MC64 is an algorithm for permuting large entries to the diagonal of a sparse matrix.
This approach can increase numerical stability of e.g. an LU factorization without pivoting. Under the assumption of working on a nonsingular square matrix, the algorithm computes a minimum weight perfect matching on a weighted edge bipartite graph of the matrix. It is described in detail in "On Algorithms for Permuting Large Entries to the
Diagonal of a Sparse Matrix" (Duff, Koster, 2001, DOI: 10.1137/S0895479899358443). There are two strategies for choosing the weights supported:
Maximizing the product of the absolute values on the diagonal. For this strategy, the weights are computed as if and otherwise. Here, a_i is the maximum absolute value in row i of the matrix A. In this case, the implementation computes a row permutation P and row and column scaling coefficients L and R such that the matrix P*L*A*R has values with unity absolute value on the diagonal and smaller or equal entries everywhere else.
Maximizing the sum of the absolute values on the diagonal. For this strategy, the weights are computed as if and otherwise. In this case, no scaling coefficients are computed.
This class creates a Combination of two ScaledPermutations representing the row and column permutation and scaling factors computed by this algorithm.
Template Parameters
ValueType
Type of the values of all matrices used in this class
IndexType
Type of the indices of all matrices used in this class
This function overrides the default LinOpFactory::generate to return a Permutation instead of a generic LinOp, which would need to be cast to ScaledPermutation again to access its indices. It is only necessary because smart pointers aren't covariant.