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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/v6.1.1/Kernel_d/include/CGAL/Linear_algebraHd.h $
// $Id: include/CGAL/Linear_algebraHd.h 08b27d3db14 $
// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s) : Michael Seel <seel@mpi-sb.mpg.de>
//---------------------------------------------------------------------
// 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 <CGAL/Kernel_d/Vector__.h>
#include <CGAL/Kernel_d/Matrix__.h>
// #define CGAL_LA_SELFTEST
namespace CGAL {
/*{\Moptions outfile=Linear_algebra.man}*/
/*{\Manpage {Linear_algebraHd}{RT}{Linear Algebra on RT}{LA}}*/
template <class RT_, class AL_ = CGAL_ALLOCATOR(RT_) >
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_<RT_,AL_> Vector;
/*{\Mtypemember the vector type.}*/
typedef Linear_Algebra::Matrix_<RT_,AL_> Matrix;
/*{\Mtypemember the matrix type.}*/
typedef AL_ allocator_type;
/*{\Mtypemember the allocator used for memory management. |\Mname| is
an abbreviation for |Linear_algebraHd<RT, ALLOC = allocator<RT,LA> >|. 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<int>& 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<int>& 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<int>& 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 <CGAL/Kernel_d/Linear_algebraHd_impl.h>
#endif // CGAL_LINALG_ALGEBRAHD_H
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