PACF: A precision-adjustable computational framework for solving singular values

Singular value decomposition (SVD) plays a significant role in matrix analysis, and the dif-ferential quotient difference with shifts (DQDS) algorithm is an important technique for solving singular values of upper bidiagonal matrices. However, ill-conditioned matrices and large-scale matrices may cause inaccurate results or long computation times when solving singular values. At the same time, it is difficult for users to effectively find the desired solution according to their needs. In this paper, we design a precision-adjustable compu-tational framework for solving singular values, named PACF. In our framework, the same solution algorithm contains three options: original mode, high-precision mode, and mixed -precision mode. The first algorithm is the original version of the algorithm. The second algorithm is a reliable numerical algorithm we designed using Error-free transformation (EFT) technology. The last algorithm is an efficient numerical algorithm we developed us-ing the mixed-precision idea. Our PACF can add different solving algorithms for different types of matrices, which are universal and extensible. Users can choose different algo-rithms to solve singular values according to different needs. This paper implements the high-precision DQDS and mixed-precision DQDS algorithms and conducts extensive exper-iments on a supercomputing platform to demonstrate that our algorithm is reliable and ef-ficient. Besides, we introduce the error analysis of the inner loop of the DQDS and HDQDS algorithms.(c) 2022 Elsevier Inc. All rights reserved.