The design of a new frontal code for solving sparse, unsymmetric systems

Reviewer: Jesse Louis Barlow

Frontal methods are an approach to implementing sparse Gaussian elimination that grew out of the finite element method. Think of solving the system Ax=b , where A is a large, sparse matrix. The matrix A can be written as the sum of frontal (or element) matrices, A= i=1 p A k , w here each A k is zero in all but a small dense block. Gaussian elimination can be performed on A while having only a few element matrices in storage at a time. Frontal code designers can take advantage of caching and can develop more efficient data structures. The authors propose a new general-purpose code for sparse Gaussian elimination based on this idea. The new code is the Harwell code MA42. It is proposed as an improvement to the previous Harwell code, MA32. The main source of improvement is the use of level 2 and level 3 BLAS (fast subroutines for dense matrix-vector and matrix-matrix multiplications) for the matrix operations involved in assembling the front. Much of the paper is about the kind of improvement one can expect from the appropriate use of BLAS in the factorization and solution phases of the frontal code. The tests show that the authors obtain good megaflop rates. The literature given in the bibliography gives a good view of the history of the Harwell frontal codes. The paper deliberately says little about the technical issues involved in these codes. A frontal code for Gaussian elimination is necessary for large-scale scientific computing. Such a code is described here.