LinearAlgebraTraits_d Concept Reference

Definition

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 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 \( c^T b \neq 0\).

Has Models:

CGAL::Linear_algebraHd<RT>

CGAL::Linear_algebraCd<FT>

Types
typedef unspecified_type	NT
	the number type of the components.
typedef unspecified_type	Vector
	the vector type.
typedef unspecified_type	Matrix
	the matrix type.

Operations
static Matrix	transpose (const Matrix &M)
	returns \( M^T\) (a M.column_dimension() \( \times\) M.column_dimension() - matrix).
static bool	inverse (const Matrix &M, Matrix &I, NT &D, Vector &c)
	determines whether M has an inverse. More...
static Matrix	inverse (const Matrix &M, NT &D)
	returns the inverse matrix of M. More...
static NT	determinant (const Matrix &M, Matrix &L, Matrix &U, std::vector< int > &q, Vector &c)
	returns the determinant \( D\) of M and sufficient information to verify that the value of the determinant is correct. More...
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.
static NT	determinant (const Matrix &M)
	returns the determinant of M. More...
static int	sign_of_determinant (const Matrix &M)
	returns the sign of the determinant of M. More...
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 \( M\cdot x = b\). More...
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. More...
static bool	linear_solver (const Matrix &M, const Vector &b, Vector &x, NT &D)
	as above, but without the witness \( c\) More...
static bool	is_solvable (const Matrix &M, const Vector &b)
	determines whether the system \( M \cdot x = b\) is solvable More...
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. More...
static int	homogeneous_linear_solver (const Matrix &M, Matrix &spanning_vecs)
	determines the solution space of the homogeneous linear system \( M\cdot x = 0\). More...
static int	independent_columns (const Matrix &M, std::vector< int > &columns)
	returns the indices of a maximal subset of independent columns of M.
static int	rank (const Matrix &M)
	returns the rank of matrix M

Member Function Documentation

determinant() [1/2]

static NT LinearAlgebraTraits_d::determinant ( const Matrix &  M, Matrix &  L, Matrix &  U, std::vector< int > &  q, Vector &  c  )

static

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 \( Q\).

Precondition

M is square.

determinant() [2/2]

static NT LinearAlgebraTraits_d::determinant ( const Matrix &  M )

static

returns the determinant of M.

Precondition

M is square.

homogeneous_linear_solver() [1/2]

static bool LinearAlgebraTraits_d::homogeneous_linear_solver ( const Matrix &  M, Vector &  x  )

static

determines whether the homogeneous linear system \( M\cdot x = 0\) has a non - trivial solution.

If yes, then \( x\) is such a solution.

homogeneous_linear_solver() [2/2]

static int LinearAlgebraTraits_d::homogeneous_linear_solver ( const Matrix &  M, Matrix &  spanning_vecs  )

static

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.

inverse() [1/2]

static bool LinearAlgebraTraits_d::inverse ( const Matrix &  M, Matrix &  I, NT &  D, Vector &  c  )

static

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 \( c^T \cdot M = 0 \).

Precondition

\( M\) is square.

inverse() [2/2]

static Matrix LinearAlgebraTraits_d::inverse ( const Matrix &  M, NT &  D  )

static

returns the inverse matrix of M.

More precisely, \( 1/D\) times the matrix returned is the inverse of M.

Precondition

determinant(M) != 0.

\( M\) is square.

is_solvable()

static bool LinearAlgebraTraits_d::is_solvable ( const Matrix &  M, const Vector &  b  )

static

determines whether the system \( M \cdot x = b\) is solvable

Precondition

M.row_dimension() = b.dimension().

linear_solver() [1/3]

static bool LinearAlgebraTraits_d::linear_solver ( const Matrix &  M, const Vector &  b, Vector &  x, NT &  D, Matrix &  spanning_vectors, Vector &  c  )

static

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.

Precondition

M.row_dimension() = b.dimension().

linear_solver() [2/3]

static bool LinearAlgebraTraits_d::linear_solver ( const Matrix &  M, const Vector &  b, Vector &  x, NT &  D, Vector &  c  )

static

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 \( c^T \cdot b \not= 0\).

Precondition

M.row_dimension() = b.dimension().

linear_solver() [3/3]

static bool LinearAlgebraTraits_d::linear_solver ( const Matrix &  M, const Vector &  b, Vector &  x, NT &  D  )

static

as above, but without the witness \( c\)

Precondition

M.row_dimension() = b.dimension().

sign_of_determinant()

static int LinearAlgebraTraits_d::sign_of_determinant ( const Matrix &  M )

static

returns the sign of the determinant of M.

Precondition

M is square.