caa a22 4500 475777026 CHVBK 20180406123633.0 cr unu---uuuuu 170329e20001201xx s 000 0 eng 10.1023/A:1011132730828 doi (NATIONALLICENCE)springer-10.1023/A:1011132730828 Solving A$$\underline x $$=$$\underline b $$ Using a Modified Conjugate Gradient Method Based on Roots of A [Elektronische Daten] [Paul Fischer, Sigal Gottlieb] We consider the modified conjugate gradient procedure for solving A $$\underline x $$ = $$\underline b $$ in which the approximation space is based upon the Krylov space associated with A 1/p and $$\underline b $$ , for any integer p. For the square-root MCG (p=2) we establish a sharpened bound for the error at each iteration via Chebyshev polynomials in $$\sqrt A$$ . We discuss the implications of the quickly accumulating effect of an error in $$\sqrt A$$ $$\underline b $$ in the initial stage, and find an error bound even in the presence of such accumulating errors. Although this accumulation of errors may limit the usefulness of this method when $$\sqrt A$$ $$\underline b $$ is unknown, it may still be successfully applied to a variety of small, "almost-SPD” problems, and can be used to jump-start the conjugate gradient method. Finally, we verify these theoretical results with numerical tests. Plenum Publishing Corporation, 2000 modified conjugate gradient method nationallicence conjugate gradient method nationallicence Krylov space nationallicence convergence rate nationallicence stability nationallicence Fischer Paul E-mail: fischer@mcs.anl.gov aut Gottlieb Sigal E-mail: sg@cfm.brown.edu aut Journal of Scientific Computing Kluwer Academic Publishers-Plenum Publishers 15/4(2000-12-01), 441-456 0885-7474 15:4<441 2000 15 10915 https://doi.org/10.1023/A:1011132730828 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1011132730828 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Fischer Paul E-mail: fischer@mcs.anl.gov aut NATIONALLICENCE 700 1- Gottlieb Sigal E-mail: sg@cfm.brown.edu aut NATIONALLICENCE 773 0- Journal of Scientific Computing Kluwer Academic Publishers-Plenum Publishers 15/4(2000-12-01), 441-456 0885-7474 15:4<441 2000 15 10915 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer