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* make LU::kernel() and LU::image() also use ReturnByValue
* make them return zero vector in the degenerate case, instead of asserting (let's stick to the principle that we only assert on memory errors)
This commit is contained in:
Benoit Jacob
parent
0ad3494bd3
commit
4f9e270343
@ -51,9 +51,9 @@ struct ei_nested<ReturnByValue<Derived>, n, PlainMatrixType>
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template<typename Derived>
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class ReturnByValue : public MatrixBase<ReturnByValue<Derived> >
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{
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typedef typename ei_traits<Derived>::ReturnMatrixType ReturnMatrixType;
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public:
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EIGEN_GENERIC_PUBLIC_INTERFACE(ReturnByValue)
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typedef typename ei_traits<Derived>::ReturnMatrixType ReturnMatrixType;
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template<typename Dest>
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inline void evalTo(Dest& dst) const
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{ static_cast<const Derived* const>(this)->evalTo(dst); }
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@ -26,88 +26,8 @@
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#define EIGEN_LU_H
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template<typename MatrixType, typename Rhs> struct ei_lu_solve_impl;
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template<typename MatrixType,typename Rhs>
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struct ei_traits<ei_lu_solve_impl<MatrixType,Rhs> >
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{
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typedef Matrix<typename Rhs::Scalar,
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MatrixType::ColsAtCompileTime,
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Rhs::ColsAtCompileTime,
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Rhs::PlainMatrixType::Options,
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MatrixType::MaxColsAtCompileTime,
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Rhs::MaxColsAtCompileTime> ReturnMatrixType;
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};
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template<typename MatrixType, typename Rhs>
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struct ei_lu_solve_impl : public ReturnByValue<ei_lu_solve_impl<MatrixType, Rhs> >
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{
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typedef typename ei_cleantype<typename Rhs::Nested>::type RhsNested;
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typedef LU<MatrixType> LUType;
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ei_lu_solve_impl(const LUType& lu, const Rhs& rhs)
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: m_lu(lu), m_rhs(rhs)
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{}
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inline int rows() const { return m_lu.matrixLU().cols(); }
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inline int cols() const { return m_rhs.cols(); }
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template<typename Dest> void evalTo(Dest& dst) const
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{
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dst.resize(m_lu.matrixLU().cols(), m_rhs.cols());
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if(m_lu.rank()==0)
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{
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dst.setZero();
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return;
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}
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/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
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* So we proceed as follows:
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* Step 1: compute c = P * rhs.
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* Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
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* Step 3: replace c by the solution x to Ux = c. May or may not exist.
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* Step 4: result = Q * c;
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*/
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const int rows = m_lu.matrixLU().rows(),
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cols = m_lu.matrixLU().cols(),
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rank = m_lu.rank();
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ei_assert(m_rhs.rows() == rows);
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const int smalldim = std::min(rows, cols);
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typename Rhs::PlainMatrixType c(m_rhs.rows(), m_rhs.cols());
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// Step 1
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for(int i = 0; i < rows; ++i)
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c.row(m_lu.permutationP().coeff(i)) = m_rhs.row(i);
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// Step 2
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m_lu.matrixLU()
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.corner(Eigen::TopLeft,smalldim,smalldim)
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.template triangularView<UnitLowerTriangular>()
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.solveInPlace(c.corner(Eigen::TopLeft, smalldim, c.cols()));
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if(rows>cols)
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{
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c.corner(Eigen::BottomLeft, rows-cols, c.cols())
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-= m_lu.matrixLU().corner(Eigen::BottomLeft, rows-cols, cols)
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* c.corner(Eigen::TopLeft, cols, c.cols());
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}
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// Step 3
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m_lu.matrixLU()
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.corner(TopLeft, rank, rank)
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.template triangularView<UpperTriangular>()
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.solveInPlace(c.corner(TopLeft, rank, c.cols()));
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// Step 4
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for(int i = 0; i < rank; ++i)
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dst.row(m_lu.permutationQ().coeff(i)) = c.row(i);
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for(int i = rank; i < m_lu.matrixLU().cols(); ++i)
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dst.row(m_lu.permutationQ().coeff(i)).setZero();
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}
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const LUType& m_lu;
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const typename Rhs::Nested m_rhs;
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};
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template<typename MatrixType> struct ei_lu_kernel_impl;
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template<typename MatrixType> struct ei_lu_image_impl;
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/** \ingroup LU_Module
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*
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@ -156,25 +76,6 @@ template<typename MatrixType> class LU
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MatrixType::MaxRowsAtCompileTime)
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};
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typedef Matrix<typename MatrixType::Scalar,
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MatrixType::ColsAtCompileTime, // the number of rows in the "kernel matrix" is the number of cols of the original matrix
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// so that the product "matrix * kernel = zero" makes sense
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Dynamic, // we don't know at compile-time the dimension of the kernel
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MatrixType::Options,
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MatrixType::MaxColsAtCompileTime, // see explanation for 2nd template parameter
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MatrixType::MaxColsAtCompileTime // the kernel is a subspace of the domain space, whose dimension is the number
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// of columns of the original matrix
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> KernelResultType;
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typedef Matrix<typename MatrixType::Scalar,
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MatrixType::RowsAtCompileTime, // the image is a subspace of the destination space, whose
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// dimension is the number of rows of the original matrix
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Dynamic, // we don't know at compile time the dimension of the image (the rank)
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MatrixType::Options,
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MatrixType::MaxRowsAtCompileTime, // the image matrix will consist of columns from the original matrix,
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MatrixType::MaxColsAtCompileTime // so it has the same number of rows and at most as many columns.
