matrixreordering

Matrix reordering refers to transforming a sparse matrix by permuting its rows and columns to produce a structure that is more favorable for subsequent computations. In practice, a permutation P is applied to A as P^T A P, where P is a permutation matrix. For symmetric matrices, the same permutation is applied to rows and columns. The goal is to rearrange nonzero entries without changing the underlying matrix values.

The main objectives are to minimize fill-in during factorization (the appearance of new nonzeros in factors), reduce bandwidth (the maximum distance from the main diagonal of nonzeros), improve data locality, and decrease memory usage and runtime in direct solvers and iterative methods. A good ordering can significantly speed up sparse LU or Cholesky factorizations and improve convergence in iterative solvers.

Common algorithms include the Cuthill-Moore and Reverse Cuthill-Moore (RCM) methods, which aim to reduce bandwidth; Minimum Degree (MD) and Approximate Minimum Degree (AMD) algorithms that target sparse elimination graphs; and Nested Dissection approaches that partition the graph to minimize cross edges. In practice, solvers often combine several strategies and tailor the reordering to the chosen factorization method and the matrix’s symmetry.

Applications of matrix reordering are widespread in scientific computing. They include finite element methods for structural and fluid dynamics, circuit simulation, and other large sparse linear systems, as well as eigenvalue problems where efficient orderings can accelerate convergence and reduce computational resources.

Limitations exist: no ordering is universally optimal, and computing a good ordering itself incurs cost. The effectiveness depends on the matrix’s sparsity pattern, structure, and the targeted factorization or solver.