Abstract
For iterative solution of symmetric systems the conjugate gradient method (CG) is commonly used when A is positive definite, while the minimum residual method (MINRES) is typically reserved for indefinite systems. We investigate the sequence of approximate solutions generated by each method and suggest that even if A is positive definite, MINRES may be preferable to CG if iterations are to be terminated early. In particular, we show for MINRES that the solution norms are monotonically increasing when A is positive definite (as was already known for CG), and the solution errors are monotonically decreasing. We also show that the backward errors for the MINRES iterates are monotonically decreasing.
Keywords
Publication details
- DOI
- 10.24200/squjs.vol17iss1pp44-62
- Journal
- Sultan Qaboos University Journal for Science, 16, 44
- Publisher
- Sultan Qaboos University
- Open access
- Gold open access
- License
- CC BY 4.0
Cite this article
APA 7
Fong, D. C. L., & Saunders, M. A. (2012). CG Versus MINRES: An Empirical Comparison. Sultan Qaboos University Journal for Science, 16, 44. https://doi.org/10.24200/squjs.vol17iss1pp44-62
MLA 9
Fong, David Chin-Lung, and Michael A. Saunders. "CG Versus MINRES: An Empirical Comparison." Sultan Qaboos University Journal for Science, vol. 16, 2012, pp. 44. https://doi.org/10.24200/squjs.vol17iss1pp44-62.
Chicago (author–date)
Fong, David Chin-Lung, and Michael A. Saunders. 2012. "CG Versus MINRES: An Empirical Comparison." Sultan Qaboos University Journal for Science 16: 44. https://doi.org/10.24200/squjs.vol17iss1pp44-62.
Harvard
Fong, D. C. L. and Saunders, M. A. (2012) 'CG Versus MINRES: An Empirical Comparison', Sultan Qaboos University Journal for Science, 16, pp. 44. doi:10.24200/squjs.vol17iss1pp44-62.
Vancouver
Fong DCL, Saunders MA. CG Versus MINRES: An Empirical Comparison. Sultan Qaboos University Journal for Science. 2012;16:44. doi:10.24200/squjs.vol17iss1pp44-62
IEEE
D. C. L. Fong, and M. A. Saunders, "CG Versus MINRES: An Empirical Comparison," Sultan Qaboos University Journal for Science, vol. 16, pp. 44, 2012, doi: 10.24200/squjs.vol17iss1pp44-62.