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> ImageResultType;
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/**
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* \brief Default Constructor.
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*
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@ -210,7 +111,15 @@ template<typename MatrixType> class LU
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ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
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return m_lu;
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}
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/** \returns a pointer to the matrix of which *this is the LU decomposition.
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*/
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inline const MatrixType* originalMatrix() const
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{
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ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
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return m_originalMatrix;
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}
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/** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed,
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* representing the P permutation i.e. the permutation of the rows. For its precise meaning,
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* see the examples given in the documentation of class LU.
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@ -235,69 +144,37 @@ template<typename MatrixType> class LU
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return m_q;
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}
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/** Computes a basis of the kernel of the matrix, also called the null-space of the matrix.
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*
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* \note This method is only allowed on non-invertible matrices, as determined by
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* isInvertible(). Calling it on an invertible matrix will make an assertion fail.
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*
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* \param result a pointer to the matrix in which to store the kernel. The columns of this
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* matrix will be set to form a basis of the kernel (it will be resized
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* if necessary).
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*
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* Example: \include LU_computeKernel.cpp
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* Output: \verbinclude LU_computeKernel.out
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*
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* \sa kernel(), computeImage(), image()
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*/
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template<typename KernelMatrixType>
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void computeKernel(KernelMatrixType *result) const;
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/** Computes a basis of the image of the matrix, also called the column-space or range of he matrix.
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*
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* \note Calling this method on the zero matrix will make an assertion fail.
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*
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* \param result a pointer to the matrix in which to store the image. The columns of this
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* matrix will be set to form a basis of the image (it will be resized
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* if necessary).
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*
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* Example: \include LU_computeImage.cpp
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* Output: \verbinclude LU_computeImage.out
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*
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* \sa image(), computeKernel(), kernel()
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*/
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template<typename ImageMatrixType>
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void computeImage(ImageMatrixType *result) const;
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/** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
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* will form a basis of the kernel.
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*
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* \note: this method is only allowed on non-invertible matrices, as determined by
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* isInvertible(). Calling it on an invertible matrix will make an assertion fail.
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*
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* \note: this method returns a matrix by value, which induces some inefficiency.
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* If you prefer to avoid this overhead, use computeKernel() instead.
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* \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.
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*
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* Example: \include LU_kernel.cpp
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* Output: \verbinclude LU_kernel.out
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*
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* \sa computeKernel(), image()
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* \sa image()
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*/
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const KernelResultType kernel() const;
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inline const ei_lu_kernel_impl<MatrixType> kernel() const
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{
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ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
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return ei_lu_kernel_impl<MatrixType>(*this);
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}
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/** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
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* will form a basis of the kernel.
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*
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* \note: Calling this method on the zero matrix will make an assertion fail.
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*
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* \note: this method returns a matrix by value, which induces some inefficiency.
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* If you prefer to avoid this overhead, use computeImage() instead.
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* \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.
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*
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* Example: \include LU_image.cpp
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* Output: \verbinclude LU_image.out
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*
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* \sa computeImage(), kernel()
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* \sa kernel()
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*/
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const ImageResultType image() const;
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inline const ei_lu_image_impl<MatrixType> image() const
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{
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ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
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return ei_lu_image_impl<MatrixType>(*this);
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}
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/** This method returns a solution x to the equation Ax=b, where A is the matrix of which
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* *this is the LU decomposition.
