1c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This file is part of Eigen, a lightweight C++ template library 2c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// for linear algebra. 3c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 4c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com> 5c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 6c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This Source Code Form is subject to the terms of the Mozilla 7c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Public License v. 2.0. If a copy of the MPL was not distributed 8c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 10c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#ifndef EIGEN_LU_H 11c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#define EIGEN_LU_H 12c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 13c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathnamespace Eigen { 14c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 15c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup LU_Module 16c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 17c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \class FullPivLU 18c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 19c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \brief LU decomposition of a matrix with complete pivoting, and related features 20c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 21c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param MatrixType the type of the matrix of which we are computing the LU decomposition 22c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is 247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * decomposed as \f$ A = P^{-1} L U Q^{-1} \f$ where L is unit-lower-triangular, U is 257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU 267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any 277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * zeros are at the end. 28c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 29c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * This decomposition provides the generic approach to solving systems of linear equations, computing 30c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * the rank, invertibility, inverse, kernel, and determinant. 31c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 32c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD 33c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, 34c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * working with the SVD allows to select the smallest singular values of the matrix, something that 35c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * the LU decomposition doesn't see. 36c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 37c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * The data of the LU decomposition can be directly accessed through the methods matrixLU(), 38c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * permutationP(), permutationQ(). 39c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 40c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * As an exemple, here is how the original matrix can be retrieved: 41c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \include class_FullPivLU.cpp 42c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Output: \verbinclude class_FullPivLU.out 43c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 44c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse() 45c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 46c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename _MatrixType> class FullPivLU 47c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 48c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath public: 49c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef _MatrixType MatrixType; 50c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath enum { 51c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RowsAtCompileTime = MatrixType::RowsAtCompileTime, 52c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ColsAtCompileTime = MatrixType::ColsAtCompileTime, 53c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Options = MatrixType::Options, 54c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 55c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime 56c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath }; 57c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename MatrixType::Scalar Scalar; 58c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; 59c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename internal::traits<MatrixType>::StorageKind StorageKind; 60c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename MatrixType::Index Index; 61c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType; 62c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType; 63c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType; 64c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType; 65c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 66c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** 67c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \brief Default Constructor. 68c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 69c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * The default constructor is useful in cases in which the user intends to 70c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * perform decompositions via LU::compute(const MatrixType&). 71c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 72c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath FullPivLU(); 73c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 74c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Default Constructor with memory preallocation 75c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 76c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Like the default constructor but with preallocation of the internal data 77c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * according to the specified problem \a size. 78c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa FullPivLU() 79c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 80c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath FullPivLU(Index rows, Index cols); 81c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 82c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** Constructor. 83c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 84c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param matrix the matrix of which to compute the LU decomposition. 85c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * It is required to be nonzero. 86c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 87c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath FullPivLU(const MatrixType& matrix); 88c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 89c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** Computes the LU decomposition of the given matrix. 90c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 91c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param matrix the matrix of which to compute the LU decomposition. 92c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * It is required to be nonzero. 93c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 94c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns a reference to *this 95c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 96c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath FullPivLU& compute(const MatrixType& matrix); 97c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 98c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the LU decomposition matrix: the upper-triangular part is U, the 99c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * unit-lower-triangular part is L (at least for square matrices; in the non-square 100c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * case, special care is needed, see the documentation of class FullPivLU). 101c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 102c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa matrixL(), matrixU() 103c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 104c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const MatrixType& matrixLU() const 105c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 106c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 107c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_lu; 108c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 109c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 110c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the number of nonzero pivots in the LU decomposition. 111c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Here nonzero is meant in the exact sense, not in a fuzzy sense. 112c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * So that notion isn't really intrinsically interesting, but it is 113c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * still useful when implementing algorithms. 