Combinatorial problems in solving linear systems

IS Duff, B Ucar
Numerical linear algebra anhd combinational optomization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some cominatorial problems, ideas, and
more » ... orithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative side, we discuss the preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a seperate part, we discuss the block trianglar form of sparse matrices.
doi:10.5286/raltr.2009016 fatcat:xnsbqqz45bfwtjznabrlnvam5m