diff options
| -rw-r--r-- | Eigen/src/QR/ColPivotingHouseholderQR.h | 159 | ||||
| -rw-r--r-- | Eigen/src/QR/FullPivotingHouseholderQR.h | 27 | ||||
| -rw-r--r-- | test/qr_colpivoting.cpp | 22 | ||||
| -rw-r--r-- | test/qr_fullpivoting.cpp | 8 |
4 files changed, 196 insertions, 20 deletions
diff --git a/Eigen/src/QR/ColPivotingHouseholderQR.h b/Eigen/src/QR/ColPivotingHouseholderQR.h index ed4b84f63..0aec6a607 100644 --- a/Eigen/src/QR/ColPivotingHouseholderQR.h +++ b/Eigen/src/QR/ColPivotingHouseholderQR.h @@ -31,14 +31,14 @@ * * \class ColPivotingHouseholderQR * - * \brief Householder rank-revealing QR decomposition of a matrix + * \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 using Householder transformations. * - * This decomposition performs full-pivoting in order to be rank-revealing and achieve optimal - * numerical stability. + * This decomposition performs column pivoting in order to be rank-revealing and improve + * numerical stability. It is slower than HouseholderQR, and faster than FullPivotingHouseholderQR. * * \sa MatrixBase::colPivotingHouseholderQr() */ @@ -82,6 +82,8 @@ template<typename MatrixType> class ColPivotingHouseholderQR /** 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. * + * \returns \c true if a solution exists, \c false if no solution exists. + * * \param b the right-hand-side of the equation to solve. * * \param result a pointer to the vector/matrix in which to store the solution, if any exists. @@ -95,7 +97,7 @@ template<typename MatrixType> class ColPivotingHouseholderQR * Output: \verbinclude ColPivotingHouseholderQR_solve.out */ template<typename OtherDerived, typename ResultType> - void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const; + bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const; MatrixType matrixQ(void) const; @@ -111,12 +113,122 @@ template<typename MatrixType> class ColPivotingHouseholderQR return m_cols_permutation; } + /** \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 is computed at the time of the construction of the QR decomposition. This + * method does not perform any further computation. + */ inline int rank() const { ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); return m_rank; } + /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition. + * + * \note Since the rank is computed at the time of the construction of the QR decomposition, this + * method almost does not perform any further computation. + */ + inline int dimensionOfKernel() const + { + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + return m_qr.cols() - m_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 Since the rank is computed at the time of the construction of the QR decomposition, this + * method almost does not perform any further computation. + */ + inline bool isInjective() const + { + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + return m_rank == m_qr.cols(); + } + + /** \returns true if the matrix of which *this is the QR decomposition represents a surjective + * linear map; false otherwise. + * + * \note Since the rank is computed at the time of the construction of the QR decomposition, this + * method almost does not perform any further computation. + */ + inline bool isSurjective() const + { + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + return m_rank == m_qr.rows(); + } + + /** \returns true if the matrix of which *this is the QR decomposition is invertible. + * + * \note Since the rank is computed at the time of the construction of the QR decomposition, this + * method almost does not perform any further computation. + */ + inline bool isInvertible() const + { + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + return isInjective() && isSurjective(); + } + + /** Computes the inverse of the matrix of which *this is the QR decomposition. + * + * \param result a pointer to the matrix into which to store the inverse. Resized if needed. + * + * \note If this matrix is not invertible, *result is left with undefined coefficients. + * Use isInvertible() to first determine whether this matrix is invertible. + * + * \sa inverse() + */ + inline void computeInverse(MatrixType *result) const + { + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + ei_assert(m_qr.rows() == m_qr.cols() && "You