114c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 115c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa rank() 116c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 117c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline Index nonzeroPivots() const 118c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 119c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 120c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_nonzero_pivots; 121c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 122c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 123c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the absolute value of the biggest pivot, i.e. the biggest 124c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * diagonal coefficient of U. 125c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 126c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar maxPivot() const { return m_maxpivot; } 127c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 128c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the permutation matrix P 129c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 130c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa permutationQ() 131c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 132c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const PermutationPType& permutationP() const 133c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 134c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 135c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_p; 136c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 137c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 138c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the permutation matrix Q 139c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 140c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa permutationP() 141c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 142c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const PermutationQType& permutationQ() const 143c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 144c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 145c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_q; 146c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 147c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 148c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix 149c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * will form a basis of the kernel. 150c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 151c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros. 152c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 153c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 154c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 155c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 156c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 157c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Example: \include FullPivLU_kernel.cpp 158c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Output: \verbinclude FullPivLU_kernel.out 159c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 160c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa image() 161c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 162c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const internal::kernel_retval<FullPivLU> kernel() const 163c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 164c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 165c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return internal::kernel_retval<FullPivLU>(*this); 166c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 167c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 168c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix 169c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * will form a basis of the kernel. 170c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 171c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param originalMatrix the original matrix, of which *this is the LU decomposition. 172c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * The reason why it is needed to pass it here, is that this allows 173c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * a large optimization, as otherwise this method would need to reconstruct it 174c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * from the LU decomposition. 175c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 176c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros. 177c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 178c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 179c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 180c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 181c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 182c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Example: \include FullPivLU_image.cpp 183c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Output: \verbinclude FullPivLU_image.out 184c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 185c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa kernel() 186c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 187c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const internal::image_retval<FullPivLU> 188c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath image(const MatrixType& originalMatrix) const 189c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 190c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 191c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return internal::image_retval<FullPivLU>(*this, originalMatrix); 192c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 193c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 194c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \return a solution x to the equation Ax=b, where A is the matrix of which 195c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * *this is the LU decomposition. 196c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 197c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix, 198c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * the only requirement in order for the equation to make sense is that 199c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition. 200c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 201c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns a solution. 202c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 203c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note_about_checking_solutions 204c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 205c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note_about_arbitrary_choice_of_solution 206c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note_about_using_kernel_to_study_multiple_solutions 207c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 208c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Example: \include FullPivLU_solve.cpp 209c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Output: \verbinclude FullPivLU_solve.out 210c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 211c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa TriangularView::solve(), kernel(), inverse() 212c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 213c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath template<typename Rhs> 214c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const internal::solve_retval<FullPivLU, Rhs> 215c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath solve(const MatrixBase<Rhs>& b) const 216c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 217c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 218c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return internal::solve_retval<FullPivLU, Rhs>(*this, b.derived()); 219c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 220c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 221c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the determinant of the matrix of which 222c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * *this is the LU decomposition. It has only linear complexity 223c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * (that is, O(n) where n is the dimension of the square matrix) 224c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * as the LU decomposition has already been computed. 225c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 226c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This is only for square matrices. 227c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 228c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers 229c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * optimized paths. 230c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 231c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \warning a determinant can be very big or small, so for matrices 232c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * of large enough dimension, there is a risk of overflow/underflow. 233c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 234c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa MatrixBase::determinant() 235c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 236c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typename internal::traits<MatrixType>::Scalar determinant() const; 237c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 238c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** Allows to prescribe a threshold to be used by certain methods, such as rank(), 239c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * who need to determine when pivots are to be considered nonzero. This is not used for the 240c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * LU decomposition itself. 241c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 242c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * When it needs to get the threshold value, Eigen calls threshold(). By default, this 243c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * uses a formula to automatically determine a reasonable threshold. 244c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Once you have called the present method setThreshold(const RealScalar&), 245c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * your value is used instead. 246c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 247c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param threshold The new value to use as the threshold. 