Cholesky decomposition of sparse matrices using permutation matrices

Question

I am interested in the Cholesky decomposition of large sparse matrices. The problem I\'m having is that the Cholesky factors are not necessarily sparse (just like the produc

Answer 1

The problem of finding an optimal permutation of rows and columns of a matrix for a minimum fill-in matrix-factorization is not a trivial trask (as pointed out in the comments). Thus, heuristic algorithms are used in praxis.

There are some libraries that implement heuristic renumbering/ordering-strategies, often based on graph-algorithms for the adjacency-graph of your matrix. One tries to reduce the bandwidth of the corresponding adjacency-matrix. An easy to implement algroithms is the Cuthill-McKee Algorithm or the Minimum-Degree Ordering algorithm. More about this problem can be found in the Book Yousef Saad: Iterative Methods for Sparse Linear Systems (2003), upon many others.

Many libraries implement heuristic algorithms, like

• Suitesparse A collection of libraries for direct solvers for largse sparse linear systems. Ordering methods implemented in the libraries AMD, CAMD, COLAMD, and CCOLAMD
• (Par-)Metis A library for Graph-partitioning, but provides Matrix reordering algorithms as well
• Boost.Graph Working on the adjacency graph directly and provides some ordering algorithms, like the mentioned Cuthill-McKee, and Minimum-Degree Ordering
• (PT-)Scotch for Graph-partitioning and sparse-matrix reordering

Some of these libraries provide also sparse Cholesky factorization methods and can be used directly.