Performance aspects of sparse matrix-vector multiplication

Sparse matrix-vector multiplication (shortly SpM×V) is an important building block in algorithms solving sparse systems of linear equations, e.g., FEM. Due to matrix sparsity, the memory access patterns are irregular and utilization of the cache can suffer from low spatial or temporal locality. Approaches to improve the performance of SpM×V are based on matrix reordering and register blocking [1,2], sometimes combined with software-pipelining [3]. Due to its overhead, register blocking achieves good speedups only for a large number of executions of SpM×V with the same matrix A. We have investigated the impact of two simple SW transformation techniques (software-pipelining and loop unrolling) on the performance of SpM×V, and have compared it with several implementation modifications aimed at reducing computational and memory complexity and improving the spatial locality. We investigate performance gains of these modifications on four CPU platforms. Terminology and notation Consider a sparse n×n matrix A with elements n j i , 1 , ; A ij ∈ . The largest distance between nonzero elements in any row is the bandwidth of matrix A and is denoted by B ω , i.e., ) 0 A ; ( min l ij i ≠ = j j ) 0 A ; ( max r ij i ≠ = j j ) 1 + l r ( max i i i B = ω Storage schemes for sparse matrices Compressed sparse row (CSR) format Matrix A is represented by 3 linear arrays A, adr, and ci (see Figure 1). Array A stores the nonzero elements of input matrix A, array adr[1,...,n] contains indexes of the initial nonzero elements of rows of A, and array ci contains column indexes of nonzero elements of A. Hence, the first nonzero element of row j is stored at index adr[j] in array A. Figure 1: The idea of the CSR format. a) A sparse matrix A in dense format. b) The CSR representation of A. Figure 2: The idea of the static L-CSR format. a) A sparse matrix A in dense format. b) The static L-CSR representation of A. Length-sorted CSR (L-CSR) storage format. The main idea is explained in [4]. The data is represented as in the CSR format, but the rows are sorted by length in increasing order. This means that the length of row i is less or equal to the length of row i+1. There are two variants: 1. Static: The rows are physically stored in the sorted order in the CSR format (see Figure 2). 2. Dynamic: The original CSR format is extended with two additional arrays (see Figure 3). Array member[1...n] contains the indexes of the rows after sorting. Array [ ] B begin ω .. 1 contains the indexes into the array member: begin[i] is the index of the first row of length i in the array member. Figure 3: The idea of the dynamic L-CSR format. a) A sparse matrix A in dense format. b) The dynamic L-CSR representation of A.