\begin{ccRefConcept}{LinearAlgebraTraits_d}\ccCreationVariable{LA}

\ccDefinition


The data type \ccc{LinearAlgebraTraits_d} encapsulates two classes
\ccc{Matrix}, \ccc{Vector} and many functions of basic linear algebra.
An instance of data type \ccc{Matrix} is a matrix of variables of type
\ccc{NT}. Accordingly, \ccc{Vector} implements vectors of variables of
type \ccc{NT}.  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 $c^T b \neq 0$.

\ccSetOneOfTwoColumns{5.5cm}
\ccTypes

\ccNestedType{NT}{the number type of the components.}
\ccNestedType{Vector}{the vector type.}
\ccNestedType{Matrix}{the matrix type.}

\ccSetTwoOfThreeColumns{2cm}{1cm}

\ccOperations

\ccMethod{static Matrix transpose(const Matrix& M);}{returns $M^T$
(a \ccc{M.column_dimension()} $\times$ \ccc{M.column_dimension()} -
matrix).  }

\ccMethod{static bool inverse(const Matrix& M, Matrix& I, NT& D,
  Vector& c);}{determines whether \ccc{M} has an inverse. It also
  computes either the inverse as $(1/D) \cdot \ccc{I}$ or when no
  inverse exists, a vector $c$ such that $c^T \cdot M = 0 $.
  \ccPrecond $M$ is square. }

\ccMethod{static Matrix inverse(const Matrix& M, NT& D) ;}{returns the
  inverse matrix of \ccc{M}. More precisely, $1/D$ times the matrix
  returned is the inverse of \ccc{M}.\\ \ccPrecond \ccc{determinant(M)
    != 0}. \ccPrecond $M$ is square. }

\ccMethod{static NT determinant (const Matrix& M, Matrix& L, Matrix&
  U, std::vector<int>& q, Vector& c);} {returns the determinant $D$ of
  \ccc{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 $Q$. \ccPrecond \ccc{M} is
  square.}

\ccMethod{static bool verify_determinant (const Matrix& M, NT D,
  Matrix& L, Matrix& U, const std::vector<int>& q, Vector&
  c);}{verifies the conditions stated above.}

\ccMethod{static NT determinant (const Matrix& M);}{returns the
  determinant of \ccc{M}.  \ccPrecond \ccc{M} is square.}

\ccMethod{static int sign_of_determinant (const Matrix& M);}{returns
  the sign of the determinant of \ccc{M}.  \ccPrecond \ccc{M} is
  square.}

\ccMethod{static bool linear_solver(const Matrix& M, const Vector& b,
  Vector& x, NT& D, Matrix& spanning_vectors, Vector& c);}{ 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 \ccc{spanning_vectors} are a maximal set of
  linearly independent solutions to the corresponding homogeneous
  system.  \ccPrecond \ccc{M.row_dimension() = b.dimension()}.  }

\ccMethod{static bool linear_solver(const Matrix& M, const Vector& b,
  Vector& x, NT& D, Vector& c) ;}{ 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 $c^T \cdot b \not= 0$.
  \ccPrecond \ccc{M.row_dimension() = b.dimension()}.  }

\ccMethod{static bool linear_solver(const Matrix& M, const Vector& b,
  Vector& x, NT& D) ;}{ as above, but without the witness $c$
  \ccPrecond \ccc{M.row_dimension() = b.dimension()}.  }

\ccMethod{static bool is_solvable(const Matrix& M, const Vector& b)
  ;}{ determines whether the system $M \cdot x = b$ is solvable \\
  \ccPrecond \ccc{M.row_dimension() = b.dimension()}.  }

\ccMethod{static bool homogeneous_linear_solver (const Matrix& M,
  Vector& x);}{determines whether the homogeneous linear system
  $M\cdot x = 0$ has a non - trivial solution. If yes, then $x$ is
  such a solution.  }

\ccMethod{static int homogeneous_linear_solver (const Matrix& M,
  Matrix& spanning_vecs);}{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 \ccc{spanning_vecs}
  span the solution space.  }

\ccMethod{static int independent_columns (const Matrix& M,
  std::vector<int>& columns);}{returns the indices of a maximal subset
  of independent columns of \ccc{M}.  }

\ccMethod{static int rank (const Matrix & M);}{returns the rank of
  matrix \ccc{M} }

\ccHasModels

\ccRefIdfierPage{CGAL::Linear_algebraHd<RT>} \\
\ccRefIdfierPage{CGAL::Linear_algebraCd<FT>} 

\end{ccRefConcept}