERIFY_IS_APPROX(cod_solution, svd_solution);
}
template<typename MatrixType> void qr()
{