QR Decomposition Based on Double-double Precision Gram-Schmidt Orthogonalization Method

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 th...

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Bibliographic Details
Published in:Ji suan ji ke xue Vol. 50; no. 6; pp. 45 - 51
Main Authors: Jin, Jiexi, Xie, Hehu, Du, Peibing, Quan, Zhe, Jiang, Hao
Format: Journal Article
Language:Chinese
Published: Chongqing Guojia Kexue Jishu Bu 01.06.2023
Editorial office of Computer Science
Subjects:
Accuracy
Algorithms
Floating point arithmetic
gram-schmidt algorithm|qr decomposition|error-free transformation|double-double precision algorithm|roundoff error|floating-point arithmetic
Numerical stability
Orthogonality
Performance tests
Roundoff error
Stability
ISSN:1002-137X
Online Access:Get full text
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Summary: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
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ISSN:1002-137X
DOI:10.11896/jsjkx.230200209