CGAL 4.10 - dD Geometry Kernel
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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 \( c^T b \neq 0\).
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 | |
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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 \( Q\).
M
is square. returns the determinant of M
.
M
is square.
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static |
determines whether the homogeneous linear system \( M\cdot x = 0\) has a non - trivial solution.
If yes, then \( x\) is such a solution.
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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.
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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 \( c^T \cdot M = 0 \).
returns the inverse matrix of M
.
More precisely, \( 1/D\) times the matrix returned is the inverse of M
.
determinant(M) != 0
.determines whether the system \( M \cdot x = b\) is solvable
M.row_dimension() = b.dimension()
.
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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.
M.row_dimension() = b.dimension()
.
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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 \( c^T \cdot b \not= 0\).
M.row_dimension() = b.dimension()
.
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static |
as above, but without the witness \( c\)
M.row_dimension() = b.dimension()
.
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static |
returns the sign of the determinant of M
.
M
is square.