Efficient multithreaded untransposed, transposed or symmetric sparse matrix–vector multiplication with the Recursive Sparse Blocks format

Abstract

In earlier work we have introduced the ''Recursive Sparse Blocks'' (RSB) sparse matrix storage scheme oriented towards cache efficient matrix-vector multiplication (SpMV) and triangular solution (SpSV) on cache based shared memory parallel computers. Both the transposed (SpMV_T) and symmetric (SymSpMV) matrix-vector multiply variants are supported. RSB stands for a meta-format: it recursively partitions a rectangular sparse matrix in quadrants; leaf submatrices are stored in an appropriate traditional format - either Compressed Sparse Rows (CSR) or Coordinate (COO). In this work, we compare the performance of our RSB implementation of SpMV, SpMV_T, SymSpMV to that of the state-of-the-art Intel Math Kernel Library (MKL) CSR implementation on the recent Intel's Sandy Bridge processor. Our results with a few dozens of real world large matrices suggest the efficiency of the approach: in all of the cases, RSB's SymSpMV (and in most cases, SpMV_T as well) took less than half of MKL CSR's time; SpMV's advantage was smaller. Furthermore, RSB's SpMV_T is more scalable than MKL's CSR, in that it performs almost as well as SpMV. Additionally, we include comparisons to the state-of-the art format Compressed Sparse Blocks (CSB) implementation. We observed RSB to be slightly superior to CSB in SpMV_T, slightly inferior in SpMV, and better (in most cases by a factor of two or more) in SymSpMV. Although RSB is a non-traditional storage format and thus needs a special constructor, it can be assembled from CSR or any other similar row-ordered representation arrays in the time of a few dozens of matrix-vector multiply executions. Thanks to its significant advantage over MKL's CSR routines for symmetric or transposed matrix-vector multiplication, in most of the observed cases the assembly cost has been observed to amortize with fewer than fifty iterations.