Accuracy of two three-term and three two-term recurrences for Krylov space solvers (Q2706251)

The authors study the numerical behaviour of the two-term and the three-term versions of some Krylov space methods for the iterative solution of linear systems \(Ax= b\). The two-term versions are usually based on the 3 two-term recurrences NEWLINE\[NEWLINEp_n= r_n+ \psi_{n-1} p_{n-1},\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEr_{n+1}= r_n- \omega_n Ap_n,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEx_{n+1}= x_n+ \omega_n p_n\tag{3}NEWLINE\]NEWLINE for the search directions \(p_n\), the residuals \(r_n= b-Ax_n\) and the iterates \(x_n\), whereas the three-term versions are based on the 2 three-term recurrences NEWLINE\[NEWLINEr_{n+1}= (\gamma_n)^{-1} (Ar_n- \alpha_n r_n- \beta_{n-1} r_{n-1}),\tag{4}NEWLINE\]NEWLINE NEWLINE\[NEWLINEx_{n+1}= -(\gamma_n)^{-1} (r_n- \alpha_n x_n- \beta_{n-1} x_{n-1})\tag{5}NEWLINE\]NEWLINE for the residuals \(r_n= b-Ax_n\) and the iterates \(x_n\), respectively, with appropriately chosen iteration parameters. The analysis of the roundoff error propagation in the recurrences gives an explanation of the different numerical behaviour of the two-term recurrences (1)--(3) and the three-term recurrences (4)--(5) observed in the numerical practice.