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@ -316,7 +193,7 @@ template<typename MatrixType> class LU
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* Example: \include LU_solve.cpp
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* Output: \verbinclude LU_solve.out
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*
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* \sa TriangularView::solve(), kernel(), computeKernel(), inverse(), computeInverse()
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* \sa TriangularView::solve(), kernel(), inverse(), computeInverse()
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*/
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template<typename Rhs>
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inline const ei_lu_solve_impl<MatrixType, Rhs>
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@ -547,71 +424,213 @@ typename ei_traits<MatrixType>::Scalar LU<MatrixType>::determinant() const
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return Scalar(m_det_pq) * m_lu.diagonal().prod();
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}
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template<typename MatrixType>
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template<typename KernelMatrixType>
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void LU<MatrixType>::computeKernel(KernelMatrixType *result) const
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{
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ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
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const int dimker = dimensionOfKernel(), cols = m_lu.cols();
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result->resize(cols, dimker);
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/********* Implementation of kernel() **************************************************/
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/* Let us use the following lemma:
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*
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* Lemma: If the matrix A has the LU decomposition PAQ = LU,
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* then Ker A = Q(Ker U).
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*
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* Proof: trivial: just keep in mind that P, Q, L are invertible.
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template<typename MatrixType>
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struct ei_traits<ei_lu_kernel_impl<MatrixType> >
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{
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typedef Matrix<
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typename MatrixType::Scalar,
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MatrixType::ColsAtCompileTime, // the number of rows in the "kernel matrix"
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// is the number of cols of the original matrix
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// so that the product "matrix * kernel = zero" makes sense
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Dynamic, // we don't know at compile-time the dimension of the kernel
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MatrixType::Options,
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MatrixType::MaxColsAtCompileTime, // see explanation for 2nd template parameter
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MatrixType::MaxColsAtCompileTime // the kernel is a subspace of the domain space,
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// whose dimension is the number of columns of the original matrix
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> ReturnMatrixType;
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};
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template<typename MatrixType>
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struct ei_lu_kernel_impl : public ReturnByValue<ei_lu_kernel_impl<MatrixType> >
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{
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typedef LU<MatrixType> LUType;
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const LUType& m_lu;
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int m_dimKer;
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ei_lu_kernel_impl(const LUType& lu) : m_lu(lu), m_dimKer(lu.dimensionOfKernel()) {}
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inline int rows() const { return m_lu.matrixLU().cols(); }
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inline int cols() const { return m_dimKer; }
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template<typename Dest> void evalTo(Dest& dst) const
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{
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typedef typename MatrixType::Scalar Scalar;
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const int rank = m_lu.rank(),
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cols = m_lu.matrixLU().cols();
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if(m_dimKer == 0)
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{
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// The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
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// avoid crashing/asserting as that depends on floating point calculations. Let's
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// just return a single column vector filled with zeros.
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dst.resize(cols,1);
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dst.setZero();
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return;
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}
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/* Let us use the following lemma:
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*
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* Lemma: If the matrix A has the LU decomposition PAQ = LU,
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* then Ker A = Q(Ker U).
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*
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* Proof: trivial: just keep in mind that P, Q, L are invertible.
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*/
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/* Thus, all we need to do is to compute Ker U, and then apply Q.
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*
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* U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
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* Thus, the diagonal of U ends with exactly
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* m_dimKer zero's. Let us use that to construct dimKer linearly
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* independent vectors in Ker U.
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*/
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dst.resize(cols, m_dimKer);
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Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
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MatrixType::MaxColsAtCompileTime, MatrixType::MaxColsAtCompileTime>
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y(-m_lu.matrixLU().corner(TopRight, rank, m_dimKer));
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m_lu.matrixLU()
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.corner(TopLeft, rank, rank)
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.template triangularView<UpperTriangular>().solveInPlace(y);
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for(int i = 0; i < rank; ++i) dst.row(m_lu.permutationQ().coeff(i)) = y.row(i);
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for(int i = rank; i < cols; ++i) dst.row(m_lu.permutationQ().coeff(i)).setZero();
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for(int k = 0; k < m_dimKer; ++k) dst.coeffRef(m_lu.permutationQ().coeff(rank+k), k) = Scalar(1);
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}
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};
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/***** Implementation of image() *****************************************************/
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template<typename MatrixType>
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struct ei_traits<ei_lu_image_impl<MatrixType> >
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{
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typedef Matrix<
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typename MatrixType::Scalar,
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MatrixType::RowsAtCompileTime, // the image is a subspace of the destination space, whose
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// dimension is the number of rows of the original matrix
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Dynamic, // we don't know at compile time the dimension of the image (the rank)
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MatrixType::Options,
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MatrixType::MaxRowsAtCompileTime, // the image matrix will consist of columns from the original matrix,
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MatrixType::MaxColsAtCompileTime // so it has the same number of rows and at most as many columns.
