Mixed precision <i>s</i> ?step Lanczos and conjugate gradient algorithms

Compared to the classical Lanczos algorithm, s-step variant has potential improve performance by asymptotically decreasing synchronization cost per iteration. However, this comes at a price; despite being mathematically equivalent, may behave quite differently in finite precision, potentially exhibiting greater loss of accuracy and slower convergence relative algorithm. It previously been shown that errors version follow same structure as but are amplified factor depending on square condition number O ( s ) -dimensional Krylov bases computed each outer loop. As these grows (in some cases very quickly) with s, limits values can be chosen thus limit attainable performance. In work, we show if select few computations performed double working error terms then depend only linearly conditioning bases. This for drastically improving numerical behavior algorithm little impact per-iteration Our experiments demonstrate improved possible mixed precision approach, also extends CG. We present preliminary results NVIDIA V100 GPUs overhead extra is minimal one uses precisions implemented hardware.