From 2840ac7e948ecb3c7b19ba8f581f829a4a4e1fea Mon Sep 17 00:00:00 2001 From: Benoit Jacob Date: Wed, 28 Oct 2009 18:19:29 -0400 Subject: big huge changes, so i dont remember everything. * renaming, e.g. LU ---> FullPivLU * split tests framework: more robust, e.g. dont generate empty tests if a number is skipped * make all remaining tests use that splitting, as needed. * Fix 4x4 inversion (see stable branch) * Transform::inverse() and geo_transform test : adapt to new inverse() API, it was also trying to instantiate inverse() for 3x4 matrices. * CMakeLists: more robust regexp to parse the version number * misc fixes in unit tests --- Eigen/src/QR/FullPivHouseholderQR.h | 429 ++++++++++++++++++++++++++++++++++++ 1 file changed, 429 insertions(+) create mode 100644 Eigen/src/QR/FullPivHouseholderQR.h (limited to 'Eigen/src/QR/FullPivHouseholderQR.h') diff --git a/Eigen/src/QR/FullPivHouseholderQR.h b/Eigen/src/QR/FullPivHouseholderQR.h new file mode 100644 index 000000000..07ec343a5 --- /dev/null +++ b/Eigen/src/QR/FullPivHouseholderQR.h @@ -0,0 +1,429 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2008-2009 Gael Guennebaud +// Copyright (C) 2009 Benoit Jacob +// +// Eigen is free software; you can redistribute it and/or +// modify it under the terms of the GNU Lesser General Public +// License as published by the Free Software Foundation; either +// version 3 of the License, or (at your option) any later version. +// +// Alternatively, you can redistribute it and/or +// modify it under the terms of the GNU General Public License as +// published by the Free Software Foundation; either version 2 of +// the License, or (at your option) any later version. +// +// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY +// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS +// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the +// GNU General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public +// License and a copy of the GNU General Public License along with +// Eigen. If not, see . + +#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H +#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H + +/** \ingroup QR_Module + * \nonstableyet + * + * \class FullPivHouseholderQR + * + * \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 + * + * This class performs a rank-revealing QR decomposition using Householder transformations. + * + * This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal + * numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR. + * + * \sa MatrixBase::fullPivHouseholderQr() + */ +template class FullPivHouseholderQR +{ + public: + + enum { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + Options = MatrixType::Options, + DiagSizeAtCompileTime = EIGEN_ENUM_MIN(ColsAtCompileTime,RowsAtCompileTime) + }; + + typedef typename MatrixType::Scalar Scalar; + typedef typename MatrixType::RealScalar RealScalar; + typedef Matrix MatrixQType; + typedef Matrix HCoeffsType; + typedef Matrix IntRowVectorType; + typedef Matrix IntColVectorType; + typedef Matrix RowVectorType; + typedef Matrix ColVectorType; + + /** \brief Default Constructor. + * + * The default constructor is useful in cases in which the user intends to + * perform decompositions via FullPivHouseholderQR::compute(const MatrixType&). + */ + FullPivHouseholderQR() : m_isInitialized(false) {} + + FullPivHouseholderQR(const MatrixType& matrix) + : m_isInitialized(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. + * + * \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. + * Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols(). + * If no solution exists, *result is left with undefined coefficients. + * + * \note The case where b is a matrix is not yet implemented. Also, this + * code is space inefficient. + * + * Example: \include FullPivHouseholderQR_solve.cpp + * Output: \verbinclude FullPivHouseholderQR_solve.out + */ + template + bool solve(const MatrixBase& b, ResultType *result) const; + + MatrixQType matrixQ(void) const; + + /** \returns a reference to the matrix where the Householder QR decomposition is stored + */ + const MatrixType& matrixQR() const + { + ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + return m_qr; + } + + FullPivHouseholderQR& compute(const MatrixType& matrix); + + const IntRowVectorType& colsPermutation() const + { + ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + return m_cols_permutation; + } + + const IntColVectorType& rowsTranspositions() const + { + ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + return m_rows_transpositions; + } + + /** \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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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; + IntColVectorType