/// <summary>
/// Calculates the Schur complement of M/C = S = A - B^T * inv(C) * B, where M = [A B; B^T C].
/// This method constructs inv(C) * B one column at a time and uses that column to calculate the superdiagonal
/// entries of the corresponding column of B^T * inv(C) * B.
/// </summary>
public static SymmetricMatrix CalcSchurComplementSymmetric(SymmetricMatrix A, CscMatrix B, ITriangulation inverseC)
{ //TODO: Unfortunately this cannot take advantage of MKL for CSC^T * vector.
double[] valuesB = B.RawValues;
int[] rowIndicesB = B.RawRowIndices;
int[] colOffsetsB = B.RawColOffsets;
var S = SymmetricMatrix.CreateZero(A.Order);
for (int j = 0; j < B.NumColumns; ++j)
{
// column j of (inv(C) * B) = inv(C) * column j of B
Vector colB = B.GetColumn(j);
double[] colInvCB = inverseC.SolveLinearSystem(colB).RawData;
// column j of (B^T * inv(C) * B) = B^T * column j of (inv(C) * B)
// However we only need the superdiagonal part of this column.
// Thus we only multiply the rows i of B^T (stored as columns i of B) with i <= j.
for (int i = 0; i <= j; ++i)
{
double dot = 0.0;
int colStart = colOffsetsB[i]; //inclusive
int colEnd = colOffsetsB[i + 1]; //exclusive
for (int k = colStart; k < colEnd; ++k)
{
dot += valuesB[k] * colInvCB[rowIndicesB[k]];
}
// Perform the subtraction S = A - (B^T * inv(C) * B) for the current (i, j)
int indexS = S.Find1DIndex(i, j);
S.RawData[indexS] = A.RawData[indexS] - dot;
}
}
return(S);
}
/// <summary>
/// Solves the linear system A * x = b, where A is the original matrix (before the factorization),
/// b = <paramref name="rhsVector"/> and x is the solution vector, which will be returned.
/// </summary>
/// <param name="rhsVector">
/// The right hand side vector. Its <see cref="IIndexable1D.Length"/> must be equal to
/// <see cref="Matrices.IIndexable2D.NumRows"/> of the original matrix A.
/// </param>
/// Thrown if the length of <paramref name="rhsVector"/> is different than <see cref="Matrices.IIndexable2D.NumRows"/>
/// of the original matrix A.
/// </exception>
public static Vector SolveLinearSystem(this ITriangulation triangulation, Vector rhsVector)
{
var solution = Vector.CreateZero(triangulation.Order);
triangulation.SolveLinearSystem(rhsVector, solution);
return(solution);
}
