Remove duplicate file (left in Kernel_d)

Solver_interface/doc/Solver_interface/Concepts/LinearAlgebraTraits_d.h

@@ -1,183 +0,0 @@
-
-/*!
-\ingroup PkgKernelDLinAlgConcepts
-\cgalConcept
-
-The data type `LinearAlgebraTraits_d` encapsulates two classes 
-`Matrix`, `Vector` and many functions of basic linear algebra. 
-An instance of data type `Matrix` is a matrix of variables of type 
-`NT`. Accordingly, `Vector` implements vectors of variables of 
-type `NT`. Most functions of linear algebra are <I>checkable</I>, 
-i.e., the programs can be asked for a proof that their output is 
-correct. For example, if the linear system solver declares a linear 
-system \f$ A x = b\f$ unsolvable it also returns a vector \f$ c\f$ such that 
-\f$ c^T A = 0\f$ and \f$ c^T b \neq 0\f$. 
-
-\cgalHasModel `CGAL::Linear_algebraHd<RT>`
-\cgalHasModel `CGAL::Linear_algebraCd<FT>`
-
-*/
-
-class LinearAlgebraTraits_d {
-public:
-
-/// \name Types 
-/// @{
-
-/*!
-the number type of the components. 
-*/ 
-typedef unspecified_type NT; 
-
-/*!
-the vector type. 
-*/ 
-typedef unspecified_type Vector; 
-
-/*!
-the matrix type. 
-*/ 
-typedef unspecified_type Matrix; 
-
-/// @} 
-
-/// \name Operations 
-/// @{
-
-/*!
-returns \f$ M^T\f$ 
-(a `M.column_dimension()` \f$ \times\f$ `M.column_dimension()` - 
-matrix). 
-*/ 
-static Matrix transpose(const Matrix& M); 
-
-/*!
-determines whether `M` has an inverse. It also 
-computes either the inverse as \f$ (1/D) \cdot I\f$ or when no 
-inverse exists, a vector \f$ c\f$ such that \f$ c^T \cdot M = 0 \f$. 
-\pre \f$ M\f$ is square. 
-*/ 
-static bool inverse(const Matrix& M, Matrix& I, NT& D, 
-Vector& c); 
-
-/*!
-returns the 
-inverse matrix of `M`. More precisely, \f$ 1/D\f$ times the matrix 
-returned is the inverse of `M`. 
-\pre `determinant(M) != 0`.
-
-\pre \f$ M\f$ is square. 
-*/ 
-static Matrix inverse(const Matrix& M, NT& D) ; 
-
-/*!
-returns the determinant \f$ D\f$ of 
-`M` and sufficient information to verify that the value of the 
-determinant is correct. If the determinant is zero then \f$ c\f$ is a 
-vector such that \f$ c^T \cdot M = 0\f$. If the determinant is non-zero 
-then \f$ L\f$ and \f$ U\f$ are lower and upper diagonal matrices respectively 
-and \f$ q\f$ encodes a permutation matrix \f$ Q\f$ with \f$ Q(i,j) = 1\f$ iff \f$ i = 
-q(j)\f$ such that \f$ L \cdot M \cdot Q = U\f$, \f$ L(0,0) = 1\f$, \f$ L(i,i) = U(i 
-- 1,i - 1)\f$ for all \f$ i\f$, \f$ 1 \le i < n\f$, and \f$ D = s \cdot U(n - 1,n - 
-1)\f$ where \f$ s\f$ is the determinant of \f$ Q\f$.
-
-\pre `M` is square. 
-*/ 
-static NT determinant (const Matrix& M, Matrix& L, Matrix& 
-U, std::vector<int>& q, Vector& c); 
-
-/*!
-verifies the conditions stated above. 
-*/ 
-static bool verify_determinant (const Matrix& M, NT D, 
-Matrix& L, Matrix& U, const std::vector<int>& q, Vector& 
-c); 
-
-/*!
-returns the 
-determinant of `M`.
-
-\pre `M` is square. 
-*/ 
-static NT determinant (const Matrix& M); 
-
-/*!
-returns 
-the sign of the determinant of `M`.
-
-\pre `M` is square. 
-*/ 
-static int sign_of_determinant (const Matrix& M); 
-
-/*!
-determines 
-the complete solution space of the linear system \f$ M\cdot x = b\f$. If 
-the system is unsolvable then \f$ c^T \cdot M = 0\f$ and \f$ c^T \cdot b 
-\not= 0\f$. If the system is solvable then \f$ (1/D) x\f$ is a solution, 
-and the columns of `spanning_vectors` are a maximal set of 
-linearly independent solutions to the corresponding homogeneous 
-system.
-
-\pre `M.row_dimension() = b.dimension()`. 
-*/ 
-static bool linear_solver(const Matrix& M, const Vector& b, 
-Vector& x, NT& D, Matrix& spanning_vectors, Vector& c); 
-
-/*!
-determines whether the linear 
-system \f$ M\cdot x = b\f$ is solvable. If yes, then \f$ (1/D) x\f$ is a 
-solution, if not then \f$ c^T \cdot M = 0\f$ and \f$ c^T \cdot b \not= 0\f$. 
-\pre `M.row_dimension() = b.dimension()`. 
-*/ 
-static bool linear_solver(const Matrix& M, const Vector& b, 
-Vector& x, NT& D, Vector& c) ; 
-
-/*!
-as above, but without the witness \f$ c\f$ 
-\pre `M.row_dimension() = b.dimension()`. 
-*/ 
-static bool linear_solver(const Matrix& M, const Vector& b, 
-Vector& x, NT& D) ; 
-
-/*!
-determines whether the system \f$ M \cdot x = b\f$ is solvable 
-
-\pre `M.row_dimension() = b.dimension()`. 
-*/ 
-static bool is_solvable(const Matrix& M, const Vector& b) 
-; 
-
-/*!
-determines whether the homogeneous linear system 
-\f$ M\cdot x = 0\f$ has a non - trivial solution. If yes, then \f$ x\f$ is 
-such a solution. 
-*/ 
-static bool homogeneous_linear_solver (const Matrix& M, 
-Vector& x); 
-
-/*!
-determines the solution space of the 
-homogeneous linear system \f$ M\cdot x = 0\f$. It returns the dimension 
-of the solution space. Moreover the columns of `spanning_vecs` 
-span the solution space. 
-*/ 
-static int homogeneous_linear_solver (const Matrix& M, 
-Matrix& spanning_vecs); 
-
-/*!
-returns the indices of a maximal subset 
-of independent columns of `M`. 
-*/ 
-static int independent_columns (const Matrix& M, 
-std::vector<int>& columns); 
-
-/*!
-returns the rank of 
-matrix `M` 
-*/ 
-static int rank (const Matrix & M); 
-
-/// @}
-
-}; /* end LinearAlgebraTraits_d */
-