Mixed-precision orthogonalization scheme and adaptive step size for CA-GMRES on GPUs

We propose a mixed-precision orthogonalization scheme that takes the input matrix in a standard 32 or 64-bit floating-point precision, but uses higher-precision arithmetics to accumulate its intermediate results. For the 64-bit precision, our scheme uses software emulation for the higher-precision arithmetics, and requires about 20x more computation but about the same amount of communication as the standard orthogonalization scheme. Since the computation is becoming less expensive compared to the communication on new and emerging architectures, the relative cost of our mixed-precision scheme is decreasing. Our case studies with CA-GMRES on a GPU demonstrate that using mixed-precision for this small but critical segment of CA-GMRES can improve not only its overall numerical stability but also, in some cases, its performance.