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> ReturnMatrixType;
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};
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template<typename MatrixType>
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struct ei_lu_image_impl : public ReturnByValue<ei_lu_image_impl<MatrixType> >
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{
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typedef LU<MatrixType> LUType;
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const LUType& m_lu;
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ei_lu_image_impl(const LUType& lu) : m_lu(lu) {}
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inline int rows() const { return m_lu.matrixLU().cols(); }
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inline int cols() const { return m_lu.rank(); }
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template<typename Dest> void evalTo(Dest& dst) const
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{
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int rank = m_lu.rank();
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if(rank == 0)
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{
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// The Image is just {0}, so it doesn't have a basis properly speaking, but let's
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// avoid crashing/asserting as that depends on floating point calculations. Let's
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// just return a single column vector filled with zeros.
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dst.resize(m_lu.originalMatrix()->rows(), 1);
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dst.setZero();
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return;
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}
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dst.resize(m_lu.originalMatrix()->rows(), rank);
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for(int i = 0; i < rank; ++i)
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dst.col(i) = m_lu.originalMatrix()->col(m_lu.permutationQ().coeff(i));
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}
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};
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/***** Implementation of solve() *****************************************************/
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template<typename MatrixType,typename Rhs>
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struct ei_traits<ei_lu_solve_impl<MatrixType,Rhs> >
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{
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typedef Matrix<typename Rhs::Scalar,
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MatrixType::ColsAtCompileTime,
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Rhs::ColsAtCompileTime,
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Rhs::PlainMatrixType::Options,
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MatrixType::MaxColsAtCompileTime,
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Rhs::MaxColsAtCompileTime> ReturnMatrixType;
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};
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template<typename MatrixType, typename Rhs>
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struct ei_lu_solve_impl : public ReturnByValue<ei_lu_solve_impl<MatrixType, Rhs> >
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{
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typedef typename ei_cleantype<typename Rhs::Nested>::type RhsNested;
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typedef LU<MatrixType> LUType;
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const LUType& m_lu;
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const typename Rhs::Nested m_rhs;
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ei_lu_solve_impl(const LUType& lu, const Rhs& rhs)
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: m_lu(lu), m_rhs(rhs)
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{}
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inline int rows() const { return m_lu.matrixLU().cols(); }
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inline int cols() const { return m_rhs.cols(); }
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template<typename Dest> void evalTo(Dest& dst) const
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{
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dst.resize(m_lu.matrixLU().cols(), m_rhs.cols());
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if(m_lu.rank()==0)
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{
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dst.setZero();
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return;
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}
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/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
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* So we proceed as follows:
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* Step 1: compute c = P * rhs.
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* Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
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* Step 3: replace c by the solution x to Ux = c. May or may not exist.
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* Step 4: result = Q * c;
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*/