248c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 249c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * A pivot will be considered nonzero if its absolute value is strictly greater than 250c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$ 251c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * where maxpivot is the biggest pivot. 252c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 253c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * If you want to come back to the default behavior, call setThreshold(Default_t) 254c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 255c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath FullPivLU& setThreshold(const RealScalar& threshold) 256c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 257c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_usePrescribedThreshold = true; 258c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_prescribedThreshold = threshold; 259c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return *this; 260c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 261c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 262c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** Allows to come back to the default behavior, letting Eigen use its default formula for 263c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * determining the threshold. 264c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 265c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * You should pass the special object Eigen::Default as parameter here. 266c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \code lu.setThreshold(Eigen::Default); \endcode 267c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 268c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * See the documentation of setThreshold(const RealScalar&). 269c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 270c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath FullPivLU& setThreshold(Default_t) 271c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 272c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_usePrescribedThreshold = false; 273c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return *this; 274c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 275c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 276c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** Returns the threshold that will be used by certain methods such as rank(). 277c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 278c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * See the documentation of setThreshold(const RealScalar&). 279c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 280c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar threshold() const 281c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 282c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized || m_usePrescribedThreshold); 283c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return m_usePrescribedThreshold ? m_prescribedThreshold 284c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // this formula comes from experimenting (see "LU precision tuning" thread on the list) 285c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // and turns out to be identical to Higham's formula used already in LDLt. 286c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : NumTraits<Scalar>::epsilon() * m_lu.diagonalSize(); 287c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 288c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 289c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the rank of the matrix of which *this is the LU decomposition. 290c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 291c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 292c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 293c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 294c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 295c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline Index rank() const 296c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 2977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 298c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 2997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold(); 300c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index result = 0; 301c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < m_nonzero_pivots; ++i) 3027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold); 303c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return result; 304c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 305c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 306c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition. 307c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 308c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 309c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 310c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 311c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 312c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline Index dimensionOfKernel() const 313c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 314c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 315c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return cols() - rank(); 316c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 317c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 318c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns true if the matrix of which *this is the LU decomposition represents an injective 319c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * linear map, i.e. has trivial kernel; false otherwise. 320c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 321c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 322c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 323c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 324c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 325c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline bool isInjective() const 326c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 327c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 328c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return rank() == cols(); 329c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 330c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 331c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns true if the matrix of which *this is the LU decomposition represents a surjective 332c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * linear map; false otherwise. 333c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 334c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 335c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 336c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 337c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 338c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline bool isSurjective() const 339c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 340c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 341c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return rank() == rows(); 342c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 343c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 344c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns true if the matrix of which *this is the LU decomposition is invertible. 345c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 346c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note This method has to determine which pivots should be considered nonzero. 347c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * For that, it uses the threshold value that you can control by calling 348c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * setThreshold(const RealScalar&). 349c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 350c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline bool isInvertible() const 351c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 352c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 353c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return isInjective() && (m_lu.rows() == m_lu.cols()); 354c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 355c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 356c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \returns the inverse of the matrix of which *this is the LU decomposition. 357c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 358c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \note If this matrix is not invertible, the returned matrix has undefined coefficients. 359c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Use isInvertible() to first determine whether this matrix is invertible. 