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static Matrix
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LA.transpose ( Matrix M)
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returns MT
(a M.column_dimension() × M.column_dimension() -
matrix).
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static bool
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LA.inverse ( Matrix M, Matrix& I, NT& D, Vector& c)
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determines whether M has an inverse. It also
computes either the inverse as (1/D) ⋅ I or when no
inverse exists, a vector c such that cT ⋅ M = 0 .
| Precondition: | M is square. |
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static Matrix
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LA.inverse ( Matrix M, NT& D)
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returns the
inverse matrix of M. More precisely, 1/D times the matrix
returned is the inverse of M.
| Precondition: | M is square. |
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static NT
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LA.determinant ( Matrix M, Matrix& L, Matrix& U, std::vector<int>& q, Vector& c)
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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 cT ⋅ 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 ⋅ M ⋅ Q = U, L(0,0) = 1, L(i,i) = U(i
- 1,i - 1) for all i, 1 ≤ i < n, and D = s ⋅ U(n - 1,n -
1) where s is the determinant of Q. | Precondition: | M is
square. |
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static bool
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LA.verify_determinant ( Matrix M, NT D, Matrix& L, Matrix& U, std::vector<int> q, Vector& c)
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verifies the conditions stated above.
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static NT
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LA.determinant ( Matrix M)
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returns the
determinant of M. | Precondition: | M is square. |
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static int
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LA.sign_of_determinant ( Matrix M)
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returns
the sign of the determinant of M. | Precondition: | M is
square. |
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static bool
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LA.linear_solver ( Matrix M, Vector b, Vector& x, NT& D, Matrix& spanning_vectors, Vector& c)
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determines
the complete solution space of the linear system M ⋅ x = b. If
the system is unsolvable then cT ⋅ M = 0 and cT ⋅ b
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(). |
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static bool
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LA.linear_solver ( Matrix M, Vector b, Vector& x, NT& D, Vector& c)
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determines whether the linear
system M ⋅ x = b is solvable. If yes, then (1/D) x is a
solution, if not then cT ⋅ M = 0 and cT ⋅ b 0.
| Precondition: | M.row_dimension() = b.dimension(). |
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static bool
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LA.linear_solver ( Matrix M, Vector b, Vector& x, NT& D)
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as above, but without the witness c
| Precondition: | M.row_dimension() = b.dimension(). |
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static bool
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LA.is_solvable ( Matrix M, Vector b)
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determines whether the system M ⋅ x = b is solvable
| Precondition: | M.row_dimension() = b.dimension(). |
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static bool
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LA.homogeneous_linear_solver ( Matrix M, Vector& x)
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determines whether the homogeneous linear system
M ⋅ x = 0 has a non - trivial solution. If yes, then x is
such a solution.
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static int
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LA.homogeneous_linear_solver ( Matrix M, Matrix& spanning_vecs)
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determines the solution space of the
homogeneous linear system M ⋅ 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 int
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LA.independent_columns ( Matrix M, std::vector<int>& columns)
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returns the indices of a maximal subset
of independent columns of M.
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static int
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LA.rank ( Matrix M)
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returns the rank of
matrix M
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