English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Since the development of the conjugate gradient (CG) method in 1952 byHestenes and Stiefel, CG, has become an indispensable tool in computationalmathematics for solving positive definite linear systems. On the other hand,the conjugate residual (CR) method, closely related CG and introduced byStiefel in 1955 for the same settings, remains relatively less known outsidethe numerical linear algebra community. Since their inception, these methods --henceforth collectively referred to as conjugate direction methods -- have beenextended beyond positive definite to indefinite, albeit consistent, settings.Going one step further, in this paper, we investigate theoretical and empiricalproperties of these methods under inconsistent systems. Among other things, weshow that small modifications to the original algorithms allow for thepseudo-inverse solution. Furthermore, we show that CR is essentially equivalentto the minimum residual method, proposed by Paige and Saunders in 1975, in suchcontexts. Lastly, we conduct a series of numerical experiments to shed lightson their numerical stability (or lack thereof) and their performance forinconsistent systems. Surprisingly, we will demonstrate that, unlike CR andcontrary to popular belief, CG can exhibit significant numerical instability,bordering on catastrophe in some instances.

Conjugate Direction Methods Under Inconsistent Systems