360c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 361c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa MatrixBase::inverse() 362c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 363c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> inverse() const 364c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 365c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 366c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!"); 367c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> 368c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols())); 369c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 370c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 371c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType reconstructedMatrix() const; 372c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 373c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline Index rows() const { return m_lu.rows(); } 374c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline Index cols() const { return m_lu.cols(); } 375c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 376c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath protected: 377c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType m_lu; 378c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath PermutationPType m_p; 379c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath PermutationQType m_q; 380c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath IntColVectorType m_rowsTranspositions; 381c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath IntRowVectorType m_colsTranspositions; 382c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index m_det_pq, m_nonzero_pivots; 383c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar m_maxpivot, m_prescribedThreshold; 384c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath bool m_isInitialized, m_usePrescribedThreshold; 385c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 386c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 387c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 388c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathFullPivLU<MatrixType>::FullPivLU() 389c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : m_isInitialized(false), m_usePrescribedThreshold(false) 390c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 391c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 392c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 393c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 394c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathFullPivLU<MatrixType>::FullPivLU(Index rows, Index cols) 395c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : m_lu(rows, cols), 396c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_p(rows), 397c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_q(cols), 398c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_rowsTranspositions(rows), 399c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_colsTranspositions(cols), 400c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_isInitialized(false), 401c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_usePrescribedThreshold(false) 402c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 403c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 404c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 405c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 406c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathFullPivLU<MatrixType>::FullPivLU(const MatrixType& matrix) 407c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : m_lu(matrix.rows(), matrix.cols()), 408c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_p(matrix.rows()), 409c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_q(matrix.cols()), 410c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_rowsTranspositions(matrix.rows()), 411c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_colsTranspositions(matrix.cols()), 412c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_isInitialized(false), 413c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_usePrescribedThreshold(false) 414c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 415c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath compute(matrix); 416c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 417c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 418c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 419c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathFullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix) 420c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 4217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez // the permutations are stored as int indices, so just to be sure: 4227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(matrix.rows()<=NumTraits<int>::highest() && matrix.cols()<=NumTraits<int>::highest()); 4237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 424c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_isInitialized = true; 425c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_lu = matrix; 426c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 427c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index size = matrix.diagonalSize(); 428c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index rows = matrix.rows(); 429c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index cols = matrix.cols(); 430c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 431c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // will store the transpositions, before we accumulate them at the end. 432c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // can't accumulate on-the-fly because that will be done in reverse order for the rows. 433c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_rowsTranspositions.resize(matrix.rows()); 434c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_colsTranspositions.resize(matrix.cols()); 435c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i 436c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 437c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case) 438c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_maxpivot = RealScalar(0); 439c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 440c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index k = 0; k < size; ++k) 441c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 442c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // First, we need to find the pivot. 443c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 444c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // biggest coefficient in the remaining bottom-right corner (starting at row k, col k) 445c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index row_of_biggest_in_corner, col_of_biggest_in_corner; 446c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar biggest_in_corner; 447c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k) 448c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .cwiseAbs() 449c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner); 450c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner, 451c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath col_of_biggest_in_corner += k; // need to add k to them. 452c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 453c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(biggest_in_corner==RealScalar(0)) 454c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 455c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // before exiting, make sure to initialize the still uninitialized transpositions 456c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // in a sane state without destroying what we already have. 457c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_nonzero_pivots = k; 458c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = k; i < size; ++i) 459c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 460c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_rowsTranspositions.coeffRef(i) = i; 461c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_colsTranspositions.coeffRef(i) = i; 462c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 463c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath break; 464c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 465c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 466c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner; 467c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 468c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Now that we've found the pivot, we need to apply the row/col swaps to 469c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // bring it to the location (k,k). 470c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 471c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner; 472c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner; 473c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(k != row_of_biggest_in_corner) { 474c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner)); 475c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ++number_of_transpositions; 476c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 477c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(k != col_of_biggest_in_corner) { 478c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner)); 479c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ++number_of_transpositions; 480c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 481c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 482c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Now that the pivot is at the right location, we update the remaining 483c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // bottom-right corner by Gaussian elimination. 