Least Squares on GPUs in Multiple Double Precision

This paper describes the application of the code generated by the CAMPARY software to accelerate the solving of linear systems in the least squares sense on Graphics Processing Units (GPUs), in double double, quad double, and octo double precision. The goal is to use accelerators to offset the cost overhead caused by multiple double precision arithmetic. For the blocked Householder QR and the back substitution, of interest are those dimensions at which teraflop performance is attained. The other interesting question is the cost overhead factor that appears each time the precision is doubled. Experimental results are reported on five different NVIDIA GPUs, with a particular focus on the P100 and the V100, both capable of teraflop performance. Thanks to the high Compute to Global Memory Access (CGMA) ratios of multiple double arithmetic, teraflop performance is already attained running the double double QR on 1,024-by-1,024 matrices, both on the P100 and the V100. For the back substitution, the dimension of the upper triangular system must be as high as 17,920 to reach one teraflops on the V100, in quad double precision, and then taking only the times spent by the kernels into account. The lower performance of the back substitution in small dimensions does not prevent teraflop performance of the solver at dimension 1,024, as the time for the QR decomposition dominates. In doubling the precision from double double to quad double and from quad double to octo double, the observed cost overhead factors are lower than the factors predicted by the arithmetical operation counts. This observation correlates with the increased performance for increased precision, which can again be explained by the high CGMA ratios.