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// Copyright (c) 1997-2000 // Utrecht University (The Netherlands), // ETH Zurich (Switzerland), // INRIA Sophia-Antipolis (France), // Max-Planck-Institute Saarbruecken (Germany), // and Tel-Aviv University (Israel). All rights reserved. // // This file is part of CGAL (www.cgal.org) // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Kernel_d/include/CGAL/Linear_algebraHd.h $ // $Id: include/CGAL/Linear_algebraHd.h b26b07a1242 $ // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Michael Seel //--------------------------------------------------------------------- // file generated by notangle from Linear_algebra.lw // please debug or modify noweb file // based on LEDA architecture by S. Naeher, C. Uhrig // coding: K. Mehlhorn, M. Seel // debugging and templatization: M. Seel //--------------------------------------------------------------------- #ifndef CGAL_LINEAR_ALGEBRAHD_H #define CGAL_LINEAR_ALGEBRAHD_H #include #include // #define CGAL_LA_SELFTEST namespace CGAL { /*{\Moptions outfile=Linear_algebra.man}*/ /*{\Manpage {Linear_algebraHd}{RT}{Linear Algebra on RT}{LA}}*/ template class Linear_algebraHd { /*{\Mdefinition The data type |\Mname| encapsulates two classes |Matrix|, |Vector| and many functions of basic linear algebra. It is parametrized by a number type |RT|. An instance of data type |Matrix| is a matrix of variables of type |RT|, the so called ring type. Accordingly, |Vector| implements vectors of variables of type |RT|. The arithmetic type |RT| is required to behave like integers in the mathematical sense. The manual pages of |Vector| and |Matrix| follow below. All functions compute the exact result, i.e., there is no rounding error. Most functions of linear algebra are \emph{checkable}, 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 $A x = b$ unsolvable it also returns a vector $c$ such that $c^T A = 0$ and $c^T b \neq 0$. All internal correctness checks can be switched on by the flag [[CGAL_LA_SELFTEST]].}*/ public: /*{\Mtypes 5.5}*/ typedef RT_ RT; /*{\Mtypemember the ring type of the components.}*/ typedef Linear_Algebra::Vector_ Vector; /*{\Mtypemember the vector type.}*/ typedef Linear_Algebra::Matrix_ Matrix; /*{\Mtypemember the matrix type.}*/ typedef AL_ allocator_type; /*{\Mtypemember the allocator used for memory management. |\Mname| is an abbreviation for |Linear_algebraHd >|. Thus |allocator_type| defaults to the standard allocator offered by the STL.}*/ /*{\Moperations 2 1}*/ static Matrix transpose(const Matrix& M); /*{\Mstatic returns $M^T$ ($m\times n$ - matrix). }*/ static bool inverse(const Matrix& M, Matrix& I, RT& D, Vector& c); /*{\Mstatic determines whether |M| has an inverse. It also computes either the inverse as $(1/D) \cdot |I|$ or when no inverse exists, a vector $c$ such that $c^T \cdot M = 0 $. }*/ static Matrix inverse(const Matrix& M, RT& D) /*{\Mstatic returns the inverse matrix of |M|. More precisely, $1/D$ times the matrix returned is the inverse of |M|.\\ \precond |determinant(M) != 0|. }*/ { Matrix result; Vector c; if (!inverse(M,result,D,c)) CGAL_error_msg("inverse(): matrix is singular."); return result; } static RT determinant (const Matrix& M, Matrix& L, Matrix& U, std::vector& q, Vector& c); /*{\Mstatic returns the determinant $D$ of |M| and sufficient information to verify that the value of the determinant is correct. If the determinant is zero then $c$ is a vector such that $c^T \cdot M = 0$. If the determinant is non-zero then $L$ and $U$ are lower and upper diagonal matrices respectively and $q$ encodes a permutation matrix $Q$ with $Q(i,j) = 1$ iff $i = q(j)$ such that $L \cdot M \cdot Q = U$, $L(0,0) = 1$, $L(i,i) = U(i - 1,i - 1)$ for all $i$, $1 \le i < n$, and $D = s \cdot U(n - 1,n - 1)$ where $s$ is the determinant of $Q$. \precond |M| is square. }*/ static bool verify_determinant (const Matrix& M, RT D, Matrix& L, Matrix& U, const std::vector& q, Vector& c); /*{\Mstatic verifies the conditions stated above. }*/ static RT determinant (const Matrix& M); /*{\Mstatic returns the determinant of |M|. \precond |M| is square. }*/ static int sign_of_determinant (const Matrix& M); /*{\Mstatic returns the sign of the determinant of |M|. \precond |M| is square. }*/ static bool linear_solver(const Matrix& M, const Vector& b, Vector& x, RT& D, Matrix& spanning_vectors, Vector& c); /*{\Mstatic determines the complete solution space of the linear system $M\cdot x = b$. If the system is unsolvable then $c^T \cdot M = 0$ and $c^T \cdot b \not= 0$. If the system is solvable then $(1/D) x$ is a solution, and the columns of |spanning_vectors| are a maximal set of linearly independent solutions to the corresponding homogeneous system. \precond |M.row_dimension() = b.dimension()|. }*/ static bool linear_solver(const Matrix& M, const Vector& b, Vector& x, RT& D, Vector& c) /*{\Mstatic determines whether the linear system $M\cdot x = b$ is solvable. If yes, then $(1/D) x$ is a solution, if not then $c^T \cdot M = 0$ and $c^T \cdot b \not= 0$. \precond |M.row_dimension() = b.dimension()|. }*/ { Matrix spanning_vectors; return linear_solver(M,b,x,D,spanning_vectors,c); } static bool linear_solver(const Matrix& M, const Vector& b, Vector& x, RT& D) /*{\Mstatic as above, but without the witness $c$ \precond |M.row_dimension() = b.dimension()|. }*/ { Matrix spanning_vectors; Vector c; return linear_solver(M,b,x,D,spanning_vectors,c); } static bool is_solvable(const Matrix& M, const Vector& b) /*{\Mstatic determines whether the system $M \cdot x = b$ is solvable \\ \precond |M.row_dimension() = b.dimension()|. }*/ { Vector x; RT D; Matrix spanning_vectors; Vector c; return linear_solver(M,b,x,D,spanning_vectors,c); } static bool homogeneous_linear_solver (const Matrix& M, Vector& x); /*{\Mstatic determines whether the homogeneous linear system $M\cdot x = 0$ has a non - trivial solution. If yes, then $x$ is such a solution. }*/ static int homogeneous_linear_solver (const Matrix& M, Matrix& spanning_vecs); /*{\Mstatic determines the solution space of the homogeneous linear system $M\cdot x = 0$. It returns the dimension of the solution space. Moreover the columns of |spanning_vecs| span the solution space. }*/ static int independent_columns (const Matrix& M, std::vector& columns); /*{\Mstatic returns the indices of a maximal subset of independent columns of |M|.}*/ static int rank (const Matrix & M); /*{\Mstatic returns the rank of matrix |M| }*/ /*{\Mimplementation The datatype |\Mname| is a wrapper class for the linear algebra functionality on matrices and vectors. Operations |determinant|, |inverse|, |linear_solver|, and |rank| take time $O(n^3)$, and all other operations take time $O(nm)$. These time bounds ignore the cost for multiprecision arithmetic operations. All functions on integer matrices compute the exact result, i.e., there is no rounding error. The implementation follows a proposal of J. Edmonds (J. Edmonds, Systems of distinct representatives and linear algebra, Journal of Research of the Bureau of National Standards, (B), 71, 241 - 245). Most functions of linear algebra are { \em checkable }, 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 $A x = b$ unsolvable it also returns a vector $c$ such that $c^T A = 0$ and $c^T b \not= 0$.}*/ }; // Linear_algebraHd } //namespace CGAL #include #endif // CGAL_LINALG_ALGEBRAHD_H