\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.5 - dD Geometry Kernel
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LinearAlgebraTraits_d Concept Reference
Linear Algebra Concepts

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

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

returns the determinant of M.

Precondition
M is square.
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.

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.

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

returns the sign of the determinant of M.

Precondition
M is square.