The design of an out-of-core multifrontal solver for the 21st century

The popularity of direct methods for solving large sparse linear systems of equations Ax = b stems from their generality and robustness. Indeed, if numerical pivoting is properly incorporated, direct solvers rarely fail for numerical reasons; the main reason for failure is a lack of memory. Although increasing amounts of main memory have enabled direct solvers to solve many problems that were previously intractable, their memory requirements generally increase much more rapidly than problem size so that they can quickly run out of memory. Buying a machine with more memory is an expensive and inflexible solution since there will always be problems that are too large for the chosen quantity of memory. Using an iterative method may be a possible alternative but for many of the “tough” systems that arise from practical applications, the difficulties involved in finding and computing a good preconditioner can make iterative methods infeasible. Another possibility is to use a direct solver that is able to hold its data structures on disk, that is, an out-of-core solver. The advantage of using disk storage is that it is many times cheaper than main memory per megabyte, making it practical and cost-effective to add tens or hundreds of gigabytes of disk space to a machine. By holding the main data structures on disk, properly implemented out-of-core direct solvers are very reliable since they are much less likely than in-core solvers to run out of memory. The idea of out-of-core linear solvers is not new (see, for example, [3],[6] and, recently, [2], [7]). Our aim is to design and develop a sparse symmetric out-ofcore solver for inclusion within the mathematical software library HSL [5]. Our new solver, which is called hsl ma77, implements an out-of-core multifrontal algorithm and is designed for the efficient solution of both positive-definite and indefinite sparse linear systems with one or more right-hand sides. Input of the system matrix A may be by rows or by square symmetric elements. The multifrontal method is a variant of sparse Gaussian elimination and involves the matrix factorization