Necessary and sufficient conditions for the simplification of generalized conjugate-gradient algorithms
The efficiency of the usual conjugate gradient method strongly depends on the possibility of obtaining the new direction of search by just taking a proper linear combination of the direction in the former step and the new gradient. In general, the recursion would require to take into consideration up to n of the foregoing directions. The authors consider generalized conjugate gradient algorithms (ORTHOMIN, ORTHORES) to solve non-Hermitian linear systems and give necessary and sufficient conditions on the matrix which allow similar simplified recursions. The paper continues research of \textit{V. Faber} and \textit{T. Manteuffel} [SIAM J. Numer. Anal. 21, 352-362 (1984; Zbl 0546.65010) and ibid. 24, 170-187 (1987; Zbl 0613.65030)] on the method ORTHODIR.
scientific article; zbMATH DE number 4028736
scientific article