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/* Thus, all we need to do is to compute Ker U, and then apply Q.
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*
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* U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
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* Thus, the diagonal of U ends with exactly
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* m_dimKer zero's. Let us use that to construct m_dimKer linearly
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* independent vectors in Ker U.
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*/
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const int rows = m_lu.matrixLU().rows(),
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cols = m_lu.matrixLU().cols(),
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rank = m_lu.rank();
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ei_assert(m_rhs.rows() == rows);
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const int smalldim = std::min(rows, cols);
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Matrix<Scalar, Dynamic, Dynamic, MatrixType::Options,
|
||||
MatrixType::MaxColsAtCompileTime, MatrixType::MaxColsAtCompileTime>
|
||||
y(-m_lu.corner(TopRight, m_rank, dimker));
|
||||
typename Rhs::PlainMatrixType c(m_rhs.rows(), m_rhs.cols());
|
||||
|
||||
m_lu.corner(TopLeft, m_rank, m_rank)
|
||||
.template triangularView<UpperTriangular>().solveInPlace(y);
|
||||
// Step 1
|
||||
for(int i = 0; i < rows; ++i)
|
||||
c.row(m_lu.permutationP().coeff(i)) = m_rhs.row(i);
|
||||
|
||||
for(int i = 0; i < m_rank; ++i) result->row(m_q.coeff(i)) = y.row(i);
|
||||
for(int i = m_rank; i < cols; ++i) result->row(m_q.coeff(i)).setZero();
|
||||
for(int k = 0; k < dimker; ++k) result->coeffRef(m_q.coeff(m_rank+k), k) = Scalar(1);
|
||||
}
|
||||
// Step 2
|
||||
m_lu.matrixLU()
|
||||
.corner(Eigen::TopLeft,smalldim,smalldim)
|
||||
.template triangularView<UnitLowerTriangular>()
|
||||
.solveInPlace(c.corner(Eigen::TopLeft, smalldim, c.cols()));
|
||||
if(rows>cols)
|
||||
{
|
||||
c.corner(Eigen::BottomLeft, rows-cols, c.cols())
|
||||
-= m_lu.matrixLU().corner(Eigen::BottomLeft, rows-cols, cols)
|
||||
* c.corner(Eigen::TopLeft, cols, c.cols());
|
||||
}
|
||||
|
||||
template<typename MatrixType>
|
||||
const typename LU<MatrixType>::KernelResultType
|
||||
LU<MatrixType>::kernel() const
|
||||
{
|
||||
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
||||
KernelResultType result(m_lu.cols(), dimensionOfKernel());
|
||||
computeKernel(&result);
|
||||
return result;
|
||||
}
|
||||
// Step 3
|
||||
m_lu.matrixLU()
|
||||
.corner(TopLeft, rank, rank)
|
||||
.template triangularView<UpperTriangular>()
|
||||
.solveInPlace(c.corner(TopLeft, rank, c.cols()));
|
||||
|
||||
template<typename MatrixType>
|
||||
template<typename ImageMatrixType>
|
||||
void LU<MatrixType>::computeImage(ImageMatrixType *result) const
|
||||
{
|
||||
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
||||
result->resize(m_originalMatrix->rows(), m_rank);
|
||||
for(int i = 0; i < m_rank; ++i)
|
||||
result->col(i) = m_originalMatrix->col(m_q.coeff(i));
|
||||
}
|
||||
// Step 4
|
||||
for(int i = 0; i < rank; ++i)
|
||||
dst.row(m_lu.permutationQ().coeff(i)) = c.row(i);
|
||||
for(int i = rank; i < m_lu.matrixLU().cols(); ++i)
|
||||
dst.row(m_lu.permutationQ().coeff(i)).setZero();
|
||||
}
|
||||
};
|
||||
|
||||
template<typename MatrixType>
|
||||
const typename LU<MatrixType>::ImageResultType
|
||||
LU<MatrixType>::image() const
|
||||
{
|
||||
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
||||
ImageResultType result(m_originalMatrix->rows(), m_rank);
|
||||
computeImage(&result);
|
||||
return result;
|
||||
}
|
||||
/******* MatrixBase methods *****************************************************************/
|
||||
|
||||
/** \lu_module
|
||||
*
|
||||
|
||||
@ -1,7 +1,7 @@
|
||||
// This file is part of Eigen, a lightweight C++ template library
|
||||
// for linear algebra.
|
||||
//
|
||||
// Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
|
||||
// Copyright (C) 2008-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
|
||||
//
|
||||
// Eigen is free software; you can redistribute it and/or
|
||||
// modify it under the terms of the GNU Lesser General Public
|
||||
@ -37,9 +37,10 @@ template<typename MatrixType> void lu_non_invertible()
|
||||
createRandomMatrixOfRank(rank, rows, cols, m1);
|
||||
|
||||
LU<MatrixType> lu(m1);
|
||||
typename LU<MatrixType>::KernelResultType m1kernel = lu.kernel();
|
||||
typename LU<MatrixType>::ImageResultType m1image = lu.image();
|
||||
typename ei_lu_kernel_impl<MatrixType>::ReturnMatrixType m1kernel = lu.kernel();
|
||||
typename ei_lu_image_impl <MatrixType>::ReturnMatrixType m1image = lu.image();
|
||||
|
||||
std::cout << "rows:" << rows << " cols:" << cols << " | " << rank << " ----- " << lu.rank() << std::endl;
|
||||
VERIFY(rank == lu.rank());
|
||||
VERIFY(cols - lu.rank() == lu.dimensionOfKernel());
|
||||
VERIFY(!lu.isInjective());
|
||||
@ -101,8 +102,6 @@ template<typename MatrixType> void lu_verify_assert()
|
||||
VERIFY_RAISES_ASSERT(lu.matrixLU())
|
||||
VERIFY_RAISES_ASSERT(lu.permutationP())
|
||||
VERIFY_RAISES_ASSERT(lu.permutationQ())
|
||||
VERIFY_RAISES_ASSERT(lu.computeKernel(&tmp))
|
||||
VERIFY_RAISES_ASSERT(lu.computeImage(&tmp))
|
||||
VERIFY_RAISES_ASSERT(lu.kernel())
|
||||
VERIFY_RAISES_ASSERT(lu.image())
|
||||
VERIFY_RAISES_ASSERT(lu.solve(tmp))
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user