484c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 485c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(k<rows-1) 486c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k); 487c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(k<size-1) 488c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1); 489c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 490c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 491c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // the main loop is over, we still have to accumulate the transpositions to find the 492c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // permutations P and Q 493c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 494c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_p.setIdentity(rows); 495c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index k = size-1; k >= 0; --k) 496c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k)); 497c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 498c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_q.setIdentity(cols); 499c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index k = 0; k < size; ++k) 500c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k)); 501c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 502c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m_det_pq = (number_of_transpositions%2) ? -1 : 1; 503c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return *this; 504c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 505c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 506c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 507c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtypename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const 508c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 509c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 510c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!"); 511c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod()); 512c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 513c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 514c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \returns the matrix represented by the decomposition, 5157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * i.e., it returns the product: \f$ P^{-1} L U Q^{-1} \f$. 5167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This function is provided for debug purposes. */ 517c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename MatrixType> 518c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathMatrixType FullPivLU<MatrixType>::reconstructedMatrix() const 519c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 520c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(m_isInitialized && "LU is not initialized."); 521c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols()); 522c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // LU 523c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType res(m_lu.rows(),m_lu.cols()); 524c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // FIXME the .toDenseMatrix() should not be needed... 525c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath res = m_lu.leftCols(smalldim) 526c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<UnitLower>().toDenseMatrix() 527c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * m_lu.topRows(smalldim) 528c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().toDenseMatrix(); 529c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 530c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // P^{-1}(LU) 531c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath res = m_p.inverse() * res; 532c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 533c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // (P^{-1}LU)Q^{-1} 534c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath res = res * m_q.inverse(); 535c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 536c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return res; 537c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 538c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 539c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/********* Implementation of kernel() **************************************************/ 540c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 541c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathnamespace internal { 542c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename _MatrixType> 543c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathstruct kernel_retval<FullPivLU<_MatrixType> > 544c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : kernel_retval_base<FullPivLU<_MatrixType> > 545c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 546c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>) 547c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 548c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( 549c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType::MaxColsAtCompileTime, 550c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType::MaxRowsAtCompileTime) 551c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath }; 552c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 553c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath template<typename Dest> void evalTo(Dest& dst) const 554c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 5557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 556c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index cols = dec().matrixLU().cols(), dimker = cols - rank(); 557c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(dimker == 0) 558c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 559c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's 560c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // avoid crashing/asserting as that depends on floating point calculations. Let's 561c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // just return a single column vector filled with zeros. 562c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dst.setZero(); 563c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return; 564c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 565c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 566c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /* Let us use the following lemma: 567c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 568c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Lemma: If the matrix A has the LU decomposition PAQ = LU, 569c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * then Ker A = Q(Ker U). 570c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 571c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Proof: trivial: just keep in mind that P, Q, L are invertible. 572c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 573c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 574c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /* Thus, all we need to do is to compute Ker U, and then apply Q. 575c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 576c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end. 577c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Thus, the diagonal of U ends with exactly 578c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * dimKer zero's. Let us use that to construct dimKer linearly 579c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * independent vectors in Ker U. 580c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 581c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 582c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); 583c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); 584c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index p = 0; 585c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < dec().nonzeroPivots(); ++i) 586c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) 587c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath pivots.coeffRef(p++) = i; 588c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_internal_assert(p == rank()); 589c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 590c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // we construct a temporaty trapezoid matrix m, by taking the U matrix and 591c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // permuting the rows and cols to bring the nonnegligible pivots to the top of 592c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // the main diagonal. We need that to be able to apply our triangular solvers. 593c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified 594c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options, 595c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime> 596c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m(dec().matrixLU().block(0, 0, rank(), cols)); 597c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < rank(); ++i) 598c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 599c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(i) m.row(i).head(i).setZero(); 600c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i); 601c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 602c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.block(0, 0, rank(), rank()); 603c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero(); 604c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < rank(); ++i) 605c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.col(i).swap(m.col(pivots.coeff(i))); 606c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 607c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // ok, we have our trapezoid matrix, we can apply the triangular solver. 608c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // notice that the math behind this suggests that we should apply this to the 609c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // negative of the RHS, but for performance we just put the negative sign elsewhere, see below. 610c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.topLeftCorner(rank(), rank()) 611c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solveInPlace( 612c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.topRightCorner(rank(), dimker) 613c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ); 614c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 615c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // now we must undo the column permutation that we had applied! 