Abstrakt
With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of total least squares problems, i.e., solving $\min_{E,f} \| [E, f]\|_F$ subject to $(A+E)x=b+f$, arises in numerous application areas. The solution to this problem requires finding the smallest singular value and corresponding right singular vector of $[A,b]$, which is challenging when $A$ is large and sparse. An efficient algorithm for this case due to Bj\"{o}rck et al. [1] is based on Rayleigh quotient iteration coupled with the conjugate gradient method preconditioned via Cholesky factors, called RQI-PCGTLS. In this talk, we introduce a mixed precision variant of this algorithm, RQI-PCGTLS-MP, which aims to improve performance without affecting the level of attainable accuracy. In addition to numerical experiments, we discuss how to choose a suitable precision for the construction of a preconditioner, and develop a theoretical performance model.
[1]Björck, Å., Heggernes, P., & Matstoms, P. (2000). Methods for large scale total least squares problems. SIAM journal on matrix analysis and applications, 22(2), 413-429.