/// <summary>
/// Computes all shortest distance between any vertices in a given graph.
/// </summary>
/// <param name="adjacence_matrix">Square adjacence matrix. The main diagonal
/// is expected to consist of zeros, any non-existing edges should be marked
/// positive infinity.</param>
/// <returns>Two matrices D and P, where D[u,v] holds the distance of the shortest
/// path between u and v, and P[u,v] holds the shortcut vertex on the way from
/// u to v.</returns>
public static MatrixQ[] Floyd(MatrixQ adjacence_matrix)
{
if (!adjacence_matrix.IsSquare())
throw new ArgumentException("Expected square matrix.");
else if (!adjacence_matrix.IsReal())
throw new ArgumentException("Adjacence matrices are expected to be real.");
int n = adjacence_matrix.RowCount;
MatrixQ D = adjacence_matrix.Clone(); // distance matrix
MatrixQ P = new MatrixQ(n);
double buf;
for (int k = 1; k <= n; k++)
for (int i = 1; i <= n; i++)
for (int j = 1; j <= n; j++) {
buf = D[i, k].Re + D[k, j].Re;
if (buf < D[i, j].Re) {
D[i, j].Re = buf;
P[i, j].Re = k;
}
}
return new MatrixQ[] { D, P };
}
/// <summary>
/// For this matrix A, this method solves Ax = b via LU factorization with
/// column pivoting.
/// </summary>
/// <param name="b">Vector of appropriate length.</param>
/// <remarks>Approximately n^3/3 + 2n^2 dot operations ~> O(n^3)</remarks>
public static MatrixQ Solve(MatrixQ A, MatrixQ b)
{
MatrixQ A2 = A.Clone();
MatrixQ b2 = b.Clone();
if (!A2.IsSquare())
throw new InvalidOperationException("Cannot uniquely solve non-square equation system.");
int n = A2.RowCount;
MatrixQ P = A2.LUSafe();
// We know: PA = LU => [ Ax = b <=> P'LUx = b <=> L(Ux) = (Pb)] since P is orthogonal
// set y := Ux, solve Ly = Pb by forward insertion
// and Ux = y by backward insertion
b2 = P * b2;
// this solves Ly = Pb
(A2.ExtractLowerTrapeze() - Diag(A2.DiagVector()) + Identity(n)).ForwardInsertion(b2);
// this solves Ux = y
(A2.ExtractUpperTrapeze()).BackwardInsertion(b2);
return b2;
}