m_rows_transpositions; + IntRowVectorType m_cols_permutation; + bool m_isInitialized; + RealScalar m_precision; + int m_rank; + int m_det_pq; +}; + +#ifndef EIGEN_HIDE_HEAVY_CODE + +template +typename MatrixType::RealScalar FullPivHouseholderQR::absDeterminant() const +{ + ei_assert(m_isInitialized && "FullPivHouseholderQR 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::RealScalar FullPivHouseholderQR::logAbsDeterminant() const +{ + ei_assert(m_isInitialized && "FullPivHouseholderQR 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 +FullPivHouseholderQR& FullPivHouseholderQR::compute(const MatrixType& matrix) +{ + int rows = matrix.rows(); + int cols = matrix.cols(); + int size = std::min(rows,cols); + m_rank = size; + + m_qr = matrix; + m_hCoeffs.resize(size); + + RowVectorType temp(cols); + + m_precision = epsilon() * size; + + m_rows_transpositions.resize(matrix.rows()); + IntRowVectorType cols_transpositions(matrix.cols()); + m_cols_permutation.resize(matrix.cols()); + int number_of_transpositions = 0; + + RealScalar biggest(0); + + for (int k = 0; k < size; ++k) + { + int row_of_biggest_in_corner, col_of_biggest_in_corner; + RealScalar biggest_in_corner; + + biggest_in_corner = m_qr.corner(Eigen::BottomRight, rows-k, cols-k) + .cwise().abs() + .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner); + row_of_biggest_in_corner += k; + col_of_biggest_in_corner += k; + if(k==0) biggest = biggest_in_corner; + + // if the corner is negligible, then we have less than full rank, and we can finish early + if(ei_isMuchSmallerThan(biggest_in_corner, biggest, m_precision)) + { + m_rank = k; + for(int i = k; i < size; i++) + { + m_rows_transpositions.coeffRef(i) = i; + cols_transpositions.coeffRef(i) = i; + m_hCoeffs.coeffRef(i) = Scalar(0); + } + break; + } + + m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner; + cols_transpositions.coeffRef(k) = col_of_biggest_in_corner; + if(k != row_of_biggest_in_corner) { + m_qr.row(k).end(cols-k).swap(m_qr.row(row_of_biggest_in_corner).end(cols-k)); + ++number_of_transpositions; + } + if(k != col_of_biggest_in_corner) { + m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner)); + ++number_of_transpositions; + } + + RealScalar beta; + m_qr.col(k).end(rows-k).makeHouseholderInPlace(&m_hCoeffs.coeffRef(k), &beta); + m_qr.coeffRef(k,k) = beta; + + m_qr.corner(BottomRight, rows-k, cols-k-1) + .applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), m_hCoeffs.coeffRef(k), &temp.coeffRef(k+1)); + } + + for(int k = 0; k < matrix.cols(); ++k) m_cols_permutation.coeffRef(k) = k; + for(int k = 0; k < size; ++k) + std::swap(m_cols_permutation.coeffRef(k), m_cols_permutation.coeffRef(cols_transpositions.coeff(k))); + + m_det_pq = (number_of_transpositions%2) ? -1 : 1; + m_isInitialized = true; + + return *this; +} + +template +template +bool FullPivHouseholderQR::solve( + const MatrixBase& b, + ResultType *result +) const +{ + ei_assert(m_isInitialized && "FullPivHouseholderQR 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); + + typename OtherDerived::PlainMatrixType c(b); + + Matrix temp(cols); + for (int k = 0; k < m_rank; ++k) + { + int remainingSize = rows-k; + c.row(k).swap(c.row(m_rows_transpositions.coeff(k))); + c.corner(BottomRight, remainingSize, cols) + .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() + .solveInPlace(c.corner(TopLeft, m_rank, c.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 */ +template +typename FullPivHouseholderQR::MatrixQType FullPivHouseholderQR::matrixQ() const +{ + ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + // compute the product H'_0 H'_1 ... H'_n-1, + // where H_k is the k-th Householder transformation I - h_k v_k v_k' + // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...] + int rows = m_qr.rows(); + int cols = m_qr.cols(); + int size = std::min(rows,cols); + MatrixQType res = MatrixQType::Identity(rows, rows); + Matrix temp(rows); + for (int k = size-1; k >= 0; k--) + { + res.block(k, k, rows-k, rows-k) + .applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), ei_conj(m_hCoeffs.coeff(k)), &temp.coeffRef(k)); + res.row(k).swap(res.row(m_rows_transpositions.coeff(k))); + } + return res; +} + +#endif // EIGEN_HIDE_HEAVY_CODE + +/** \return the full-pivoting Householder QR decomposition of \c *this. + * + * \sa class FullPivHouseholderQR + */ +template +const FullPivHouseholderQR::PlainMatrixType> +MatrixBase::fullPivHouseholderQr() const +{ + return FullPivHouseholderQR(eval()); +} + +#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H -- cgit v1.2.3