616c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = rank()-1; i >= 0; --i) 617c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath m.col(i).swap(m.col(pivots.coeff(i))); 618c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 619c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // see the negative sign in the next line, that's what we were talking about above. 620c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker); 621c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero(); 622c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1); 623c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 624c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 625c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 626c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/***** Implementation of image() *****************************************************/ 627c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 628c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename _MatrixType> 629c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathstruct image_retval<FullPivLU<_MatrixType> > 630c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : image_retval_base<FullPivLU<_MatrixType> > 631c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 632c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>) 633c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 634c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( 635c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType::MaxColsAtCompileTime, 636c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType::MaxRowsAtCompileTime) 637c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath }; 638c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 639c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath template<typename Dest> void evalTo(Dest& dst) const 640c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 6417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 642c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(rank() == 0) 643c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 644c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // The Image is just {0}, so it doesn't have a basis properly speaking, but let's 645c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // avoid crashing/asserting as that depends on floating point calculations. Let's 646c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // just return a single column vector filled with zeros. 647c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dst.setZero(); 648c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return; 649c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 650c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 651c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); 652c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); 653c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index p = 0; 654c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < dec().nonzeroPivots(); ++i) 655c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) 656c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath pivots.coeffRef(p++) = i; 657c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_internal_assert(p == rank()); 658c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 659c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < rank(); ++i) 660c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i))); 661c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 662c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 663c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 664c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/***** Implementation of solve() *****************************************************/ 665c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 666c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename _MatrixType, typename Rhs> 667c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathstruct solve_retval<FullPivLU<_MatrixType>, Rhs> 668c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath : solve_retval_base<FullPivLU<_MatrixType>, Rhs> 669c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 670c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs) 671c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 672c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath template<typename Dest> void evalTo(Dest& dst) const 673c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 674c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}. 675c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * So we proceed as follows: 676c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Step 1: compute c = P * rhs. 677c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible. 678c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Step 3: replace c by the solution x to Ux = c. May or may not exist. 679c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Step 4: result = Q * c; 680c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 681c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 682c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index rows = dec().rows(), cols = dec().cols(), 683c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath nonzero_pivots = dec().nonzeroPivots(); 684c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(rhs().rows() == rows); 685c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const Index smalldim = (std::min)(rows, cols); 686c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 687c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(nonzero_pivots == 0) 688c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 689c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dst.setZero(); 690c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return; 691c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 692c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 693c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typename Rhs::PlainObject c(rhs().rows(), rhs().cols()); 694c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 695c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Step 1 696c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath c = dec().permutationP() * rhs(); 697c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 698c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Step 2 699c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dec().matrixLU() 700c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .topLeftCorner(smalldim,smalldim) 701c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<UnitLower>() 702c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .solveInPlace(c.topRows(smalldim)); 703c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if(rows>cols) 704c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 705c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath c.bottomRows(rows-cols) 706c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath -= dec().matrixLU().bottomRows(rows-cols) 707c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * c.topRows(cols); 708c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 709c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 710c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Step 3 711c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dec().matrixLU() 712c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .topLeftCorner(nonzero_pivots, nonzero_pivots) 713c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>() 714c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .solveInPlace(c.topRows(nonzero_pivots)); 715c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 716c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Step 4 717c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = 0; i < nonzero_pivots; ++i) 718c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i); 719c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i) 720c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath dst.row(dec().permutationQ().indices().coeff(i)).setZero(); 721c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 722c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 723c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 724c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} // end namespace internal 725c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 726c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/******* MatrixBase methods *****************************************************************/ 727c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 728c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \lu_module 729c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 730c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \return the full-pivoting LU decomposition of \c *this. 731c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 732c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa class FullPivLU 733c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 734c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename Derived> 735c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathinline const FullPivLU<typename MatrixBase<Derived>::PlainObject> 736c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathMatrixBase<Derived>::fullPivLu() const 737c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 738c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return FullPivLU<PlainObject>(eval()); 739c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 740c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 741c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} // end namespace Eigen 742c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 743c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#endif // EIGEN_LU_H 744