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In this paper we introduce a parameter dependent class of Krylov-basedmethods, namely CD, for the solution of symmetric linear systems. We giveevidence that in our proposal we generate sequences of conjugate directions,extending some properties of the standard Conjugate Gradient (CG) method, inorder to preserve the conjugacy. For specific values of the parameters in ourframework we obtain schemes equivalent to both the CG and the scaled-CG. Wealso prove the finite convergence of the algorithms in CD, and we provide someerror analysis. Finally, preconditioning is introduced for CD, and we show thatstandard error bounds for the preconditioned CG also hold for thepreconditioned CD.

A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization