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

摘要

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. (C) 2014 Elsevier B.V. All rights reserved.