/// <summary>
/// Performs the operation: <paramref name="result"/> = generalized_inverse(A) * <paramref name="vector"/>
/// </summary>
/// <param name="vector">The vector that will be multiplied. Its <see cref="IIndexable1D.Length"/> must be equal to
/// <see cref="IIndexable2D.NumRows"/> of the original matrix A.
/// </param>
/// <param name="result">
/// Output vector that will be overwritten with the solution of the linear system. Its <see cref="IIndexable1D.Length"/>
/// must be equal to <see cref="IIndexable2D.NumColumns"/> of the original matrix A.
/// </param>
/// <exception cref="NonMatchingDimensionsException">
/// Thrown if <paramref name="vector"/> or <paramref name="result"/> violate the described constraints.
/// </exception>
/// <exception cref="IndefiniteMatrixException">
/// Thrown if the original skyline matrix turns out to not be symmetric positive semi-definite.
/// </exception>
public void MultiplyGeneralizedInverseMatrixTimesVector(Vector vector, Vector result)
{
Preconditions.CheckSystemSolutionDimensions(Order, vector.Length);
Preconditions.CheckMultiplicationDimensions(Order, result.Length);
LdlSkyline.Solve(Order, values, diagOffsets, vector.RawData, result.RawData);
}
public static Matrix SolveLinearSystems(this LdlSkyline triangulation, Matrix rhsMatrix)
{
var solution = Matrix.CreateZero(triangulation.Order, rhsMatrix.NumColumns);
triangulation.SolveLinearSystems(rhsMatrix, solution);
return(solution);
}
/// <summary>
/// Applies the LDL factorization to the independent columns of a symmetric positive semi-definite matrix,
/// sets the dependent ones equal to columns of the identity matrix and return the nullspace of the matrix. Requires
/// extra memory for the basis vectors of the nullspace.
/// </summary>
/// <param name="order">The number of rows/ columns of the square matrix.</param>
/// <param name="skyValues">
/// The non-zero entries of the original <see cref="SkylineMatrix"/>. This array will be overwritten during the
/// factorization.
/// </param>
/// <param name="skyDiagOffsets">
/// The indexes of the diagonal entries into <paramref name="skyValues"/>. The new
/// <see cref="SemidefiniteLdlSkyline"/> instance will hold a reference to <paramref name="skyDiagOffsets"/>.
/// However they do not need copying, since they will not be altered during or after the factorization.
/// </param>
/// <param name="pivotTolerance">
/// If a diagonal entry is <= <paramref name="pivotTolerance"/> it means that the corresponding column is dependent
/// on the rest. The Cholesky factorization only applies to independent column, while dependent ones are used to compute
/// the nullspace. Therefore it is important to select a tolerance that will identify small pivots that result from
/// singularity, but not from ill-conditioning.
/// </param>
public static SemidefiniteLdlSkyline Factorize(int order, double[] skyValues, int[] skyDiagOffsets,
double pivotTolerance = SemidefiniteLdlSkyline.PivotTolerance)
{
(List <int> dependentColumns, List <double[]> nullSpaceBasis) =
LdlSkyline.FactorizeInternal(order, skyValues, skyDiagOffsets, pivotTolerance);
Debug.Assert(dependentColumns.Count == nullSpaceBasis.Count);
return(new SemidefiniteLdlSkyline(order, skyValues, skyDiagOffsets, dependentColumns, nullSpaceBasis));
}