QR Decomposition Based on Double-double Precision Gram-Schmidt Orthogonalization Method - Jin, Jiexi; Xie, Hehu; Du, Peibing; Quan, Zhe; Jiang, Hao (Ji suan ji ke xue, 2023)
The Gram-Schmidt orthogonalization algorithm and its related modified algorithms often show numerical instability when computing ill-conditioned or large-scale matrices.To solve this problem, this paper explores the cumulative effect of round-off errors of modified Gram-Schmidt algorithm（MGS）,and then designs and implements a double-double precision modified Gram-Schmidt orthogonalization algorithm（DDMGS） based on the error-free transformation technology and double-double precision algorithm.A variety of accuracy tests illustrate that DDMGS algorithm has better numerical stability than the varients of BMGS＿SVL,BMGS＿CWY,BCGS＿PIP and BCGS＿PIO algorithms, which proves that DDMGS algorithm can effectively reduce the loss of orthogonality of matrix, improve the numerical accuracy, and demonstrate the stability of our algorithm.In the performance test, the floating point computations（flops） of different algorithms are calculated and then compared DDMGS algorithm with the modified Gram-Schmidt algorithm on ARM and I