Extending hierarchical probing

Abstract. In this paper we investigate improvements to Hierarchical Probing. Probing is a method for estimating the trace of the inverse of a large sparse matrix by attempting to determine the important elements of its inverse. This is typically done through taking powers of An, since this will match the structure of a polynomial approximation to A 1. This is equivalent to determining which nodes are connected to each other at distance n. However, if the matrix being probed is a lattice, then the distances are known a priori and more e cient algorithms can be designed. This was behind the idea of Hierarchical Probing, which allows for the e cient generation of probing vectors for lattices. Not only is the generation of these vectors very e cient, but the earlier vectors are subsets of vectors generated later in the process, meaning that it is simple to continue probing if additional accuracy is found to be needed. However this method had the drawback that it only worked on lattices whose dimensions were powers of two. In this paper we improve the method to work on lattices of di↵erent dimensions. Additionally, we expand the idea of Hierarchical Probing to develop a probing heuristic that will generate probing vectors for arbitrary matrices, while still retaining the much lower memory requires of our algorithm.