can't take the inverse of a non-square matrix!"); + solve(MatrixType::Identity(m_qr.rows(), m_qr.cols()), result); + } + + /** \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. + * + * \sa computeInverse() + */ + inline MatrixType inverse() const + { + MatrixType result; + computeInverse(&result); + return result; + } + protected: MatrixType m_qr; HCoeffsType m_hCoeffs; @@ -130,6 +242,22 @@ template<typename MatrixType> class ColPivotingHouseholderQR #ifndef EIGEN_HIDE_HEAVY_CODE template<typename MatrixType> +typename MatrixType::RealScalar ColPivotingHouseholderQR<MatrixType>::absDeterminant() const +{ + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + ei_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!"); + return ei_abs(m_qr.diagonal().prod()); +} + +template<typename MatrixType> +typename MatrixType::RealScalar ColPivotingHouseholderQR<MatrixType>::logAbsDeterminant() const +{ + ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + ei_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!"); + return m_qr.diagonal().cwise().abs().cwise().log().sum(); +} + +template<typename MatrixType> ColPivotingHouseholderQR<MatrixType>& ColPivotingHouseholderQR<MatrixType>::compute(const MatrixType& matrix) { int rows = matrix.rows(); @@ -199,12 +327,23 @@ ColPivotingHouseholderQR<MatrixType>& ColPivotingHouseholderQR<MatrixType>::comp template<typename MatrixType> template<typename OtherDerived, typename ResultType> -void ColPivotingHouseholderQR<MatrixType>::solve( +bool ColPivotingHouseholderQR<MatrixType>::solve( const MatrixBase<OtherDerived>& b, ResultType *result ) const { ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized."); + result->resize(m_qr.cols(), b.cols()); + if(m_rank==0) + { + if(b.squaredNorm() == RealScalar(0)) + { + result->setZero(); + return true; + } + else return false; + } + const int rows = m_qr.rows(); const int cols = b.cols(); ei_assert(b.rows() == rows); @@ -219,13 +358,21 @@ void ColPivotingHouseholderQR<MatrixType>::solve( .applyHouseholderOnTheLeft(m_qr.col(k).end(remainingSize-1), m_hCoeffs.coeff(k), &temp.coeffRef(0)); } + if(!isSurjective()) + { + // is c is in the image of R ? + RealScalar biggest_in_upper_part_of_c = c.corner(TopLeft, m_rank, c.cols()).cwise().abs().maxCoeff(); + RealScalar biggest_in_lower_part_of_c = c.corner(BottomLeft, rows-m_rank, c.cols()).cwise().abs().maxCoeff(); + if(!ei_isMuchSmallerThan(biggest_in_lower_part_of_c, biggest_in_upper_part_of_c, m_precision)) + return false; + } m_qr.corner(TopLeft, m_rank, m_rank) .template triangularView<UpperTriangular>() .solveInPlace(c.corner(TopLeft, m_rank, c.cols())); - result->resize(m_qr.cols(), b.cols()); for(int i = 0; i < m_rank; ++i) result->row(m_cols_permutation.coeff(i)) = c.row(i); for(int i = m_rank; i < m_qr.cols(); ++i) result->row(m_cols_permutation.coeff(i)).setZero(); + return true; } /** \returns the matrix Q */ diff --git a/Eigen/src/QR/FullPivotingHouseholderQR.h b/Eigen/src/QR/FullPivotingHouseholderQR.h index 77a0abedc..77b664f6e 100644 --- a/Eigen/src/QR/FullPivotingHouseholderQR.h +++ b/Eigen/src/QR/FullPivotingHouseholderQR.h @@ -31,7 +31,7 @@ * * \class FullPivotingHouseholderQR * - * \brief Householder rank-revealing QR decomposition of a matrix + * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting * * \param MatrixType the type of the matrix of which we are computing the QR decomposition * @@ -62,12 +62,11 @@ template<typename MatrixType> class FullPivotingHouseholderQR typedef Matrix<Scalar, 1, ColsAtCompileTime> RowVectorType; typedef Matrix<Scalar, RowsAtCompileTime, 1> ColVectorType; - /** - * \brief Default Constructor. - * - * The default constructor is useful in cases in which the user intends to - * perform decompositions via FullPivotingHouseholderQR::compute(const MatrixType&). - */ + /** \brief Default Constructor. + * + * The default constructor is useful in cases in which the user intends to + * perform decompositions via FullPivotingHouseholderQR::compute(const MatrixType&). + */ FullPivotingHouseholderQR() : m_qr(), m_hCoeffs(), m_isInitialized(false) {} FullPivotingHouseholderQR(const MatrixType& matrix) @@ -81,6 +80,8 @@ template<typename MatrixType> class FullPivotingHouseholderQR /** 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. * + * \returns \c true if a solution exists, \c false if no solution exists. + * * \param b the right-hand-side of the equation to solve. * * \param result a pointer to the vector/matrix in which to store the solution, if any exists. @@ -345,7 +346,16 @@ bool FullPivotingHouseholderQR<MatrixType>::solve( ) const { ei_assert(m_isInitialized && "FullPivotingHouseholderQR is not initialized."); - if(m_rank==0) return false; + result->resize(m_qr.cols(), b.cols()); + if(m_rank==0) + { + if(b.squaredNorm() == RealScalar(0)) + { + result->setZero(); + return true; + } + else return false; + } const int rows = m_qr.rows(); const int cols = b.cols(); @@ -374,7 +384,6 @@ bool FullPivotingHouseholderQR<MatrixType>::solve( .template triangularView<UpperTriangular>() .solveInPlace(c.corner(TopLeft, m_rank, c.cols())); - result->resize(m_qr.cols(), b.cols()); for(int i = 0; i < m_rank; ++i) result->row(m_cols_permutation.coeff(i)) = c.row(i); for(int i = m_rank; i < m_qr.cols(); ++i) result->row(m_cols_permutation.coeff(i)).setZero(); return true; diff --git a/test/qr_colpivoting.cpp b/test/qr_colpivoting.cpp index 3a07c7131..d190bce73 100644 --- a/test/qr_colpivoting.cpp +++ b/test/qr_colpivoting.cpp @@ -28,7 +28,6 @@ template<typename MatrixType> void qr() { - /* this test covers the following files: QR.h */ int rows = ei_random<int>(20,200), cols = ei_random<int>(20,200), cols2 = ei_random<int>(20,200); int rank = ei_random<int>(1, std::min(rows, cols)-1); @@ -39,6 +38,10 @@ template<typename MatrixType> void qr() createRandomMatrixOfRank(rank,rows,cols,m1); ColPivotingHouseholderQR<MatrixType> qr(m1); VERIFY_IS_APPROX(rank, qr.rank()); + VERIFY(cols - qr.rank() == qr.dimensionOfKernel()); + VERIFY(!qr.isInjective()); + VERIFY(!qr.isInvertible()); + VERIFY(!qr.isSurjective()); MatrixType r = qr.matrixQR(); // FIXME need better way to construct trapezoid @@ -54,14 +57,17 @@ template<typename MatrixType> void qr() MatrixType m2 = MatrixType::Random(cols,cols2); MatrixType m3 = m1*m2; m2 = MatrixType::Random(cols,cols2); - qr.solve(m3, &m2); + VERIFY(qr.solve(m3, &m2)); VERIFY_IS_APPROX(m3, m1*m2); + m3 = MatrixType::Random(rows,cols2); + VERIFY(!qr.solve(m3, &m2)); } template<typename MatrixType> void qr_invertible() { - /* this test covers the following files: RRQR.h */ typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; + typedef typename MatrixType::Scalar Scalar; + int size = ei_random<int>(10,50); MatrixType m1(size, size), m2(size, size), m3(size, size); @@ -78,6 +84,16 @@ template<typename MatrixType> void qr_invertible() m3 = MatrixType::Random(size,size); qr.solve(m3, &m2); VERIFY_IS_APPROX(m3, m1*m2); + + // now construct a matrix with prescribed determinant + m1.setZero(); + for(int i = 0; i < size; i++) m1(i,i) = ei_random<Scalar>(); + RealScalar absdet = ei_abs(m1.diagonal().prod()); + m3 = qr.matrixQ(); // get a unitary + m1 = m3 * m1 * m3; + qr.compute(m1); + VERIFY_IS_APPROX(absdet, qr.absDeterminant()); + VERIFY_IS_APPROX(ei_log(absdet), qr.logAbsDeterminant()); } template<typename MatrixType> void qr_verify_assert() diff --git a/test/qr_fullpivoting.cpp b/test/qr_fullpivoting.cpp index a6430e6f1..bdebea7cc 100644 --- a/test/qr_fullpivoting.cpp +++ b/test/qr_fullpivoting.cpp @@ -28,7 +28,6 @@ template<typename MatrixType> void qr() { - /* this test covers the following files: QR.h */ int rows = ei_random<int>(20,200), cols = ei_random<int>(20,200), cols2 = ei_random<int>(20,200); int rank = ei_random<int>(1, std::min(rows, cols)-1); @@ -44,7 +43,6 @@ template<typename MatrixType> void qr() VERIFY(!qr.isInvertible()); VERIFY(!qr.isSurjective()); - MatrixType r = qr.matrixQR(); // FIXME need better way to construct trapezoid for(int i = 0; i < rows; i++) for(int j = 0; j < cols; j++) if(i>j) r(i,j) = Scalar(0); @@ -110,6 +108,12 @@ template<typename MatrixType> void qr_verify_assert() VERIFY_RAISES_ASSERT(qr.matrixR()) VERIFY_RAISES_ASSERT(qr.solve(tmp,&tmp)) VERIFY_RAISES_ASSERT(qr.matrixQ()) + VERIFY_RAISES_ASSERT(qr.dimensionOfKernel()) + VERIFY_RAISES_ASSERT(qr.isInjective()) + VERIFY_RAISES_ASSERT(qr.isSurjective()) + VERIFY_RAISES_ASSERT(qr.isInvertible()) + VERIFY_RAISES_ASSERT(qr.computeInverse(&tmp)) + VERIFY_RAISES_ASSERT(qr.inverse()) } void test_qr_fullpivoting() |