In this paper we show that if the step (displacement) vectors generated by the preconditioned conjugate gradient algorithm are scaled appropriately they may be used to solve equations whose coefficient matrices are the preconditioning matrices of the original equations. The dual algorithms thus obtained are shown to be equivalent to the reverse algorithms of Hegedüs and are subsequently generalised to their block forms. It is finally shown how these may be used to construct dual (or reverse) algorithms for solving equations involving nonsymmetric matrices using only short recurrences, and reasons are suggested why some of these algorithms may be more numerically stable than their primal counterparts.
Broyden, C.G., Foschi, P. (1999). Duality in conjugate gradient methods. NUMERICAL ALGORITHMS, 22(2), 113-128 [10.1023/A:1019154707108].
Duality in conjugate gradient methods
Foschi P.
1999
Abstract
In this paper we show that if the step (displacement) vectors generated by the preconditioned conjugate gradient algorithm are scaled appropriately they may be used to solve equations whose coefficient matrices are the preconditioning matrices of the original equations. The dual algorithms thus obtained are shown to be equivalent to the reverse algorithms of Hegedüs and are subsequently generalised to their block forms. It is finally shown how these may be used to construct dual (or reverse) algorithms for solving equations involving nonsymmetric matrices using only short recurrences, and reasons are suggested why some of these algorithms may be more numerically stable than their primal counterparts.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
